Linear Models and Analysis of
Variance:
CONCEPTS, MODELS, AND APPLICATIONS
Volume II
First Edition
David W. Stockburger
Southwest Missouri State
University
@Copyright 1993
TABLE OF CONTENTS
Title Page
EXPERIMENTAL DESIGNS.............................................................................. 132
Notation.................................................................................................... 133
Kinds of Factors........................................................................................ 134
Treatment...................................................................................... 134
Group Factors............................................................................... 135
Trials Factors................................................................................ 135
Blocking........................................................................................ 135
Unit Factors.................................................................................. 137
Error Factors................................................................................. 137
Fixed and Random Factors........................................................................ 137
Fixed Factors................................................................................ 137
Random Factors............................................................................ 138
Relationships Between Factors................................................................... 139
Crossed........................................................................................ 139
Nested.......................................................................................... 140
An Example Design................................................................................... 142
A Second Example Design......................................................................... 144
A Third Example Design............................................................................ 146
Determining the Number of Subjects and
Measures per Subject................. 148
Setting up the Data Matrix......................................................................... 148
A Note of Caution..................................................................................... 149
One Between Group ANOVA.............................................................................. 150
Why Multiple Comparisons Using t-tests is NOT
the Analysis of Choice..... 150
The Bottom Line - Results and Interpretation
of ANOVA.......................... 151
HYPOTHESIS TESTING THEORY UNDERLYING ANOVA.............. 153
The Sampling Distribution Reviewed.............................................. 153
Two Ways of Estimating the Population
Parameter σX˛................... 154
The F-ratio and F-distribution.................................................................... 157
Non-significant and Significant F-ratios....................................................... 159
Similarity of ANOVA and t-test................................................................. 162
EXAMPLE OF A NON-SIGNIFICANT ONE-WAY ANOVA.............. 164
EXAMPLE OF A SIGNIFICANT ONE-WAY ANOVA........................ 164
USING MANOVA.................................................................................. 164
The Data....................................................................................... 165
Example Output............................................................................. 166
Dot Notation............................................................................................. 168
POST-HOC Tests of Significance.............................................................. 170
Example SPSS Program using MANOVA................................................. 173
Interpretation of Output............................................................................ 174
Graphs of Means....................................................................................... 175
The ANOVA Summary Table................................................................... 176
Main Effects.................................................................................. 177
Simple Main Effects....................................................................... 178
Interaction Effects.......................................................................... 178
Example Data Sets, Means, and Summary Tables...................................... 180
No Significant Effects..................................................................... 180
Main Effect of A............................................................................ 181
Main Effect of B............................................................................ 182
AB Interaction............................................................................... 183
Main Effects of A and B................................................................ 184
Main effect of A, AB Interaction ................................................... 185
Main Effect of B, AB Interaction.................................................... 186
Main Effects of A and B, AB Interaction........................................ 187
No Significant Effects..................................................................... 188
Dot Notation Revisited.............................................................................. 189
Nested Two Factor Between Groups Designs B(A)............................................... 192
The Design................................................................................................ 192
The Data................................................................................................... 192
SPSS commands....................................................................................... 193
The Analysis.............................................................................................. 193
The Table of Means....................................................................... 194
Graphs.......................................................................................... 194
The ANOVA Table....................................................................... 195
Interpretation of Output............................................................................. 195
Similarities to the A X B Analysis............................................................... 195
Contrasts, Special and Otherwise........................................................................... 197
Definition................................................................................................... 197
Sets of Contrasts....................................................................................... 198
Orthogonal Contrasts..................................................................... 198
Non-orthogonal Contrasts............................................................. 199
Sets of Orthogonal Contrasts......................................................... 199
Finding Sets of Orthogonal Contrasts............................................ 199
The Data................................................................................................... 201
SPSS commands....................................................................................... 202
The Analysis.............................................................................................. 202
The Table of Means....................................................................... 202
The ANOVA table........................................................................ 203
Interpretation of Output............................................................................. 203
Constants.................................................................................................. 203
Contrasts, Designs, and Effects.................................................................. 205
Non-Orthogonal Contrasts........................................................................ 208
Smaller than Total Sum of Squares................................................. 208
Larger than Total Sum of Squares.................................................. 209
Standard Types of Orthogonal Contrasts.................................................... 209
DIFFERENCE.............................................................................. 210
SIMPLE....................................................................................... 210
POLYNOMIAL........................................................................... 210
Conclusion................................................................................................ 213
ANOVA and Multiple Regression.......................................................................... 214
ONE FACTOR ANOVA......................................................................... 214
ANOVA and Multiple Regression.................................................. 214
Example........................................................................................ 215
Example Using Contrasts............................................................... 216
Dummy Coding............................................................................. 216
ANOVA, Revisited....................................................................... 218
TWO FACTOR ANOVA........................................................................ 219
Example........................................................................................ 220
Example Using Contrasts............................................................... 220
Regression Analysis using Dummy Coding...................................... 221
Conclusion................................................................................................ 223
Unequal Cell Frequencies...................................................................................... 225
Equal Cell Frequency - Independence of
Effects......................................... 225
Unequal Cell Frequency - Dependent Effects............................................. 226
Solutions for Dealing with Dependent Effects.............................................. 229
UNEQUAL CELL SIZES FROM A MULTIPLE REGRESSION
VIEWPOINT 231
REGRESSION ANALYSIS OF UNEQUAL N ANOVA........................ 234
RECOMMENDATIONS......................................................................... 237
Subjects Crossed With Treatments S X A............................................................. 238
The Design................................................................................................ 238
The Data................................................................................................... 239
SPSS commands....................................................................................... 239
The Correlation Matrix.................................................................. 240
The Table of Means....................................................................... 240
Graphs.......................................................................................... 241
The ANOVA Table....................................................................... 241
Interpretation of Output................................................................. 242
Additional Assumptions for Univariate S X A
Designs................................ 244
SS, MS, and Expected Mean Squares (EMS)................................ 245
Subjects Crossed With Two Treatments -S X A X
B............................................. 249
The Design................................................................................................ 249
The Data................................................................................................... 249
SPSS commands....................................................................................... 250
The Correlation Matrix.................................................................. 251
The Table of Means....................................................................... 251
Graphs.......................................................................................... 253
The ANOVA Table....................................................................... 253
Interpretation of Output................................................................. 253
Additional Assumptions for Univariate S X A X
B Designs......................... 258
SS, MS, and Expected Mean Squares (EMS)................................ 258
Mixed Designs - S ( A ) X B................................................................................. 261
The Design................................................................................................ 261
The Data................................................................................................... 261
SPSS commands....................................................................................... 262
The Table of Means....................................................................... 263
Graphs.......................................................................................... 264
Interpretation of Output................................................................. 264
Expected Mean Squares (EMS).................................................... 267
Three Factor ANOVA.......................................................................................... 268
Effects....................................................................................................... 268
Main Effects.................................................................................. 269
Two-Way Interactions................................................................... 269
Three-Way Interaction................................................................... 271
Additional Examples.................................................................................. 272
All Effects Significant..................................................................... 272
Example 3 - B, AC, and BC......................................................... 274
Two More Examples..................................................................... 275
Expected Mean Squares............................................................................ 276
Tests of Significance.................................................................................. 279
Error Terms................................................................................... 279
SPSS Output................................................................................. 279
Examples................................................................................................... 279
S ( A X B X C)............................................................................. 279
S ( A X B ) X C............................................................................ 280
S ( A ) X B X C............................................................................ 280
BIBLIOGRAPHY................................................................................................. 287
INDEX................................................................................................................. 289
Chapter
8
EXPERIMENTAL
DESIGNS
Experimental design refers to the manner in which the
experiment was set up. Experimental design includes the way the treatments were
administered to subjects, how subjects were grouped for analysis, how the
treatments and grouping were combined.
In ANOVA there is a single dependent variable or score. In Psychology the dependent measure is
usually some measure of behavior. If
more than one measure of behavior is taken, multivariate analysis of variance,
or MANOVA, may be the appropriate analysis.
Because the ANOVA model breaks the score into component parts, or
effects, which sum the total score, the one must assume the interval property
of measurement for this variable. Since
in real life the interval property is never really met, one must be satisfied
that at least an approximation of an interval scale exists for the dependent
variable. To the extent that this
assumption is unwarranted, the ANOVA hypothesis testing procedure will not
work.
In ANOVA there is at least one independent variable or factor. There are different kinds of factors; treatment, trial, blocking, and group. Each will be discussed in the following
section. All factors, however, have
some finite number of different levels.
Each level is the same in either some quality or quantity. The only restriction on the number of
levels is that there are fewer levels than scores, although in practice one
seldom sees more than ten levels in a factor unless the data set is very large. It is not necessary that the independent
variables or factors be measured on an interval scale. If the factors are measured on an
(approximate) interval scale, then some flexibility in analysis is gained. The continued popularity of ANOVA can
partially be explained by the lack of the necessity of the interval assumption
for the factors.
Notation
Every writer of an introductory, intermediate, or advanced
statistics text has his or her own pet notational system. I have taught using a number of different
systems and have unabashedly borrowed the one to be described below from Lee
(1975). In my opinion it is the easiest
for students to grasp.
The dependent variable or score will be symbolized by the
letter X. Subscripts (usually multiple)
will be tagged on this letter to differentiate the different scores. For example, to designate a single score
from a group of scores a single subscript would be necessary and the symbol Xs
could be used. In this case X1
would indicate the first subject, X2 the second, X3 the
third, and so forth.
When it is desired to indicate a single score belonging to a
given combination of factors, multiple subscripts must be used. For example, Xabs would describe
a given score for a combination of a and b.
Thus, X236 would describe the sixth score when a=2 and
b=3. Another example, X413,
would describe the third score when a=4 and b=1.
Bolded capital letters will be used to symbolize
factors. Example factors are A, B,
C, ..., Z. Some factor
names are reserved for special factors.
For example, S will always refer to the subject factor, E will always be the error factor, and
G will be the group factor.
Small letters with a numerical subscript are used to
indicate specific levels of a factor.
For example c1 will indicate the first level of factor C,
while cc will indicate a specific level of factor C, but the
level is unspecified. The number of
levels of a factor are given by the unbolded capital letter of that
factor. For example there are 1, 2,
..., C levels of factor C.
In an example experiment, let X, the score, be the dollar
amount after playing WindowsTM Solitaire for an hour. In this experiment the independent variable
(factor) is the amount of practice, called factor A. Let nine subjects each participate in one of
four (A=4) levels of training. The
first level, a1, consists on no practice, a2 = one hour
of practice, a3 = five hours of practice, and a4 = twenty
hours of practice. A given score
(dollar amount) would be symbolized by Xas, where X35
would be the fifth subject in the group that received five hours of practice.
Kinds
of Factors
Treatment
Treatments will be defined as quantitatively or
qualitatively different levels of experience.
For example, in an experiment on the effects of caffeine, the treatment
levels might be exposure to different amounts of caffeine, from none to .0375
milligrams. In a very simple experiment
there are two levels of treatment, none, called the control condition, and
some, called the experimental condition.
Treatment factors are usually the main focus of the
experiment. A treatment factor is
characterized by the following two attributes (Lee, 1975):
1. An investigator could assign any of his
experimental subjects to any one of the levels of the factor.
2. The different levels of the factor
consist of explicitly distinguishable stimuli or situations in the environment
of the subject.
In the solitaire example, practice time would be a treatment
factor if the experimenter controlled the amount of time that the
subject practiced. If subject's came to
the experiment having already practiced a given amount, then the experimenter
could not arbitrarily or randomly assign that subject to a given practice
level. In that case the factor would no
longer be considered a treatment factor.
In an experiment where subjects are run in groups, it
sometimes is valuable to treat each group as a separate level of a factor. There might be, for example, an obnoxious
subject who affects the scores of all other subjects in that group. In this case the second attribute would not
hold and the factor would be called a group factor.
Group Factors
As described above, a group factor is one in which the
subjects are arbitrarily assigned to a given group which differs from other
groups only in that different subjects are assigned to it. If each group had some type of
distinguishing feature, other than the subjects assigned to it, then it would
no longer be considered as a group factor.
If a group factor exists in an experimental design, it will be
symbolized by G.
Trials Factors
If each subject is scored more than once under the same
condition and the separate scores are included in the analysis, then a trials
factor exists. If the different scores
for a subject are found under different levels of a treatment, then the factor would
be called a treatment factor rather than a trials factor. Trials factors will be denoted by T.
Trials factors are useful in examining practice or fatigue
effects. Any change in scores over time
may be attributed to having previously experienced similar conditions.
Blocking
If subjects are grouped according to some pre-existing subject similarity, then that grouping is
called a blocking factor. The
experimenter has no choice but to assign the subject to one or the other of the
levels of a blocking factor. For
example, gender (sex) is often used as a blocking factor. A subject enters the experiment as either a
male or female and the experimenter may not arbitrarily (randomly) assign that
individual to one gender or the other.
Because the experimenter has no control over the assignment
of subjects to a blocking factor, causal inference is made much more
difficult. For example, if in the
solitaire experiment, the practice factor was based on a pre-existing
condition, then any differences between the groups may be due either to
practice or to the fact that some subjects liked to play solitaire, were
better at the game and thus practiced more.
Since the subjects are self-selected, it is not possible to attribute
the differences between groups to practice, enjoyment of the game, natural skill
in playing the game, or some other reason.
It is possible, however, to say that the groups differed.
Even though causal inference is not possible, blocking
factor can be useful. A factor
which accounts for differences in the
scores adds power to the experiment.
That is, a blocking factor which explains some of the differences
between scores may make it more likely to find treatment effects. For example, if males and females performed significantly different in the
solitaire experiment, it might be useful to include sex as a blocking factor
because differences due to gender would be included in the error variance
otherwise.
In other cases blocking factors are interesting in their own
right. It may be interesting to know
that freshmen, sophomores, juniors, and seniors differ in attitude toward
university authority, even though causal inferences may not be made.
In some cases the pre-existing condition is quantitative, as
in an IQ score or weigh. In these cases
it is possible to use a median split where the scores above the median are placed
in one group and the scores below the median are placed in another. Variations of this procedure divide the
scores into three, four, or more approximately equal sized groups. Such procedures are not recommended as there
are better ways of handling such data (Edwards, 1985).
Unit Factors
The unit factor is the entity from which a score is
taken. In experimental psychology, the
unit factor is usually a subject (human or animal), although classrooms,
dormitories, or other units may serve the same function. In this text, the unit factor will be
designated as S, with the understanding that it might be some other type
of unit than subject.
Error Factors
The error factor, designated as E, is not a factor in
the sense of the previous factors and is not included in the experimental
design. It is necessary for future
theoretical development.
Fixed
and Random Factors
Each factor in the design must be classified as either a
fixed or random factor. This is
necessary in order to find the correct error term for each effect. The MANOVA program in SPSS does not require
that the user designate the type for each factor. If the user is willing to accept the program defaults, which are
correct in most cases, no problem is encountered. There are situations, however, where the program defaults are
incorrect and additional coding is necessary to do the correct hypothesis
tests.
Fixed Factors
A factor is fixed if
(Lee, 1975)
1. The results of the factor generalize only
to the levels that were included in the experimental design. The experimenter may wish to generalize to
other levels not included in the factor, but it is done at his or her own
peril.
2. Any procedure is allowable to select
the levels of the factor.
3. If the experiment were replicated, the
same levels of that factor would be included in the new experiment.
Random Factors
A factor is random if
1. The results of the factor generalize to
both levels that were included in the factor and levels which were not. The experimenter wishes to generalize to a
larger population of possible factor levels.
2. The levels of the factor used in the
experiment were selected by a random procedure.
3. If the experiment were replicated,
different levels of that factor would be included in the new experiment.
In many cases an exact determination of whether a factor is
fixed or random is not possible. In
general, the subjects (S) and groups (G) factors will always be a
random factor and all other factors will be considered fixed. The default designation of MANOVA will set
the subjects factor as random and all other factors as fixed.
Some reflection on the assumption of a random selection of
subjects may cause the experimenter to question whether it is in fact a random
factor. Suppose, as often happens,
subjects volunteered to participate in the experiment. In this case the assumptions underlying the
ANOVA are violated, but the procedure is used anyway. Seldom, if ever, will all the assumptions necessary to do an
ANOVA be completely satisfied. The
experimenter must examine how badly the assumptions were violated and then make
a decision as to whether or not the ANOVA is useful.
In general, when in doubt as to whether a factor is fixed or
random, consider it fixed. One should
never have so much doubt, however, as to consider the subjects factor as a
fixed factor.
Relationships
Between Factors
The following two relationships between factors describe a
large number of useful designs. Not all
possible experimental designs fit neatly into categories described by the
following two relationships, but most do.
Crossed
When two factors are crossed, each level of each factor
appears with each level of the other factor.
A crossing relationship is indicated by an "X".
For example, consider two factors, A and B,
were A is gender (a1 = Females, a2 = Males) and B
is practice (b1 = none, b2 = one hour, b3 =
five hours, and b4 = twenty hours).
If gender was crossed with practice, A X B, then both
males and females would participate in all four levels of practice. There would be eight groups of subjects
including: ab11, females who
had no practice, ab12, females who had one hour of practice, and so
forth to ab24, males who practiced twenty hours. An additional factor may be added to the
design, say handedness (C), where c1 = right handed and c2
= left handed. If the design of the
experiment was A X B X C, then there would be sixteen groups, including abc231,
left-handed males who practiced five hours.
If subjects (S) are crossed with treatments (A),
S X A, each subject sees each level of the treatment conditions. In a very simple experiment such as the
effects of caffeine on alertness (A), each subject would be exposed to
both a caffeine condition (a1) and a no caffeine condition (a2). For example, using the members of a
statistics class as subjects, the experiment might be conducted as
follows. On the first day of the
experiment the class is divided in half with one half of the class getting
coffee with caffeine and the other half getting coffee without caffeine. A measure of alertness is taken for each
individual, such as the number of yawns during the class period. On the second day the conditions are
reversed, that is, the individuals who received coffee with caffeine are now
given coffee without and vice-versa.
The distinguishing feature of crossing subjects with
treatments is that each subject will have more than one score. This feature is sometimes used in referring
to this class of designs as repeated measures designs. The effect also occurs within each subject,
thus these designs are sometimes referred to as within subjects
designs.
Crossing subjects with treatments has two advantages. One, they generally require fewer subjects,
because each subject is used a number of times in the experiment. Two, they are more likely to result in a
significant effect, given the effects are real. This is because the effects of individual differences between
subjects is partitioned out of the error term.
Crossing subjects with treatments also has
disadvantages. One, the experimenter
must be concerned about carry-over effects.
For example, individuals not used to caffeine may still feel the effects
of caffeine on the second day, when they did not receive the drug. Two, the first measurements taken may
influence the second. For example, if
the measurement of interest was score on a statistics test, taking the test
once may influence performance the second time the test is taken. Three, the assumptions necessary when more
than two treatment levels are employed in a crossing subjects with
treatments may be restrictive.
When a factor is a blocking factor, it is not possible to
cross that factor with subjects. It is
difficult to find subjects for a S X A design where A is
gender. I generally will take points
off if a student attempts such a design.
Nested
Factor B is said to be nested within factor if each
meaningful level of factor B occurs in conjunction with only one level
of A. This relationship is
symbolized a B(A), and is read as "B nested within
A". Note that B(A)
is considerably different from A(B). In the latter, each meaningful level of A would occur in
one and only one level of B.
These types of designs are also
designated as hierarchical designs in some textbooks.
A B(A) design occurs, for example, when the
first three levels of factor B (b1 ,b3, and b3)
appear only under level a1 of factor A and the next three levels of
B (b4 ,b5, and b6) appear only under level a2 of
factor A. Depending upon the labelling
scheme, b4 ,b5, and b6 may also be called b1
,b3, and b3, respectively. It is understood by the design designation that the b1
occurring under a1 is different from the b1 occurring
under a2.
Nested or hierarchical designs can appear because many
aspects of society are organized
hierarchically. For example
within the university, classes (sections) are nested within courses, courses
are nested within departments, departments within colleges, and colleges within
the university..
In experimental research it is also possible to nest
treatment conditions within other treatment conditions. For example, suppose a researcher was
interested in the effect of diet on health in hamsters. One factor (A) might be a high
cholesterol (a1) or low cholesterol (a2) diet. A second factor (B) might be type of
food, peanut butter (b1), cheese (b2), red meat (b3),
chicken (b4), fish (b5), or vegetables(b6). Because type of food may be categorized as
being either high or low in cholesterol, a B(A) experimental
design would result. Chicken, fish, and
vegetables would be relabelled as b1 ,b3, and b3,
respectively, but it would be clear from the experimental design specification
that peanut butter and chicken, cheese and fish, and red meat and vegetables,
were qualitatively different, even though they all share the same label.
While any factor may possibly be nested within any other
factor, the critical nesting relationship is with respect to subjects. If S is nested within some
combination of other factors, then each subjects appear under one, and only
one, combination of factors within which they are nested. These effects are often called the Between
Subjects effects. If S is
crossed with come combination of other factors, then each subject see all
combinations of factors with which they are crossed. These effects are referred to as Within Subjects effects.
As mentioned earlier subjects are necessarily nested within
blocking factors. Subjects are
necessarily nested within the effects of gender and current religious
preference, for example.
Treatment factors, however, may be nested or crossed with
subjects. The effect of caffeine on
alertness could be studied by dividing the subjects into two groups, with one
receiving a beverage with caffeine and one group not. This design would nest subjects with caffeine and be specified as
S(A), or simply A, as the S is often dropped when
the design is completely between subjects.
If subjects appeared under both caffeine conditions,
receiving caffeine on one day and no caffeine on the other, then subjects would
be crossed with caffeine. The design
would be specified as S X A.
In this case the S would remain in the design.
An Example
Design
A psychologist (McGuire, 1993) was interested in studying
adults' memory for medical information presented by a videotape. She included one-hundred and four
participants in which sixty-seven
ranged in age from 18 to 44 years and thirty seven ranged in age from 60
to 82 years. Participants were randomly
assigned to one of two conditions, either an organized presentation condition
or an unorganized presentation
condition. Following observation of the
videotape, each participant completed an initial recall sequence consisting of
free-recall and probed recall retrieval tasks. A probed recall is like a multiple-choice test and a free-recall
is like an essay test. Following a
one-week interval, participants completed the recall sequence again.
This experimental design provides four factors in addition
to subjects (S). The age factor
(A) has two level a1=young and a2=old and would
necessarily be a blocking factor. The
type of videotape factor (B) would be a treatment factor and would
consist of two levels b1=organized and b2=unorganized. The recall method factor (C) would be
a form of trials factor and would have two levels c1=free-recall and
c2=probed recall. The forth
factor (D) would be another trials factor where d1=immediate
and d2=one week delay.
Each level of B appears with each level of A,
thus A is crossed with B.
Since each subject appears in one and only one combination of A
and B, subjects are nested within A X B. That is, each subject is either young or old
and sees either an organized or unorganized videotape. The design notation thus far would be S
( A X B ).
Each type of recall (C) was done by each subject at
both immediate and delayed intervals (D). Thus subjects would be crossed with recall method and
interval. The complete design
specification would be S ( A X B ) X C X D. In words this design would be subjects
nested within A and B and crossed with C and D.
In preparation for entering the data into a data file, the
design could be viewed in a different perspective. Listing each subject as a row and each measure as a column, the
design would appear as follows:
Immediate Week Later
Age Videotape Subject Free Probed Free Probed
S1
Organized S2
Young ...
Unorganized S1
...
S1
Organized S2
Old ...
Unorganized S1
...
In this design,
two variables would be needed. One to classify each subjects as either young or
old, and one to document which type of videotape the subject saw. In addition to the classification variables,
each subject would require four variables to record the two types of measures
taken at the two different times.
A score taken
from the design presented above could be represented as Xabscd. For example, the immediate probed test score
taken from the third subject in the old group who viewed an organized videotape
would be X21312.
A
Second Example Design
The Lombard
effect is a phenomenon in which a speaker or singer involuntarily raises his or
her vocal intensity in the presence of high levels of sound. In a study of the Lombard effect in choral
singing (modified from Tonkinson, 1990), twenty-seven subjects, some
experienced choral singers and some not,
were asked to sing the national anthem along with a choir heard through
headphones. The performances were
recorded and vocal intensity readings from three selected places in the song
were obtained from a graphic level recorder chart. Each subject sang the song four times: with a none, or a soft, medium, or loud choir accompaniment. After some brief instructions to resist
increasing vocal intensity as the choir increased, each subject again sang the
national anthem four times with the four different accompaniments. The order of accompaniments was
counterbalanced over subjects.
In this design,
there would be four factors in addition to subjects. Subjects would be nested within experience level (A), with
a1=inexperienced and a2=experienced choral singers. This factor would be a blocking
factor. Subjects would be crossed with
instructions (B), where b1=no instructions and b2=resist
Lombard effect. In addition, subjects
would be crossed with accompaniment (C) and place in song (D).
The accompaniment factor would include four levels c1=soft, c2=medium,
c3=loud, and c4=none.
This factor would be considered a treatment factor. The place in song factor could be considered
a trial factor and would have three levels.
The
experimental design could be written as S ( A ) X B X C
X D. In words, subjects were
nested within experience level and crossed with instructions, accompaniment,
and place in song. In this design, one
variable would be needed for the classification of each subject and twenty-four
variables would be needed for each subject, one for each combination of
instructions, accompaniment, and place in song. The design could be written:
No
Instructions Resist
Lombard Effect
Soft Medium Loud
None Soft Medium
Loud None
Exp S 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 1
1 2
...
2 1
2 2
...
A
Third Example Design
From the
Springfield News-Leader, March 1, 1993:
Images of
beauty such at those shown by Sports Illustrated's annual swimsuit issue, are
harmful to the self-esteem of all women and contribute to the number of eating
disorder cases in the U. S., says a St. Louis professor who researches women's
health issues.
In a recent
study at Washington University, two groups of women - one with bulimia and one
without - watched videotapes of SI models in swimsuits.
Afterwards,
both groups reported a more negative self-image than they did before watching
the tape, describing themselves as "feeling fat and flabby" and
"feeling a great need to diet."
The experiment
described above has a number of inadequacies, the lack of control conditions
being the most obvious. The original
authors, unnamed in the article, may have designed a much better experiment
than is described in the popular press.
In any case, this experiment will now be expanded to illustrate a
complex experimental design.
The dependent
measure, apparently a rating of "feeling fat and flabby" and
"feeling a great need to diet", will be retained. In addition, two neutral questions will be
added, say "feeling anxious" and "feeling good about the
environment." These four
statements will be rated by all subjects, thus subjects will be crossed with
ratings. The first two statements deal
with body image and diet and the last two do not, thus they will form a factor
in the design (called D). Since
the statements within each of body image factor share no similarity across
levels of D, these statements (A) are nested within D. For
example, the rating of "feeling a great need to diet" and
"feeling good about the environment" share no qualitative
relationship. At this point the design
may be specified as S X A(D).
Suppose the
researcher runs the subjects in groups of six to conserve time and effort, thus
creating a groups (G) factor.
In addition to the two groups, with bulimia and without (B),
suppose the subjects viewed one of the following videotapes (V): SI models, Rosanne Barr, or a show about the
seals of the great northwest. Assuming
that all the subjects in each level of group either had bulimia or did not,
then the design could be specified as
S(G(B
X V)).
The factor B
is crossed with V because each level of B appears with each level
of V. That is, subjects with and
without bulimia viewed all three videotapes. Because each group viewed only a single videotape and was composed
of subjects either with bulimia or without, the groups factor is nested within
the cross of B and V.
Because subjects appeared in only one group, subjects are nested within
groups.
Combining the
between subjects effects, S(G(B X V)), and the
within subjects effects, A(D), yields the complete design
specification
S(G(B
X V)) X A(D).
Determining
the Number of Subjects and Measures per Subject
It is important
to be able to determine the number of subjects and the number of measures per
subject for practical reasons, namely, is the experiment feasible? After listening to a student propose an
experiment and a little figuring, I remarked "according to my
calculations, you should be able to complete the experiment sometime near the
middle of the next century." If an
experimenter is limited in the time a subject is available, then the number of
measures per subject is another important consideration.
To determine
the number of subjects, multiply the number of levels of the between subjects
factor together. In the previous
example, S = 6 because the subjects were run in groups of six. Let G=4, or there be four groups of six each
of combinations of bulimia and videotape.
Since there were two levels of bulimia, B=2, and three levels of
videotape, V=3. Since S(G(B
X V)), then the total number of subjects needed would be S * G * B * V
or 6*4*2*3 or 144. Since half of the
subjects must have bulimia, the question of whether or not 72 subjects with
bulimia are available must be asked before the experiment proceeds.
To find the
number of measures per subject, multiply the number of levels of the within
subjects factors together. In the
previous example A(D), where A=2 and D=2, there would be A * D or
2 * 2 or 4 measures per subject.
Setting
up the Data Matrix
A few rules
simplify setting up the data matrix.
First, each subject appears on a single row of the data matrix. Second, each measure or combination of
within subjects factors appears in a column of data. Third, each subject must be identified as to the combination of
between subjects factors which he or she appears.
1 1 1 2 5 5 3
1 1 1 5 5 5 4
1 1 1 3 2 1 3
1 1 1 2 5 3 1
1 1 1 3 5 4 3
2 1 1 5 4 5 5
...
4 2 3 3 5 5 4
|
In the previous
example, since there would be 144 subjects in the experiment, there would be
144 rows of data. Each subject would be
identified as to the level of G,
B, and V to which she belonged.
For example, a subject who appeared under g3 of b1
and v4 would be labelled as 3 1 4.
Since there are four measures per subject, these would appear as columns
in addition to the identifiers. An
example data matrix might appear as follows.
In this example, the level of G is in the first column, B in the second,
and V in the third. The four
combinations of within subjects factors appear next as ad11 ad12
ad12 ad22.
A Note
of Caution
It is fairly
easy to design complex experiments.
Running the experiments and interpreting the results are a different
matter. Many complex experiments are
never completed because of such difficulties.
This is from personal experience.
Chapter
9
One Between
Group ANOVA
Why Multiple Comparisons Using
t-tests is NOT the Analysis of Choice
Group
Therapy Method _ SX SX˛
|
1
Reality 20.53 3.45
11.9025
2
Behavior 16.32 2.98
8.8804
3
Psychoanalysis 10.39 5.89
35.7604
4
Gestalt 24.65 7.56
57.1536
5
Control 10.56 5.75
33.0625
|
Suppose a researcher has performed a study of
the effectiveness of various methods of individual therapy. The methods used were: Reality Therapy, Behavior Therapy,
Psychoanalysis, Gestalt Therapy, and, of course, a control group. Twenty patients were randomly assigned to
each group. At the conclusion of the
study, changes in self-concept were found for each patient. The purpose of the study was to determine if
one method was more or less effective than the other methods.
At the conclusion of the experiment the
researcher organizes the collected data in the following manner:
The researcher wishes to compare the means of
the groups with each other to decide about the effectiveness of the
therapy.
One method of performing this analysis is by
doing all possible t-tests, called multiple t-tests. That is, Reality Therapy is first compared with Behavior Therapy,
then Psychoanalysis, then Gestalt Therapy, then the Control Group. Behavior Therapy is then individually
compared with the last three groups, and so on. Using this procedure there would be ten different t-tests
performed. Therein lies the difficulty
with multiple t-tests.
First, because the number of t-tests
increases geometrically as a function of the number of groups, analysis becomes
cognitively difficult somewhere in the neighborhood of seven different
tests. An analysis of variance
organizes and directs the analysis, allowing easier interpretation of results.
Secondly, by doing a greater number of
analyses the probability of committing at least one type I error somewhere in
the analysis greatly increases. The
probability of committing at least one type I error in an analysis is called
the experiment-wise error rate. The researcher may desire to perform a fewer
number of hypothesis tests in order to reduce the experiment-wise error
rate. The ANOVA procedure performs this
function.
The Bottom Line - Results and
Interpretation of ANOVA
Results of an ANOVA are usually presented in
an ANOVA table. This table
contains columns labelled
"Source", "SS or Sum of Squares", "df - for degrees of
freedom", "MS - for mean square", "F or F-ratio", and
"p, prob, probability, sig., or
sig. of F". The only columns that
are critical for interpretation are the first and the last, the others are used
mainly for intermediate computational purposes.
Source SS df MS F sig of F
|
BETWEEN
5212.960 4 1303.240 4.354 .0108
WITHIN
5986.400 20 299.320
TOTAL
11199.360 24
|
An example of an ANOVA table appears below:
The row labelled "BETWEEN" under
"Source", having a probability value associated with it, is the only
one of any any great importance at this time.
The other rows are used mainly for computational purposes. The researcher then would most probably
first look at the value ".0108" located under the "sig of
F" column.
Of all the information presented in the ANOVA
table, the major interest of the researcher will most likely be focused on the
value located in the "sig of F." column. If the number (or numbers) found in this column is (are) less
than the critical value (α) set by the experimenter, then the effect is said to be
significant. Since this value is
usually set at .05, any value less than this will result in significant
effects, while any value greater than this value will result in nonsignificant
effects.
If the effects are found to be significant
using the above procedure, it implies that the means differ more than would be
expected by chance alone. In terms of
the above experiment, it would mean that the treatments were not equally
effective. This table does not tell the
researcher anything about what the effects were, just that there most likely
were real effects.
If the effects are found to be
nonsignificant, then the differences between the means are not great enough to
allow the researcher to say that they are different. In that case no further interpretation is attempted.
When the effects are significant, the means
must then be examined in order to determine the nature of the effects. There are procedures called "post-hoc
tests" to assist the researcher in this task, but often the analysis is
fairly evident simply by looking at the size of the various means. For example, in the preceding analysis
Gestalt and Reality Therapy were the most effective in terms of mean
improvement.
In the case of significant effects, a
graphical presentation of the means can sometimes assist in analysis. For example, in the preceding analysis, the
graph of mean values would appear as follows:
HYPOTHESIS TESTING THEORY UNDERLYING
ANOVA
The Sampling
Distribution Reviewed
In order to explain why the above procedure
may be used to simultaneously analyze a number of means, the following presents
the theory on ANOVA in relation to the hypothesis testing approach discussed in
earlier chapters.
First, a review of the sampling distribution
is necessary. If you have difficulty
with this summary, please go back and read the more detailed chapter on the sampling
distribution.
A sample is a finite number (N) of
scores. Sample statistics are numbers
which describe the sample. Example
statistics are the mean (_), mode (Mo), median (Md), and
standard deviation (sX).
Probability models exist in a theoretical
world where complete information is unavailable. As such, they can never be known except in the mind of the
mathematical statistician. If an
infinite number of infinitely precise scores were taken, the resulting distribution
would be a probability model of the population. Population models are characterized by parameters. Two common parameters are µX and σX.
Sample statistics are used as estimators of
the corresponding parameters in the population model. For example, the mean and standard deviation of the sample are
used as estimates of the corresponding population parameters µX and σX.
The sampling distribution is a distribution
of a sample statistic. It is a model of
a distribution of scores, like the population distribution, except that the
scores are not raw scores, but statistics.
It is a thought experiment; "what would the world be like if a
person repeatedly took samples of size N from the population distribution and
computed a particular statistic each time?" The resulting distribution of statistics is called the sampling
distribution of that statistic.
The sampling distribution of the mean is a special case of a sampling
distribution. It is a distribution of
sample means, described with the parameters µ_ and σ_. These parameters are closely related to the
parameters of the population distribution, the relationship being described by
the CENTRAL LIMIT THEOREM. The
CENTRAL LIMIT THEOREM essentially states that the mean of the sampling
distribution of the mean (µ_) equals the mean of the population (µX)
and that the standard error of the mean (σ_) equals the
standard deviation of the population (σX) divided by the
square root of N. These relationships
may be summarized as follows:
Two Ways of Estimating the
Population Parameter σX˛
When the data have been collected from more
than one sample, there exists two independent methods of estimating the
population parameter σX˛, called respectively the between and the
within method. The collected data are
usually first described with sample statistics as demonstrated in the following
example:
Group Therapy
Method _ SX SX˛
|
1 Reality 20.53 3.45 11.9025
2 Behavior 16.32 2.98 8.8804
3 Psychoanalysis 10.39
5.89 35.7604
4 Gestalt 24.65 7.56 57.1536
5 Control 10.56 5.75 33.0625
Mean 16.49 29.3519
Variance 38.83 387.8340
|
THE WITHIN METHOD
Since each of the sample variances may be
considered an independent estimate of the parameter σX˛, finding the mean of the variances provides a method of combining the
separate estimates of σX˛ into a single value. The resulting statistic is called the MEAN
SQUARES WITHIN, often represented by MSW. It is called the within method because it computes the estimate
by combining the variances within each sample. In the above example, the Mean Squares Within would be equal to
29.3519.
THE BETWEEN METHOD
The parameter σX˛ may also be
estimated by comparing the means of the different samples, but the logic is
slightly less straightforward and employs both the concept of the sampling
distribution and the Central Limit Theorem.
Sampling Distribution Actual Data
|
_ _
_ _
_ _
. _
. _
.
Mean µ_ __
Variance σ_˛ s_˛
|
First, the standard error of the mean squared
(σ_˛) is the population variance of a distribution of sample means. In real life in the situation where there is
more than one sample, the variance of the sample means may be used as an
estimate of the standard error of the mean squared (σ_˛). This is analogous to the
situation where the variance of the sample (sX˛) is used as an
estimate of σ_˛. The relationship is
demonstrated below:
In this case the Sampling Distribution
consists of an infinite number of means and the real life data consists of A
(in this case 5) means. The computed
statistic s_˛ is thus an estimate of the theoretical parameter σ_˛.
The relationship expressed in the Central Limit Theorem may now be used
to obtain an estimate of σ˛.
Thus
the variance of the population may be found by multiplying the standard error
of the mean squared (σ_˛) by N, the size of each sample.
Since the variance of the means, s_˛,
is an estimate of the standard error of the mean squared, σ_˛, the variance of the population, σX˛, may be
estimated by multiplying the size of each sample, N, by the variance of the
means. This value is called the Mean
Squares Between and is often symbolized by MSB. The computational procedure for MSB
is presented below:
MSB = N * s_˛
MSB
= 20 * 38.83
MSB
= 776.60
|
The expressed value is called the Mean
Squares Between because it uses the variance between the samples, that
is the sample means, to compute the estimate.
Using the above procedure on the example data yields:
At this point it has been established that
there are two methods of estimating σX˛, Mean Squares
Within and Mean Squares Between. It could
also be demonstrated that these estimates are independent. Because of this independence, when both are
computed using the same data, in almost all cases different values will result. For example, in the presented data MSW=29.3519
while MSB=776.60. This
difference provides the theoretical background for the F-ratio and ANOVA.
The F-ratio and F-distribution
A new statistic, called the F-ratio is computed by dividing the MSB
by MSW. This is illustrated
below:
Using the example data described earlier the
computed F-ratio becomes:
Fobs = MSB
/ MSW
Fobs = 776.60 /
29.3519
Fobs = 26.4582
The F-ratio can be thought of as a measure of
how different the means are relative to the variability within each
sample. The larger this value, the
greater the likelihood that the differences between the means are due to
something other than chance alone, namely real effects. The size of the F-ratio necessary to make a
decision about the reality of effects is the next topic of discussion.
If the difference between the means means is
due only to chance, that is, there are no real effects, then the expected value
of the F-ratio would be one (1.00).
This is true because both the numerator and the denominator of the
F-ratio are estimates of the same parameter, σX˛. Seldom will the F-ratio be exactly equal to
1.00, however, because the numerator and the denominator are estimates rather
than exact, known values. Therefore,
when there are no effects the F-ratio will sometimes be greater than one, and
other times less than one.
To review, the basic procedure used in
hypothesis testing is that a model is created in which the experiment is
repeated an infinite number of times when there are no effects. A sampling distribution of a statistic is
used as the model of what the world would look like if there were no
effects. The results of the experiment,
a statistic, is compared with what would be expected given the model of no
effects was true. If the computed
statistic is unlikely given the model, then the model is rejected along with
the hypothesis that there were no effects.
In an ANOVA the F-ratio is the statistic used
to test the hypothesis that the effects are real, in other words, that the
means are significantly different from one another. Before the details of the hypothesis test may be presented, the
sampling distribution of the F-ratio must be discussed.
If
the experiment were repeated an infinite number of times, each time
computing the F-ratio, and there were no effects, the resulting distribution
could be described by the F-distribution.
The F-distribution is a theoretical probability distribution
characterized by two parameters, df1 and df2, both of
which affect the shape of the distribution.
Since the F-ratio must always be positive, the F-distribution is
non-symmetrical, skewed in the positive direction.
Two examples of an F-distribution are
presented below; the first with df1=1 and df2=5, and the
second with df1=10 and df2=25.
The
F-distribution has a special relationship to the t-distribution described
earlier. When df1=1, the
F-distribution is equal to the t-distribution squared (F=t˛). Thus the t-test and the ANOVA will always
return the same decision when there are two groups. That is, the t-test is a special case of ANOVA.
Non-significant and Significant
F-ratios
Theoretically, when there are no real effects, the F-distribution is an
accurate model of the distribution of F-ratios. The F-distribution will have the parameters df1=a-1
(where a-1 is the number of different groups minus one) and df2=a(N-1),
where a is the number of groups and N is the number in each group. In this case
an assumption is made that sample size is equal for each group. For example, if
five groups of five subjects each were run in an experiment and there were no
effects, then the F-ratios would be distributed with df1=k-1=5-1=4
and df2=k(n-1)=5(5-1)=5*4=20.
A visual representation of the preceding appears as follows:
The F-ratio in the above which cuts off
various proportions of the distributions may be computed for different values
of α. These F-ratios are called Fcrit
values. In the above example the Fcrit
value for α=.25 is 1.46, for α=.10 results in a value of 2.25, for α=.05 the value is 2.87, and for α=.01 the value is 4.43. These values are illustrated in the figure
below:
When there are real effects, that is, the
means of the groups are different due to something other than chance, then the
F-distribution no longer describes the distribution of F-ratios. In almost all cases the observed F-ratio
will be larger than would be expected when there were no effects. The rationale for this situation is
presented below.
First, an assumption is made that any effects
are an additive transformation of the score.
That is, the scores for each group can be modelled as a constant ( aa
- the effect) plus error (eae).
The scores appear as follows:
Xae = aa + eae
where X is the score, aa is the
treatment effect, and eae is the error. The eea, or error, is different for each subject,
while aa is constant within a given group.
As described in the chapter on
transformations, an additive transformation changes the mean, but not the
standard deviation or the variance.
Because the variance of each group is not changed by the nature of the
effects, the Mean Square Within, as the mean of the variances, is not
affected. The Mean Square Between, as N
time the variance of the means, will in most cases become larger because the
variance of the means will most likely become larger.
Imagine three individuals taking a test. An instructor first finds the variance of
the three score. He or she then adds
five points to one random individual and subtracts five from another random
individual. In most cases the variance
of the three test score will increase, although it is possible that the
variance could decrease if the points were added to the individual with the
Group Mean Variance
Group Mean Variance
1 µ σ˛ 1 µ + a1
σ˛
2 µ σ˛ 2 µ + a2
σ˛
3 µ σ˛ 3 µ + a3
σ˛
4 µ σ˛ 4 µ + a4
σ˛
5 µ σ˛ 5 µ + a5
σ˛
Mean µ σ˛ µ σ˛
Variance σ˛/N >σ˛/N
|
lowest score and subtracted from the
individual with the highest score. If
the constant added and subtracted was 30 rather than 5, then the variance would
almost certainly be increased. Thus,
the greater the size of the constant, the greater the likelihood of a larger
increase in the variance.
With respect to the sampling distribution, the model differs depending
upon whether or not there are effects.
The difference is presented below:
Since
the MSB usually increases and MSW remains the same, the
F-ratio (F=MSB/MSW) will most likely increase. Thus, if there are real effects, then the
F-ratio obtained from the experiment will most likely be larger than the
critical level from the F-distribution.
The greater the size of the effects, the larger the obtained F-ratio is
likely to become.
Thus, when there are no effects, the obtained
F-ratio will be distributed as an F-distribution which may be specified. If effects exist, then the obtained F-ratio
will most likely become larger. By
comparing the obtained F-ratio with that predicted by the model of no effects,
an hypothesis test may be performed to decide on the reality of effects. If the obtained F-ratio is greater than the
critical F-ratio, then the decision will be that the effects are real. If not, then no decision about the reality
of effects can be made.
Similarity of ANOVA and t-test
When the number of groups (A) equals two (2),
an ANOVA and t-test will give similar results, with tCRIT˛=FCRIT
and tOBS˛=FOBS.
This equality is demonstrated in the example below:
Given the following numbers for two
groups:
Mean
Variance
Group 1 - 12 23 14 21 19 23 26 11 16
18.33 28.50
Group 2 - 10 17 20 14 23 11 14 15 19
15.89 18.11
Computing the t-test
s_1-_2
= Ö (s1˛ + s2˛)/ 9 = Ö (28.50 + 18.11)/9 = Ö 5.18 =
2.28
tOBS
= ( _1-_2 ) / s_1-_2 = 18.33 - 15.89 / 2.28 =
1.07
t(df=16) = 2.12 for α=.05 and two-tailed test
Computing the ANOVA
MSBETWEEN = N * s_˛ = 9 * 2.9768 = 26.7912
MSWITHIN = Mean of
the Variances = ( 28.50 + 18.11 ) / 2 = 23.305
FOBS
= MSBETWEEN/MSWITHIN = 1.1495
F(1,16) = 4.41 for α=.05 - two-tailed test is assumed
Comparing the results
tOBS˛
= 1.1449 FOBS = 1.1449
t(16)˛ = 4.49 F(1,16) = 4.49
The differences between the predicted and
observed results can be attributed to rounding error (close enough for
government work).
Because the t-test is a special case of the
ANOVA and will always yield similar results, most researchers perform the ANOVA
because the technique is much more powerful in complex experimental designs.
EXAMPLE OF A NON-SIGNIFICANT ONE-WAY
ANOVA
7 7 5 4 2 7 5 4 1 7 5 6 7 6 3 5 2 5 1
4 4.65 4.03
6 9 3 6 9 4 9 8 9 3 4 4 7 2 2 7 7 7 9
3 5.90 6.52
5 5 2 5 6 2 3 3 6 8 2 1 1 2 5 7 9 6 5
7 4.50 5.63
4 1 4 8 9 5 2 8 6 8 2 9 6 6 7 8 4 3 1
4 5.25 6.93
3 6 1 2 3 5 8 4 1 5 4 5 6 9 4 2 4 8 9
3 4.60 6.04
Computing
the ANOVA
MSBETWEEN = N * s_˛
= 20 * .351 = 7.015
MSWITHIN = Mean of the Variances = 5.83
FOBS = MSBETWEEN/MSWITHIN
= 1.20
F(4,95) = 2.53 for α=.05 - non-directional test is
assumed
|
Given the following data for five groups, perform an ANOVA:
Since the FCRIT is greater than
the FOBS, the means are not significantly different and no effects
are said to be discovered.
EXAMPLE OF A SIGNIFICANT ONE-WAY
ANOVA
Given the following data for five groups,
perform an ANOVA. Note that the numbers
are similar to the previous example except that one has been subtracted from
all scores in Group 3 and one has been added to all scores in Group 4.
In this case the FOBS is greater
than FCRIT, thus the means are significantly different and we decide
that the effects are real.
1 31
1 25
1 29
1 30
1 28
1 31
1 31
1 33
2 32
2 28
2 36
2 34
2 41
2 35
2 32
2 28
2 31
|
USING MANOVA
7 7 5 4 2 7 5 4 1 7 5 6 7 6 3 5
2 5 1 4 4.65 4.03
6 9 3 6 9 4 9 8 9 3 4 4 7 2 2 7
7 7 9 3 5.90 6.52
4 4 1 4 5 1 2 2 5 7 1 0 0 1 4 6
8 5 4 6 3.50 5.63
5 2 5 9 10 6 3 9 7 9 3 10 7 7 8 9 5 4 2
4 6.25 6.93
3 6 1 2 3 5 8 4 1 5 4 5 6 9 4 2
4 8 9 3 4.60 6.04
Computing
the ANOVA
MSBETWEEN = N * s_˛
= 20 * 1.226 = 24.515
MSWITHIN = Mean of the Variances = 5.83
FOBS = MSBETWEEN/MSWITHIN
= 4.20
F(4,95) = 2.53 for α=.05 - two-tailed test is assumed
|
While an single factor between groups ANOVA
may be done using the MEANS command in SPSS, the MANOVA command is a general purpose
command which allows the statistician to do almost any type of multifactor
univariate or multivariate ANOVA.
The Data
The data is entered into a data file
containing two columns. One column
contains the level of the factor to which the observation belongs and the
second the score for the dependent variable.
A third column containing the observation number, in the example a
number from one to nine, is optional.
As in all SPSS data files, the number of rows in the data file
corresponds to the number of subjects and each variable is lined up neatly in
each row. In the example data file
presented to the right, there are two groups of nine each. The level of the independent variable is
given in the first column of the data file, a space is entered, and the
dependent variable is entered in columns 3 and 4.
RUN NAME EXAMPLE FOR ANOVA BOOK - DESIGN A.
|
DATA LIST FILE='DESIGNA
DATA A' /1 A 1 X 3‑4.
VALUE LABELS
A 1 'BLUE BOOK' 2
'COMPUTER'
LIST.
MANOVA X by A(1,2)
/PRINT CELLINFO(MEANS)
/DESIGN.
|
The RUN NAME command of the example program
gives a general description of the purpose of the program. The second command reads in the data
file. Note that the group factor is
called "A" and the dependent variable is called "X". The value label command then describes the
different levels of the group variable.
The LIST command gives a description of the data as the computer
understands it.
The MANOVA command is followed by the name of
the dependent variable, here X, and the keyword BY . The factor name "A" is then entered, followed by the
the beginning and ending levels of that factor. In this case there were only two levels, defined by a beginning
value of "1", and an ending value of "2". The second line on the command is preceded
by a slash "/'" and then the subcommand PRINT=CELLINFO(MEANS). This command will print the means of the
respective groups. The last subcommand,
"/DESIGN" , is optional at this point, but not including it will
generate a WARNING. Nothing is altered
with the WARNING, but it is not neat.
Example Output
The output produced by the example MANOVA
command is presented on the next page.
The default error term in MANOVA has been changed
from WITHIN CELLS to WITHIN+RESIDUAL.
Note that these are the same for all full factorial designs.
|
* *
* * * * A n a l y s i s o f V a r i a n c e * * * * *
18 cases accepted.
0 cases rejected because of out‑of‑range
factor values.
0 cases rejected because of missing
data.
2 non‑empty cells.
1 design will be processed.
Cell Means and Standard Deviations
Variable .. X
FACTOR CODE Mean Std. Dev. N
A 1
29.000 3.202 9
A 2
33.000 4.093 9
For entire sample 31.000
4.116 18
* *
* * * * A n a l y s i s o f V a r i a n c e ‑‑
design
Tests of Significance for X using UNIQUE
sums of squares
Source of Variation SS DF MS F
Sig of F
WITHIN CELLS 216.00
16 13.50
A 72.00
1 72.00 5.33
.035
(Model) 72.00
1 72.00 5.33
.035
(Total) 288.00
17 16.94
R‑Squared = .250
Adjusted R‑Squared = .203
|
Note that the F-ratio was significant with a
value of .03. The means for the
two groups were 29 and 33 respectively.
Dot Notation
In order to simplify the notational system
involving the summation sign, a notational system call dot notation has
been introduced. Dot notation places a period in place of subscript to mean
summation. For example:
The symbol X. means that the variable X has
been summed over whatever counter variable was used as a subscript. In a like manner, if a bar is placed over a
variable, it refers to a mean over the dotted counter variable(s). For example:
where _. means the same thing as _. (The dot
notation does become a bit tricky where real periods are involved.)
The real advantage is apparent when two or
more subscripts are used. For example
and
or if a=1 then
Using the same notational system, means may
be given. For example:
and
or if a=1 then
The difference between _a. and _1.
is that the second is a special case of the first when a=1.
For example, if A=3, B=4 and X11=5,
X12=8, X13=6, X14=9, X21=7, X22=10,
X23=5, X24=9, X31=6, X32=4, X33=7,
X34=3, then
X.. = 79,
_.. = 79/12 = 6.5833,
X1. = 28,
_1. = 28/4 = 7.
POST-HOC Tests of Significance
If the results of the ANOVA are significant,
it indicates that there are real effects between the means of the groups. The nature of the effects are not specified
by the ANOVA. For example, an effect
could be significant because the mean of group three was larger than the means
of the rest of the groups. In another
case, the means of groups one, four and five might be significantly smaller
than the means of groups two, three, and six.
Often the pattern of results can be determined by a close examination of
the means. In other instances, the
reason for the significant differences is not apparent. To assist the statistician in interpreting
effects in significant ANOVAs, post-hoc tests of significance were developed.
A post-hoc (after the fact) test of significance
is employed only if the results of the overall significance test are
significant. A post-hoc test is
basically a multiple t-test procedure with some attempt to control for the
increase in the experiment wide error rate when doing multiple significance
tests. A number of different procedures
are available to perform post-hoc tests differing in the means of control of
the increase in error rates . The
different procedures include Duncan’s Multiple Range test, the Newman-Keuls
procedure, and a procedure developed by Sheffe’. The interested reader is referred to Winer (1971) or Hays (1981)
for a thorough discussion of these methods.
My personal feeling is that post-hoc tests
are not all that useful. Most often,
the reason for the significant results is obvious from a close observation of
the means of the groups. Better
procedures are available using pre-planned contrasts to test patterns of
results. The use of pre-planned
contrasts requires that the statistician have the type of comparisons in mind
before doing the analysis. This is the
difference between data-driven (post-hoc) and theory-driven (pre-planned)
analysis. If a choice is possible, the
recommendation is for theory-driven analysis.
Chapter
10
TWO Between
Groups ANOVA (A x B)
The Design
In this design there are two independent
factors, A and B, crossed with each other. That is, each level of A appears in
combination with each level of B.
Subjects are nested within the combined levels of A and B
such that the full design would be written as S ( A X B ).
Because the Subjects (S) term is confounded with the error term,
it is dropped from the description of the design.
1 1
23
1 1
32
1 1
25
1 2
29
1 2
30
1 2
34
1 3
31
1 3
36
1 3
33
2 1
32
2 1
26
2 1
26
2 2
34
2 2
41
2 2
35
2 3
24
2 3
27
2 3
31
|
Suppose a statistics teacher gave an essay
final to his class. He randomly divides
the classes in half such that half the class writes the final with a blue-book
and half with notebook computers. In
addition the students are partitioned into three groups, no typing ability,
some typing ability, and highly skilled at typing. Answers written in blue-books will be transcribed to word
processors and scoring will be done blindly.
Not with a blindfold, but the instructor will not know the method or
skill level of the student when scoring the final. The dependent measure will be the score on the essay part of the
final exam.
The first factor (A) will be called Method and will have two levels, a1=blue-book
and a2 = computer. The second factor (B) will be
designated as Ability and will have three levels: b1=none, b2=some, and b3=lots. Because each level of A appears with
each level of B, A is said to be crossed with B (AXB). Since different subjects will appear in each
combination of A and B, subjects are nested within AXB. Each subject will be measured a single
time. Any effects discovered will
necessarily be between subjects or groups and hence the designation
"between groups" designs.
The Data
The data file for the A X B
design is similar to the data file for design A with the addition of the
second descriptive variable, B, for each subject. In the case of the example data, the A
factor has two levels while the B factor has three. The X variable is the score on the final
exam. The example data file appears in
the text box on the right.
Example SPSS Program using MANOVA
The SPSS commands necessary to do the
analysis for an A X B design are given in the text box below.
RUN NAME EXAMPLES FOR ANOVA BOOK ‑ A X B DESIGNS
|
DATA LIST FILE='AXB
DATA A' / A 3 B 6 X 15‑19.
VARIABLE LABELS
A 'METHOD OF WRITING EXAM'
B 'KEYBOARD EXPERIENCE'
X 'SCORE ON FINAL EXAM'.
VALUE LABELS
A 1 'BLUE‑BOOK' 2 'COMPUTER'/
B 1 'NONE' 2 'SOME' 3 'LOTS'.
LIST.
MANOVA X BY A (1,2) B
(1,3)
/PRINT CELLINFO(MEANS)
/DESIGN.
|
Note that the DATA LIST command must read in
a variable to code each factor. In this
case the variables were named A and B to correspond with the factor names,
although in most real-life situations more descriptive names will be used. The addition of the optional VALUE LABELS
command will label the output from the PRINT CELLINFO(MEANS) command, making
the output easier to interpret.
The MANOVA command is followed by the name of
the dependent variable, in this case X and then the variable names of the
factors. As in the previous MANOVA
commands, the factor names are each followed by the beginning and ending levels
enclosed in parentheses. In this case
the A factor has two levels beginning at level 1 and the B factor
has three. The PRINT CELLINFO(MEANS)
command is optional, but usually included because the means are the central
focus of the analysis. The DESIGN
command is optional, but excluding it will generate a warning when the program
is run, so it is usually included for the sake of neatness.
All of the analyses on the following pages
were generated from the SPSS program presented above. The program will not be included as part of the interpretation of
the output.
Interpretation of Output
Cell Means and Standard Deviations
|
Variable .. X
SCORE ON FINAL EXAM
FACTOR CODE Mean Std. Dev. N 95 percent Conf. Interval
A BLUE‑BOO
B NONE 26.717 4.822 3 14.739 38.695
B SOME 30.923 2.418 3 24.916 36.930
B LOTS 33.277 2.157 3 27.919 38.634
A COMPUTER
B NONE 28.057 3.285 3 19.896 36.217
B SOME 36.627 3.413 3 28.148 45.106
B LOTS 27.080 3.673 3
17.955 36.205
For
entire sample 30.447 4.675 18
28.122 32.771
|
The interpretation of the output from the
MANOVA command will focus on two parts:
the table of means and the ANOVA summary table. The table of means is the primary focus of
the analysis while the summary table directs attention to the interesting or
statistically significant portions of the table of means.
A table of means generated using the example
data and the PRINT CELLINFO(MEANS) subcommand in MANOVA is presented below:
Often the means are organized and presented
in a slightly different manner than the form of the output from the MANOVA
command. The table of means may be
rearranged and presented as follows:
b1
b2 b3
┌────────┬────────┬────────┐
a1 │ 26.72 │ 30.92 │ 33.28 │ 30.31
├────────┼────────┼────────┤
a2 │ 28.06 │ 36.62 │ 27.08 │ 30.58
└────────┴────────┴────────┘
27.39
33.78 30.18 30.447
|
The means inside the boxes are called cell
means, the means in the margins are called marginal means, and the
number on the bottom right-hand corner is called the grand mean. An analysis of these means reveals that there is very little difference between
the marginal means for the different levels of A across the levels of B
(30.31 vs. 30.58). The marginal means
of B over levels of A are different (27.39 vs. 33.78 vs. 30.18)
with the mean for b2 being the highest. The cell means show an increasing pattern for levels of B
at a1 (26.72 vs. 30.92 vs. 33.28) and a different pattern for levels
of B at a2 (28.06 vs. 36.62 vs. 27.08).
Graphs of Means
Graphs of means are often used to present
information in a manner which is easier to comprehend than the tables of
means. One factor is selected for
presentation as the X-axis and its levels are marked on that axis. Separate lines are drawn the height of the
mean for each level of the second factor.
In the following graph, the B, or keyboard experience, factor was
selected for the X-axis and the A, or method, factor was selected for
the different lines. Presenting the
information in an opposite fashion would be equally correct, although some
graphs are more easily understood than others, depending upon the values for
the means and the number of levels of each factor. The second possible graph is presented below. It is recommended that if there is any doubt
that both versions of the graphs be attempted and the one which best
illustrates the data be selected for inclusion into the statistical
report. It is hopefully obvious that
the graph with B on the X-axis is easier to understand than the one with A on
the X-axis.
Because the interpretation of the graph of
the interaction depends upon the results of the analysis, the ANOVA summary
table will now be presented. Following
this, the graph of the interaction will be reanalyzed.
The ANOVA Summary Table
The results of the A X B ANOVA
are presented in the ANOVA summary table by MANOVA. An example of this table is presented below:
Tests of Significance for X using UNIQUE sums of squares
|
Source SS DF MS F
Sig of F
WITHIN CELLS
139.36 12 11.61
A .36 1
.36 .03 .863
B 123.08 2
61.54 5.30 .022
A BY B 108.73 2 54.36 4.68 .031
|
The items of primary interest in this table
are the effects listed under the "Source" column and the values under
the "Sig of F" column. As in
the previous hypothesis test, if the value of "Sig of F" is less than
the value of α as set by the experimenter, then that effect
is significant. If α=.05, then the B main effect and the A BY B
interaction would be significant in this table.
Main Effects
Main
effects are differences in means over levels of one factor collapsed over
levels of the other factor. This is
actually much easier than it sounds.
For example, the main effect of A is simply the difference
between the means of final exam score for the two levels of Method, ignoring or
collapsing over experience. As seen in
the second method of presenting a table of means, the main effect of A
is whether the two marginal means associated with the A factor are
different. In the example case these means
were 30.31 and 30.58 and the differences between these means was not
statistically significant.
As can be seen from the summary table, the
main effect of B is significant.
This effect refers to the differences between the three marginal means
associated with factor B. In
this case the values for these means were 27.39, 33.78, and 30.18 and the
differences between them may be attributed to a real effect.
Simple Main Effects
A simple main effect is a main effect of one
factor at a given level of a second factor. In the example data it would be possible to talk about the simple
main effect of B at a1.
That effect would be the difference between the three cell means at
level a1 (26.72, 30.92, and 33.28).
One could also talk about the simple main effect of A at b3
(33.28 and 27.08). Simple main effects
are not directly tested in design A X B, however they are
necessary to understand an interaction.
Interaction Effects
An interaction effect is a change in the
simple main effect of one variable over levels of the second. An AB or A BY B interaction is a change
in the simple main effect of B over levels of A or the change in
the simple main effect of A over levels of B. In either case the cell means cannot be
modelled simply by knowing the size of the main effects. An additional set of parameters must be used
to explain the differences between the cell means. These parameters are collectively called an interaction.
The change in the simple main effect of one
variable over levels of the other is most easily seen in the graph of the
interaction. If the lines describing
the simple main effects are not parallel, then a possibility of an interaction
exists. As can be seen from the graph
of the example data, the possibility of a significant interaction exists
because the lines are not parallel. The
presence of an interaction was confirmed by the significant interaction in the
summary table.
The following graph overlays the main effect
of B on the graph of the interaction.
Two things can be observed from this presentation. The first is that the main effect of B is
possibly significant, because the means are different heights. Second, the interaction is possibly significant
because the simple main effects of B at a1 and a2 are
different from the main effect of B.
One method of understanding how main effects
and interactions work is to observe a wide variety of data and data
analysis. With three effects, A,
B, and AB, which may or may not be significant there are eight
possible combinations of effects. All
eight are presented on the following pages.
Example Data Sets, Means, and
Summary Tables
No Significant Effects
A B X
1 1 22
1 1 24
1 1 25
1 2 26
1 2 29
1 2 23
1 3 21
1 3 25
1 3 22
2 1 21
2 1 26
2 1 25
2 2 24
2 2 20
2 2 24
2 3 23
2 3 26
2 3 20
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE 23.667 1.528 3 19.872 27.461
B
SOME
26.000 3.000 3 18.548 33.452
B
LOTS
22.667 2.082 3 17.495 27.838
A COMPUTER
B
NONE
24.000 2.646 3 17.428 30.572
B
SOME
22.667 2.309 3 16.930 28.404
B
LOTS 23.000 3.000 3 15.548 30.452
For entire sample 23.667 2.401 18 22.473 24.861
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS
F Sig of F
WITHIN CELLS
74.00 12 6.17
A
3.56 1 3.56 .58 .462
B
7.00 2 3.50 .57 .581
A BY B
13.44 2 6.72 1.09 .367
Main Effect of A
A B X
1 1 32
1 1 34
1 1 35
1 2 36
1 2 39
1 2 33
1 3 31
1 3 35
1 3 32
2 1 21
2 1 26
2 1 25
2 2 24
2 2 20
2 2 24
2 3 23
2 3 26
2 3 20
Cell Means and Standard
Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
33.667 1.528 3 29.872 37.461
B
SOME
36.000 3.000 3 28.548 43.452
B
LOTS
32.667 2.082 3 27.495 37.838
A COMPUTER
B
NONE
24.000 2.646 3 17.428 30.572
B
SOME
22.667 2.309 3 16.930 28.404
B
LOTS
23.000 3.000 3 15.548 30.452
For entire sample 28.667 6.078 18
25.644 31.689
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
74.00 12 6.17
A
533.56 1 533.56
86.52 .000
B
7.00 2 3.50 .57 .581
A BY B
13.44 2 6.72 1.09 .367
Main Effect of B
A B X
1 1 42
1 1 44
1 1 45
1 2 36
1 2 39
1 2 33
1 3 21
1 3 25
1 3 22
2 1 41
2 1 46
2 1 45
2 2 34
2 2 30
2 2 34
2 3 23
2 3 26
2 3 20
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
43.667 1.528 3 39.872 47.461
B
SOME
36.000 3.000 3 28.548 43.452
B
LOTS
22.667 2.082 3 17.495 27.838
A COMPUTER
B
NONE 44.000 2.646 3 37.428 50.572
B
SOME
32.667 2.309 3 26.930 38.404
B
LOTS
23.000 3.000 3 15.548 30.452
For entire sample 33.667 9.133 18 29.125 38.208
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
74.00 12
6.17
A
3.56 1 3.56 .58 .462
B
1327.00 2 663.50
107.59 .000
A BY B
13.44 2 6.72 1.09 .367
AB Interaction
A B X
1 1 42
1 1 44
1 1 45
1 2 36
1 2 39
1 2 33
1 3 21
1 3 25
1 3 22
2 1 21
2 1 26
2 1 25
2 2 34
2 2 30
2 2 34
2 3 43
2 3 46
2 3 40
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
43.667 1.528 3
39.872 47.461
B
SOME
36.000 3.000 3 28.548 43.452
B
LOTS
22.667 2.082 3 17.495 27.838
A COMPUTER
B NONE 24.000
2.646 3 17.428 30.572
B
SOME
32.667 2.309 3 26.930 38.404
B
LOTS
43.000 3.000 3 35.548 50.452
For entire sample 33.667 8.738 18 29.321 38.012
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS 74.00 12 6.17
A
3.56 1 3.56 .58 .462
B
7.00 2 3.50 .57 .581
A BY B
1213.44 2 606.72
98.39 .000
Main Effects of A and B
A B X
1 1 52
1 1 54
1 1 55
1 2 46
1 2 49
1 2 43
1 3 31
1 3 35
1 3 32
2 1 41
2 1 46
2 1 45
2 2 34
2 2 30
2 2 34
2 3 23
2 3 26
2 3 20
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE 53.667 1.528 3 49.872 57.461
B
SOME
46.000 3.000 3 38.548 53.452
B
LOTS
32.667 2.082 3 27.495 37.838
A COMPUTER
B
NONE
44.000 2.646 3 37.428 50.572
B
SOME
32.667 2.309 3 26.930 38.404
B
LOTS 23.000 3.000 3 15.548 30.452
For entire sample 38.667
10.705 18 33.343 43.990
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
74.00 12 6.17
A
533.56 1 533.56
86.52 .000
B
1327.00 2 663.50
107.59 .000
A BY B
13.44 2 6.72 1.09 .367
Main effect of A, AB Interaction
A B X
1 1 52
1 1 54
1 1 55
1 2 46
1 2 49
1 2 43
1 3 31
1 3 35
1 3 32
2 1 21
2 1 26
2 1 25
2 2 34
2 2 30
2 2 34
2 3 43
2 3 46
2 3 40
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
53.667 1.528 3 49.872 57.461
B
SOME
46.000 3.000 3 38.548 53.452
B
LOTS
32.667 2.082 3 27.495 37.838
A COMPUTER
B
NONE
24.000 2.646 3 17.428 30.572
B
SOME
32.667 2.309 3 26.930 38.404
B
LOTS
43.000 3.000 3 35.548 50.452
For entire sample 38.667
10.370 18
33.510 43.823
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
74.00 12 6.17
A
533.56 1 533.56
86.52 .000
B
7.00 2 3.50 .57 .581
A BY B
1213.44 2 606.72
98.39 .000
Main Effect of B, AB Interaction
A B X
1 1 32
1 1 34
1 1 35
1 2 46
1 2 49
1 2 43
1 3 21
1 3 25
1 3 22
2 1 31
2 1 36
2 1 35
2 2 34
2 2 30
2 2 34
2 3 33
2 3 36
2 3 30
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
33.667 1.528 3
29.872 37.461
B
SOME
46.000 3.000 3 38.548 53.452
B
LOTS
22.667 2.082 3 17.495 27.838
A COMPUTER
B NONE 34.000
2.646 3 27.428 40.572
B
SOME
32.667 2.309 3 26.930 38.404
B
LOTS
33.000 3.000 3 25.548 40.452
For entire sample 33.667 7.268 18 30.052 37.281
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS 74.00 12 6.17
A
3.56 1 3.56 .58 .462
B
397.00 2 198.50
32.19 .000
A BY B
423.44 2 211.72
34.33 .000
Main Effects of A and B, AB Interaction
A B X
1 1 22
1 1 24
1 1 25
1 2 36
1 2 39
1 2 33
1 3 41
1 3 45
1 3 42
2 1 41
2 1 46
2 1 45
2 2 44
2 2 40
2 2 44
2 3 43
2 3 46
2 3 40
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B NONE 23.667
1.528 3 19.872 27.461
B
SOME
36.000 3.000 3 28.548 43.452
B
LOTS
42.667 2.082 3 37.495 47.838
A COMPUTER
B
NONE
44.000 2.646 3 37.428 50.572
B
SOME
42.667 2.309 3 36.930 48.404
B LOTS 43.000
3.000 3 35.548 50.452
For entire sample 38.667 7.700 18 34.837 42.496
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
74.00 12 6.17
A
373.56 1 373.56
60.58 .000
B
247.00 2 123.50
20.03 .000
A BY B
313.44 2 156.72
25.41 .000
No Significant Effects
A B X
1 1 32
1 1 24
1 1 15
1 2 46
1 2 39
1 2 23
1 3 31
1 3 45
1 3 52
2 1 31
2 1 46
2 1 55
2 2 34
2 2 40
2 2 54
2 3 33
2 3 46
2 3 50
Cell Means and Standard Deviations
Variable .. X
SCORE ON FINAL EXAM
FACTOR
CODE Mean Std. Dev. N 95 percent
Conf. Interval
A BLUE‑BOO
B
NONE
23.667 8.505 3 2.539 44.794
B
SOME
36.000 11.790 3 6.712 65.288
B
LOTS 42.667 10.693 3 16.104 69.229
A COMPUTER
B
NONE
44.000 12.124 3 13.881 74.119
B
SOME
42.667 10.263 3
17.171 68.162
B
LOTS
43.000 8.888 3 20.920 65.080
For entire sample 38.667
11.499 18 32.948 44.385
Tests of Significance for X using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
1314.00 12 109.50
A
373.56 1 373.56 3.41 .090
B
247.00 2 123.50 1.13 .356
A BY B
313.44 2 156.72 1.43 .277
Note that the means and graphs of the last
two example data sets were identical.
The ANOVA table, however, provided a quite different analysis of each
data set. The data in this final set
was constructed such that there was a large standard deviation within each
cell. In this case the marginal and
cell means were not different enough to warrant rejecting the hypothesis of no
effects, thus no significant effects were observed.
Dot Notation Revisited
The reader may recall from the previous
chapter that placing dots instead of subscripts is a shorthand notation for
summation. For example
When two subscripts are involved the notation
becomes somewhat more complicated (and powerful). For example
and
or if a=1 then
When three subscripts are involved as is
necessary in an A X B design the notation involves even more
summation signs. For example
and, for example
where one sums over the subscript containing
the dot.
Using the dot notation with means rather than
sums is a relatively simple extension of the dot notation. The mean is found by
dividing the sum by the number of scores which were included in the sum. For example, the grand mean can be found as
follows:
and the cell means
All this is most easily understood in the
context of a data table.
b1 b2 b3
X111=10 X11.=33 X121=22 X12.=72 X131=14 X13.=45 X1..=150
a1 X112=11
_11.=11 X122=24 _12.=24 X132=15 _13.=15 _1..=16.67
X113=12
X123=26 X133=16
X211=20 X21.=63 X221=21 X22.=66 X231=18 X23.=57 X2..=186
a2 X212=21
_21.=21 X222=22 _22.=22 X232=19 _23.=19 _2..=20.67
X213=22
X223=23 X233=20
X.1..=96 X.2..=138 X.3..=102 X...=336
_.1..=16 _.2..=23 _.3..=17 _...=18.67
Chapter
11
Nested Two
Factor Between Groups Designs B(A)
The
Design
Factor B is
said to be nested within factor A when a given level of B appears under
a single level of A. This occurs, for
example, when the first three levels of factor B (b1 ,b3,
and b3) appear only under level a1 of factor A and the
next three levels of B (b4
,b5, and b6)
appear only under level a2 of factor A. These types of designs are also designated as hierarchical designs in
some textbooks.
1 1
1
1 1
2
1 1
3
1 2
5
1 2
6
1 2
7
1 3
9
1 3 10
1 3 11
2 1
1
2 1
2
2 1
3
2 2
1
2 2
2
2 2
3
2 3
1
2 3
2
2 3
3
|
Nested or
hierarchical designs can appear because many aspects of society are
organized hierarchically. For example within the university, classes
(sections) are nested within courses, courses are nested within instructors,
and instructors are nested within departments.
In experimental
research it is also possible to nest treatment conditions within other
treatment conditions. For example in
studying the addictive potential of drugs, drug type could be nested within
drug function, i. e. Librium, Valium,
and Xanax are drugs used to treat anxiety, Prozac is used to treat depression,
and Halcion sold as a sleeping pill.
Here each drug appears in one and only one level of drug function.
The
Data
The data are
organized similarly to the A x B design.
Note that the data in the following table is identical to that in Table
6.1.
SPSS
commands
The SPSS
commands to do this analysis are
identical to those necessary to do the A x B analysis with one exception; the
DESIGN subcommand must completely specify the design. In this case the design can be specified with the effects A and B
WITHIN A, where A corresponds to the main effect of A, and B WITHIN A, the nested main effect of
B. The following text box contains the
SPSS program to run the analysis.
The
Analysis
RUN NAME PSYCHOLOGY 460 ‑ ASSIGNMENT
4 ‑ DAVID W. STOCKBURGER
|
DATA LIST FILE='ANOVA2 DATA A'/A B X X2 X3 X4 1‑12
LIST.
MANOVA X BY A(1,2) B(1,3)
/PRINT=CELLINFO(MEANS)
/DESIGN A B WITHIN A.
|
The analysis of
the B(A) design is similar to the A x B design. The Tables of Means will be identical, but the graphs drawn from
the means will be done differently than the A x B design. The ANOVA table will contain only two
effects and interpretation will be discussed.
Cell Means and
Standard Deviations
|
FACTOR CODE Mean Std.Dev. N
A 1
B 1 2.000 1.000 3
B 2 6.000 1.000 3
B 3 10.000 1.000 3
A 2
B 1 2.000 1.000 3
B 2 2.000 1.000 3
B 3 2.000 1.000 3
|
The
Table of Means
The Table of
Means produced in this analysis is identical to that produced in the A x B
design.
Graphs
Because the
qualitative or quantitative meaning of levels of a at each level of b is
different, the graph of the means must be modified. The nested main effects of b are presented side-by-side rather
than on top of one another. This is
necessary because b1 is different depending on whether it appears
under a1 or a2.
The
ANOVA Table
The ANOVA table
produced by design B(A) is presented below.
Source of Variation
SS DF MS
F Sig of F
|
WITHIN CELLS 12.00 12 1.00
A 72.00 1
72.00 72.00 .000
B WITHIN A 96.00 4 4.00 24.00
.000
|
Interpretation
of Output
The
interpretation of the ANOVA table is straightforward. The WITHIN CELLS and A main effect SS's, DF's, and MS's are
identical to the analysis done in design A x B and are interpreted similarly.
The B WITHIN A term is called a nested main
effect and is the sum of the two simple B main effects. If this term is significant, then the graph
of the simple main effects should be drawn (Figure 8.1). What significance means is that the points
WITHIN each line are not the same height (value). That is, the means for b1, b2, and b3
within levels a1 and a2 are different. In the example graph the simple main effect
of B under a1 would be significant while the simple main effect of B
under a2 would not. Because
the B WITHIN A effect is the sum of both simple effects, the combined effect
was found to be significant in this case.
Similarities
to the A X B Analysis
As noted
earlier, the data files for the A x B and B(A) designs are identical. Likewise, the MANOVA commands are similar
except that the DESIGN subcommand on the B(A) design must be specified because
it is not a completely crossed design.
The table of means of the B(A) design will be identical to the A x B
design, the difference being that they will be plotted differently. The ANOVA source table will slightly differ
for the two types of designs. Both are
presented below to allow them to be contrasted.
Source Table
for the B(A) Design
|
Source of Variation SS DF MS
F Sig of F
WITHIN CELLS
12.00 12 1.00
A 72.00
1 72.00 72.00
.000
B WITHIN A 96.00 4 4.00
24.00 .000
Source Table for the A x B Design
Source of Variation SS DF MS
F Sig of F
WITHIN
CELLS 12.00 12
1.00
A 72.00 1
72.00 72.00 .000
B 48.00 2
24.00 24.00 .000
A BY
B 48.00 2
24.00 24.00 .000
|
Note that the
WITHIN CELLS and A effects are identical in both analyses. Note also that both the SS and DF for B and
A BY B in the A x B design together add up to the SS and DF for B WITHIN A for
the B(A) design. What is happening here
is that the B main effect and the A x B interaction in the A x B design is
collapsed into the nested main effect of B in the B(A) design.
Chapter
12
Contrasts,
Special and Otherwise
Understanding how contrasts may be employed
in the analysis of experiments gives the researcher considerably flexibility in
the specification of
"effects".
In addition, the study of contrasts leads to
even greater flexibility if multiple regression models are employed. Using contrasts, the researcher can test
specific “theory-driven” comparisons between groups.
Definition
A contrast is a set of numbers. When a set of means are being contrasted, a
contrast for the set of means will contain the same number of numbers as the
number of means. For example, if six
means are being contrasted, a contrast for the set of means would contain six
numbers. The order of the numbers in
the contrast corresponds to the order of the means. Examination of the signs of the numbers in the contrast describes
which means are being contrasted.
For example, the following contrast
1 1 1 -1 -1 -1
would compare the first three means with the
last three means.
A contrast of the form
2 2 -1 -1 -1 -1
would compare the first two means with the
last four means. The manner in which
the means are compared is determined by similar number. In the above example, the first two groups
share the same number (2), while the last four groups share the number minus
one (-1). All groups sharing the same
positive number are compared with all groups sharing the same negative
number. If a contrast contains zeros,
then those groups with zeros are not included in the contrast. For example:
0 0 3 -1 -1 -1
Would compare the third mean with the last
three means. The first and second means
would not be included in the calculation or interpretation of the contrast.
Sets of Contrasts
A set of contrasts is simply a number
of contrasts considered simultaneously.
For example, the three contrasts presented above could be combined into
a set of contrasts as follows:
contrast 1 2 2 -1 -1 -1 -1
contrast 2 0 0
3 -1 -1 -1
contrast 3 1 1
1 -1 -1 -1
Orthogonal Contrasts
An orthogonal contrast is two contrasts such
that when corresponding numbers in the contrasts are multiplied together and
then the products are summed, the sum of the products is zero. For example, in the preceding example,
contrasts 1 and 2 are orthogonal as can be seen in the following:
contrast 1 2 2 -1 -1 -1 -1
contrast 2 x
0 x 0 x 3 x -1 x
-1 x -1
products =
0 = 0 = -3 = 1 = 1 = 1
The sum of the products, 0 + 0 + -3 + 1 + 1 +
1 , equals zero, thus contrast 1 and contrast 2 are orthogonal contrasts.
Non-orthogonal Contrasts
Non-orthogonal contrasts are two contrasts such that when
corresponding numbers in the contrasts are multiplied together and then the
products are summed, the sum of the products is not zero. For example, in the preceding example,
contrasts 2 and 3 are non-orthogonal as can be seen in the following:
contrast 3 1 1 1 -1 -1 -1
contrast 2 x
0 x 0 x 3 x -1 x
-1 x -1
products =
0 = 0 = 3 =
1 = 1 = 1
The sum of the products, 0 + 0 + 3 + 1 + 1 +
1 , equals six, thus contrast 2 and contrast 3 are non-orthogonal
contrasts. In a similar manner
contrasts 1 and 3 are non-orthogonal.
Sets of Orthogonal Contrasts
The guiding principle is that the number
of possible orthogonal contrasts will always equal the number of means being
contrasted. If six means are being
contrasted, there will be no more than six contrasts which will be orthogonal
to one another. If six means are being
contrasted and five orthogonal contrasts have already been found, then there
exists a contrast which is orthogonal to the first five contrasts. That contrast is not always easy to find.
Finding Sets of Orthogonal Contrasts
In ANOVA the first contrast, which will be
referred to as contrast 0, is a set of 1's.
This contrast will be seen to be equivalent to the m term in the score
model. For a contrast comparing six
means, the first contrast would be:
contrast 0 1 1 1 1 1 1
In order for any following contrasts to be
orthogonal with contrast 0, it can be seen that the sum of the numbers in the
contrast must equal zero. Both contrasts 1 and 2 described earlier fit this
criterion, and since they are orthogonal to one another, they will be included
in the current set of orthogonal contrasts.
contrast 1 2 2 -1 -1 -1 -1
contrast 2 0 0 3 -1 -1 -1
In finding a fourth contrast which is
orthogonal to the first three, look for patterns of means in the preceding
contrasts which have the same number.
In both contrasts 1 and 2, the first and second means have the same
number, 2 in contrast 1 and 0 in contrast 2.
Working with this subset, finding numbers which sum to zero, and setting
all other numbers to zero , the following contrast is found:
contrast 3 1 -1 0 0 0 0
The student should verify that this contrast
is indeed orthogonal to all three preceding contrasts.
Using the same logic, the fifth contrast will
involve the fourth, fifth, and sixth means.
There are any number of possibilities, but the following will be used:
contrast 4 0 0 0 -2 1 1
The final contrast will compare the fifth
mean and the sixth mean. Any two
numbers which sum to zero (i. e. 2.34
and -2.34) could be used without changing the analysis or interpretation, but 1
and -1 make the contrasts parsimonious.
contrast 5 0 0 0 0 1 -1
Putting the six contrasts together as a set
results in the following:
contrast 0 1
1 1 1 1 1
contrast 1 2
2 -1 -1 -1 -1
contrast 2 0
0 3 -1 -1 -1
contrast 3 1 -1 0
0 0 0
contrast 4 0
0 0 -2 1
1
contrast 5 0
0 0 0 1 -1
1 1
1 2
1 3
2 5
2 6
2 7
3 9
3 10
3 11
4 1
4 2
4 3
5 1
5 2
5 3
6 1
6 2
6 3
|
Again, the student should verify that all the
contrasts in the set are orthogonal to one another. There is no other contrast available which is orthogonal to all
six of the preceding contrasts. The
non-believing student is welcome to search for such a contrast. I wish him or her well and hope that they
have a lot of spare time on their hands.
The Data
SPSS requires that the data for contrasts be
set up as a single factor design. The
example data is set up for a single factor between groups design with six
levels to the factor. The data appears
in the following text box.
RUN NAME PSYCHOLOGY 460 ‑ ASSIGNMENT 3
|
DATA LIST FILE='ANOVA2
DTA A'/G A B X X2 X3 X4 1‑14.
LIST.
MANOVA X BY G(1,6)
/PRINT=CELLINFO(MEANS)
/PRINT=PARAM(ESTIM)
/CONTRAST(G)=SPECIAL
( 1
1 1 1 1 1
2
2 ‑1 ‑1 ‑1 ‑1
0
0 3 ‑1 ‑1 ‑1
‑1 1 0 0
0 0
0
0 0 2 ‑1 ‑1
0
0 0 0 1 ‑1 )
/DESIGN G(1) G(2) G(3) G(4) G(5).
|
SPSS commands
The SPSS language requires that the contrasts
be specified as a subcommand on the MANOVA command. The following text box presents the appropriate commands to run
contrasts of means.
The Analysis
The Table of Means
Cell Means and Standard Deviations
|
Variable .. X
FACTOR CODE Mean
Std. Dev. N
G 1 2.000
1.000 3
G 2 6.000
1.000 3
G 3 10.000
1.000 3
G 4 2.000
1.000 3
G 5 2.000
1.000 3
G 6 2.000
1.000 3
4.000
3.254 18
|
The table of means is presented below:
The ANOVA table
The ANOVA table lists each contrast as an
effect.
* * A N A L Y S I S O F V A R I A N C E ‑‑
DESIGN 1 * *
|
Tests of Significance for X using UNIQUE
sums of squares
Source SS DF MS
F Sig of F
WITHIN CELLS 12.00 12 1.00
G(1) .00
1 .00 .00
1.000
G(2) 144.00
1 144.00 144.00 .000
G(3) 24.00
1 24.00 24.00
.000
G(4) .00
1 .00 .00
1.000
G(5) .00
1 .00 .00
1.000
|
Interpretation of Output
Note that only contrasts 2 and 3 are
significant. Contrast 2, comparing the
mean of Group 3 with Groups 4, 5, and 6 has an F-ratio of 144.00,
significant. Contrast 3, comparing the
means of Groups 1 and 2, is also significant.
The other three contrasts are all non-significant, comparing groups with
the same mean.
Constants
Compare the above ANOVA table with one
produced by treating the design as a single factor ANOVA (below).
Source of Variation SS
DF MS F
Sig of F
|
WITHIN CELLS
12.00 12 1.00
G
168.00 5 33.60
33.60 .000
|
One can observe that the SS for the single
factor - G (168.00) is the sum of the SS for the five contrasts listed above
(.00 + 144.00 + 24.00 + .00 + .00 = 168.00).
In general it can be stated The sum of the SS for any set of
orthogonal contrasts will equal the SS for that effect. The degrees of freedom (DF) for any effect
will be the number of contrasts summed for that
effect.
To demonstrate this constancy, using the
following set of orthogonal contrasts:
contrast 0 1
1 1 1 1 1
contrast 1 5 -1 -1 -1 -1 -1
contrast 2 0
4 -1 -1 -1 -1
contrast 3 0
0 3 -1 -1 -1
contrast 4 0
0 0 2 -1 -1
contrast 5 0
0 0 0 1 -1
The following ANOVA table is produced:
Source SS DF MS F
Sig of F
|
WITHIN CELLS 12.00 12 1.00
G(1) 14.40
1 14.40 14.40 .003
G(2) 9.60
1 9.60 9.60 .009
G(3) 144.00
1 144.00 144.00 .000
G(4) .00
1 .00 .00
1.000
G(5) .00
1 .00 .00
1.000
|
Note that in this case the first three
contrasts are significant and that the sum of the SS for the five contrasts is
equal to 168.00.
Contrasts, Designs, and Effects
It will now be demonstrated that the
different models for the analysis of
experiments are simply special cases of
contrasts.
For example, the following set of contrasts
contrast 0
1 1 1 1
1 1
contrast 1 -1 -1
-1 1 1 1
contrast 2 2 -1
-1 2 -1 -1
contrast 3 0 -1 1 0 -1 1
contrast 4 0 -1 1 0
1 -1
contrast 5 2 -1
-1 -2 1 1
produces the following ANOVA table
Note that in this case all contrasts
are significant. Comparing this with
the ANOVA table produced by an A x B design with A possessing 2 levels and B 3
levels yields the following:
Source
of Variation SS DF
MS F Sig of F
|
WITHIN CELLS 12.00 12 1.00
A 72.00
1 72.00 72.00
.000
B 48.00
2 24.00 24.00
.000
A BY B 48.00
2 24.00 24.00
.000
|
Source SS DF MS F
Sig of F
|
WITHIN CELLS
12.00 12 1.00
G(1)
72.00 1 72.00
72.00 .000
G(2)
36.00 1 36.00
36.00 .000
G(3)
12.00 1 12.00
12.00 .005
G(4)
12.00 1 12.00
12.00 .005
G(5) 36.00 1 36.00 36.00
.000
|
Note that the A main effect corresponds to
the first contrast and has one degree of freedom. The B main effect corresponds to the sum of contrasts 2 and 3 and
has two degrees of freedom. Finally,
the A x B interaction corresponds to the sum of contrasts 4 and 5 with two
degrees of freedom.
The preceding contrasts were a special form
of the general form of main effects and interactions. For a 2 x 2 design
contrast 0 1 1
1 1
A main effect 1 1 -1 -1
B main effect 1 -1
1 -1
A x B interaction 1 -1 -1
1
In a like manner, contrasts corresponding to
the B(A) design may be specified. From
the general form for a 2 x 2 design
contrast 0 1 1
1 1
A main effect 1 1 -1 -1
simple B1 main effect 1 -1 0
0
simple B2 main effect 0
0 1 -1
the set of contrasts for a 2 x 3 design may
be found as follows
contrast 0
1 1 1
1 1 1
contrast 1 -1 -1 -1
1 1 1
Contrast 2 2 -1
-1 0 0 0
contrast 3 0 -1 1 0
0 0
contrast 4 0
0 0 2 -1 -1
contrast 5 0
0 0 0 -1 1
which produce the following ANOVA table.
Source SS DF
MS F Sig of F
|
WITHIN CELLS 12.00
12 1.00
G(1) 72.00 1 72.00 72.00
.000
G(2) 72.00 1 72.00 72.00
.000
G(3) 24.00 1 24.00 24.00
.000
G(4) .00 1 .00 .00
1.000
G(5) .00 1 .00 .00
1.000
|
Comparing this to the ANOVA table produced by
design B(A) yields
Source of Variation
SS DF MS
F Sig of F
|
WITHIN CELLS 12.00 12 1.00
A 72.00
1 72.00 72.00
.000
B WITHIN A 96.00 4 4.00
24.00 .000
|
Contrast 1 corresponds to the A main effect,
contrasts 2 and 3 correspond to the simple main effect of B1 and
contrasts 4 and 5 to the simple main effect of B2.
Non-Orthogonal Contrasts
When the specified set of contrasts is
non-orthogonal, the sum of the sum of squares for the contrasts will not equal
the total sum of squared for the effects.
In some cases the sum of the contrast sum of squares will be greater
than the total sum of squares, in others, the sum of the contrast sum of
squares will be less. An example of
each will now be given.
Smaller than Total Sum of Squares
The following set of contrasts produces a sum
of contrast sum of square which is less than the total sum of squares (168.00)
for a set of non-orthogonal contrasts.
Note that while the WITHIN SS remains constant, the sum of the contrast
SS is only 24.00. This was accomplished
by contrasting groups with similar means.
/CONTRAST(G)=SPECIAL
( 1 1
1 1 1 1
2 2 ‑1 ‑1 ‑1 ‑1
0 0
0 1 0 ‑1
0 0
0 1 ‑1 0
1 ‑1 0
0 0 0
1 0
0 0 0 ‑1 )
/DESIGN G(1)
G(2) G(3) G(4) G(5).
Source of Variation SS DF MS F Sig of F
WITHIN CELLS
12.00 12 1.00
G(1) .00 1
.00 .00 1.000
G(2) .00 1
.00 .00 1.000
G(3) .00 1
.00 .00 1.000
G(4) 24.00 1
24.00 24.00 .000
G(5) .00 1
.00 .00 1.000
|
Larger than Total Sum of Squares
On the other hand, if non-orthogonal
contrasts are specified to contrast groups which have different means, then the
sum of the SS's will be larger than the
total SS. In the following case, the
sum of the SS's is 424.00 compared with a total SS for orthogonal effects of
168.00. Thus, if non-orthogonal effects
are specified, then the ANOVA procedure will distort the nature of the effects
depending upon how the effects were specified.
/CONTRAST(G)=SPECIAL
( 1 1
1 1 1 1
2 2 ‑1 ‑1 ‑1 ‑1
1 ‑1 0
0 0 0
0 0
3 ‑1 ‑1 ‑1
0 0
2 ‑1 ‑1 0
0 0
2 ‑1 0 ‑1 )
/DESIGN G(1)
G(2) G(3) G(4) G(5).
Source of Variation
SS DF MS F Sig of F
WITHIN
CELLS 12.00 12
1.00
G(1) .00 1
.00 .00 1.000
G(2) 24.00 1
24.00 24.00 .000
G(3) 144.00 1
144.00 144.00 .000
G(4) 128.00 1
128.00 128.00 .000
G(5) 128.00 1
128.00 128.00 .000
|
Standard Types of Orthogonal
Contrasts
A number of standard orthogonal contrasts are
built into MANOVA. Instead of
specifying "/CONTRAST(G)=SPECIAL(
)", a standard contrast may be done by using "/CONTRAST(G)=XXX",
where the "XXX" is one of the following:
DIFFERENCE
This is a difference or reverse Helmert
contrast which compares levels of a factor with the mean of the previous levels
of the factor. For the six level
design, the contrast would appear as follows:
contrast 0 1 1 1 1 1 1
contrast 1 1 -1
0 0 0 0
contrast 2 1 1 -2 0 0 0
contrast 3 1 1 1 -3
0 0
contrast 4 1 1 1 1 -4 0
contrast 5 1 1 1 1 1 -5
SIMPLE
Compares each level of a factor to the last
level. For the six level design, the
contrast would appear as follows:
contrast 0 1 1 1 1 1 1
contrast 1 1 0
0 0 0 -1
contrast 2 0 1 0 0 0 -1
contrast 3 0 0 1 0
0 -1
contrast 4 0 0 0 1
0 -1
contrast 5 0 0 0 0 1 -1
POLYNOMIAL
This contrast requires that the levels
of the independent variable be measured on at least a good approximation to an interval
scale. If this is the case, then
the polynomial contrasts tests for trends.
The first contrast tests for a linear
trend. That is, as the value of the
independent variable increases, the dependent variable increases or decreases
at a steady rate. The increase or
decrease may be described with a straight line.
The second contrast tests for quadratic
trends. The quadratic trend my be
described with a parabola, it changes direction twice. The third trend, the cubic, changes
directions three times. For each trend
the number of changes in direction is the size of the trend. The number of trends for a given set of
contrasts will be one less than the number of groups being contrasted. An illustration of each of the five trends
for six groups is given below.
The set of orthogonal contrasts which tests
for polynomial trends for six means is presented below:
contrast 0 1 1 1 1 1 1
contrast 1
-5 -3 -1 1
3 5
contrast 2 5 -1 -4 -4 -1 5
contrast 3 -5 7 4 -4 -7 5
contrast 4 1 -3 2 2 -3 1
contrast 5 1 -5 10 -10 5 -1
Example of Polynomial
Contrasts
Given that the data produced the following
table of means and resulting graph.
The MANOVA of the data using a polynomial
contrast would produce the following source table.
FACTOR CODE Mean Std.
Dev. N
G 1 2.000 1.000 3
G 2 6.000 1.000 3
G 3 8.000 1.000 3
G 4 8.000 1.000 3
G 5 7.000 1.000 3
G 6 10.000 1.000 3
For entire sample 6.833 2.684 18
|
It
can be seen that the first three contrasts; the linear, the quadratic, and the
cubic, were significant, while the last two were not. The graph of the means changes direction three times, with a
slight overall rise (linear trend). A
model of the following form would be used to describe the relationship between
groups (X) and the dependent measure (Y).
Y' = m
+ c1 * X + c2
* X2 + c3 * X3
Only significant contrasts would be included
in the model.
/CONTRAST(G)=POLYNOMIAL
/PRINT = CELLINFO(MEANS)
/DESIGN G(1) G(2) G(3) G(4) G(5).
Source
of Variation SS DF
MS F Sig of F
WITHIN
CELLS 12.00 12
1.00
G(1) 79.24 1
79.24 79.24 .000
G(2) 10.32 1
10.32 10.32 .007
G(3) 18.15 1
18.15 18.15 .001
G(4) 2.68 1
2.68 2.68 .128
G(5) .11 1
.11 .11 .749
|
Other types of contrast may be
available. Check the SPSS manual.
Conclusion
Contrasts of means provide a very general
mechanism for testing hypotheses about data.
The two-factor and nested main effect designs were seen to be special
cases of orthogonal contrasts.
Contrasts are probably not used as much in ANOVA as they could be.
Chapter
13
ANOVA and
Multiple Regression
Analysis of variance may be conceptualized
and performed as a special case of multiple regression. Doing so provides the user with a flexible
analytical tool which can be applied to a wide variety of data types. In addition,
conceptualization of ANOVA in this manner
permits a greater understanding of the issues involved when unequal number of
subjects appear in between subjects designs with two or more factors.
A review of a one factor ANOVA will be
discussed first, with an additional discussion of orthogonal contrasts. The concept of orthogonal transformations
translates easily into “dummy coding” and the use of multiple regression for
ANOVA. After the discussion of one
factor ANOVA, the conceptualization will be extended to two factor ANOVA in a
similar manner.
ONE FACTOR ANOVA
ANOVA and Multiple Regression
Up until this point, the stated purpose of
multiple regression was the prediction of a continuous dependent variable (Y)
from one or more continuous variables (X’s).
Using the multiple regression approach allowed the testing of hypotheses
about whether a variable or set of variables (X’s) significantly predicted the
variance of the dependent variable (Y).
The purpose of one factor ANOVA was to make a decision as to whether the
differences between means on a continuous dependent variable (Y) for levels of
an independent variable, or factor, were large enough to attribute the
differences between the means to something other than chance.
In order to extend the multiple regression
approach to incorporate ANOVA, the requirement that the independent variable(s)
be continuous must be extended to the case of discrete variables. This is accomplished by recoding the single
independent variable of ANOVA into multiple independent variables for inclusion
into a multiple regression analysis.
This process is called “dummy coding” and if one understands orthogonal
contrasts, the transition is straightforward.
1 26
1 31
2 34
2 32
2 38
3 45
3 41
3 49
4 18
4 24
4 23
5 31
5 21
5 27
6 28
6 34
6 22
|
Example
x BY group(1 6)
/DESIGN .
GROUP MEAN
1 26.67
2 34.67
3 45.00
4 21.67
5 26.33
6 28.00
|
Start with a one factor between subjects
design, A, where there are six levels of A (A=6) and three subjects in
each group (S=3). An example data file
is presented below, with the SPSS commands, means, and ANOVA table
following. It can be seen that the
means are significantly different from each other with F(5,15)=10.92,
p<.05. The sum of squares for the treatment
effect and error are equal to 1031.61 and 226.67 respectively.
* * * * * * A n a l y s I
s o f V a r I a n c e ‑‑ design 1 * * * * * *
|
Tests of
Significance for X using UNIQUE sums of squares
Source of
Variation SS DF MS F Sig of F
WITHIN+RESIDUAL
226.67 12 18.89
GROUP 1031.61 5
206.32 10.92 .000
(Model)
1031.61 5 206.32 10.92 .000
(Total)
1258.28 17 74.02
R‑Squared
= .820
Adjusted R‑Squared
= .745
|
Example Using Contrasts
x BY group(1 6)
/CONTRAST
(GROUP) SPECIAL
( 1
1 1 1 1 1
1 1 1 1 ‑2 ‑2
1 1 ‑1 ‑1 0
0
1 ‑1 0
0 0 0
0 0 1 ‑1 0 0
0 0 0 0 1 ‑1 )
/DESIGN
GROUP(1) GROUP(2)
GROUP(3)
GROUP(4) GROUP(5) .
|
As discussed in the previous chapter, the
single factor ANOVA may be performed as a series of orthogonal contrasts. Any orthogonal set of contrasts will account
for exactly the same amount of variance and produce the identical decision as
the single factor ANOVA. As a
demonstration of this equality, the example data presented in the previous
section was re-analyzed using the set of orthogonal contrasts specified below.
The end results of this analysis are
identical to the results of the previous analysis. Note that the sum of the sum of squares for the various contrasts
equals the sum of squares for the Groups factor.
* * A n a l y s I s o
f V a r I a n c e * *
|
Tests of
Significance for X using UNIQUE sums of squares
Source of
Variation SS DF
MS F Sig of F
WITHIN+RESIDUAL
226.67 12 18.89
GROUP(1)
93.44 1 93.44 4.95 .046
GROUP(2)
21.33 1
21.33 1.13 .309
GROUP(3)
96.00 1 96.00 5.08 .044
GROUP(4)
816.67 1 816.67
43.24 .000
GROUP(5)
4.17 1 4.17 .22 .647
(Model)
1031.61 5 206.32
10.92 .000
(Total)
1258.28 17 74.02
R‑Squared
= .820
Adjusted R‑Squared
= .745
|
Dummy Coding
To use a multiple regression approach to
perform an ANOVA, the single independent variable with K levels must be recoded
as K-1 independent variables. The form
of the recoding is identical to the orthogonal contrast. For the examination of the total effect, any
set of orthogonal contrasts will yield the same overall result. Partitioning a single categorical variable
into a set of variables suitable for multiple regression is called “dummy
coding.”
1 1
1 1 0 0 23
1 1
1 1 0 0 26
1 1
1 1 0 0 31
2 1
1 ‑1 0 0 34
2 1
1 ‑1 0 0 32
2 1
1 ‑1 0 0 38
3 1 ‑1 0 1 0 45
3 1 ‑1 0 1 0 41
3 1 ‑1 0 1 0 49
4 1 ‑1 0 ‑1 0 18
4 1 ‑1 0 ‑1 0 24
4 1 ‑1 0 ‑1 0 23
5 ‑2
0 0 0 1 31
5 ‑2
0 0 0 1 21
5 ‑2
0 0 0 1 27
6 ‑2
0 0 0 ‑1 28
6 ‑2
0 0 0 ‑1 34
6 ‑2
0 0 0 ‑1 22
|
The dummy codes may be created by turning the
set of orthogonal contrasts on their side (called the transpose of a matrix)
and assigning each subject in the group the same codes. For example, dummy coding the example data
set with the example set of contrasts produces the following data set. Note that there are five new variables
created, one for each of the orthogonal contrasts. The first contrast, which is a row of ones, is not included,
because the constant term (b0) in the multiple regression procedure
does the same function.
The dummy coded variables will be orthogonal,
or uncorrelated with one another, if there are an equal number of subjects in
each group. In the preceding example
this was the case and the correlation matrix which results is as follows. Note that all off-diagonal entries are zero.
‑ ‑ Correlation Coefficients
‑ ‑
|
D1 D2 D3 D4 D5
D1
1.0000 .0000 .0000 .0000 .0000
D2
.0000 1.0000 .0000 .0000 .0000
D3
.0000 .0000 1.0000 .0000 .0000
D4
.0000 .0000 .0000 1.0000 .0000
D5
.0000 .0000 .0000 .0000 1.0000
|
ANOVA, Revisited
/DEPENDENT x
/METHOD=ENTER d1 d2 d3 d4 d5.
|
The ANOVA may now be done by regressing the
dummy coded variables as independent variables, in this case D1, D2, . . . , on the dependent variable, in this
case X. The REGRESSION command in SPSS
does this function and an example of the use of this command is given below.
The output from this command is presented
below:
* * * * M U L
T I P L E R E G R E S S I O N * * * *
|
Block Number
1. Method: Enter
D1 D2 D3 D4 D5
Multiple R .90546
R Square .81986
Adjusted R Square .74480
Standard Error 4.34613
Analysis of Variance
DF
Sum of Squares Mean Square
Regression 5
1031.61111 206.32222
Residual 12
226.66667 18.88889
F =
10.92294 Signif F = .0004
‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑
Variables in the Equation ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑
Variable B SE
B Beta T Sig T
D5
‑.833333 1.774302 ‑.057545 ‑.470 .6470
D4
11.666667 1.774302 .805627 6.575 .0000
D3
‑4.000000
1.774302 ‑.276215 ‑2.254 .0436
D2
‑1.333333
1.254621 ‑.130209 ‑1.063 .3088
D1
1.611111 .724356 .272514 2.224 .0461
(Constant)
30.388889 1.024394 29.665 .0000
|
Note that the values of DF, Sum of Squares,
Mean Square, F, and Sig of F for the “Regression” and “Residual” effects in the
source table are identical to those of the “GROUP” and “WITHIN+RESIDUAL”
effects in the output of the MANOVA command.
A one-factor ANOVA has in effect been performed.
TWO FACTOR ANOVA
In the same manner that orthogonal contrasts
may be used to perform a two factor ANOVA, dummy codes may be selected such
that a multiple regression analysis may perform the same function. The general procedure is identical to dummy
coding procedure for a one factor ANOVA, except that different dummy codes are
used. The recoded variables are
partitioned into subsets and each subset is included as a hypothesis test in
the regression model.
1 1 23
1 1 26
1 1 31
1 2 34
1 2 32
1 2 38
1 3 45
1 3 41
1 3 49
2 1 18
2 1 24
2 1 23
2 2 31
2 2 21
2 2 27
2 3 28
2 3 34
2 3 22
|
Example
The raw data, SPSS MANOVA command, and SPSS
output is presented below. The dependent
variable is identical to that in the one factor case and the six levels of the
single factor have been partitioned into two level of a factor called “A” and
three levels of a factor called “B.”
x BY a(1
2) b(1 3)
/DESIGN .
|
A n a l y s I s
o f V a r I a n c e
|
Tests of
Significance for X using UNIQUE sums of squares
Source of
Variation SS DF
MS F Sig of F
WITHIN+RESIDUAL
226.67 12 18.89
A 460.06 1
460.06 24.36 .000
B 456.44 2
228.22 12.08 .001
A BY
B 115.11 2
57.56 3.05 .085
(Model)
1031.61 5 206.32
10.92 .000
(Total)
1258.28 17 74.02
R‑Squared
= .820
Adjusted R‑Squared
= .745
|
Example Using Contrasts
The set of orthogonal contrasts may be
selected such that the output of the MANOVA command produces results similar to
those in the above example. The raw
data for this analysis is that presented in the one factor ANOVA section in
this chapter. The MANOVA command and
results of this command are presented below:
x BY group(1 6)
/CONTRAST(GROUP) SPECIAL
( 1
1 1 1 1 1
1 1 1 ‑1 ‑1 ‑1
1 1 ‑2 1
1 ‑2
1 ‑1 0
1 ‑1 0
1 1 ‑2 ‑1 ‑1 2
1 ‑1 0 ‑1 1 0 )
/DESIGN
GROUP(1) GROUP(2)+GROUP(3) GROUP(4)+GROUP(5).
|
A n a l y s I s
o f V a r I a n c e
|
Tests of
Significance for X using UNIQUE sums of squares
Source of
Variation SS DF
MS F Sig of F
WITHIN+RESIDUAL
226.67 12 18.89
GROUP(1)
460.06 1 460.06
24.36 .000
GROUP(2) +
GROUP(3) 456.44 2
228.22 12.08 .001
GROUP(4) +
GROUP(5) 115.11 2
57.56 3.05 .085
(Model)
1031.61 5 206.32
10.92 .000
(Total) 1258.28
17 74.02
R‑Squared
= .820
Adjusted R‑Squared
= .745
|
Note that the results of this analysis are
identical to that of the two factor ANOVA.
Regression Analysis using Dummy
Coding
A
B DA DB1 DB2 DAB1 DAB2 X
|
1 1
1 1 1 1 1
23
1 1
1 1 1 1 1
26
1 1
1 1 1 1 1
31
1 2
1 1 ‑1 1 ‑1
34
1 2
1 1 ‑1 1 ‑1
32
1 2
1 1 ‑1 1 ‑1
38
1 3
1 ‑2 0
‑2 0 45
1 3
1 ‑2 0
‑2 0 41
1 3
1 ‑2 0
‑2 0 49
2 1
1 1 1 ‑1 ‑1
18
2 1 ‑1 1 1 ‑1
‑1 24
2 1 ‑1 1 1 ‑1
‑1 23
2 2 ‑1 1 ‑1 ‑1 1 31
2 2 ‑1 1 ‑1 ‑1 1 21
2 2 ‑1 1 ‑1 ‑1 1 27
2 3 ‑1 ‑2 0 2
0 28
2 3 ‑1 ‑2 0 2
0 34
2 3 ‑1 ‑2 0 2
0 22
|
In a manner analogous to the dummy coding of
the one factor ANOVA, the contrast matrix is transposed and expanded such that
each subject in a group is given the same X variables. The variables are labeled “DA, DB1, DB2,
DAB1, and DAB2" where the D indicates that it is a dummy code and the
letter or letters which follow indicate the name of the effect. Because the B and AB factors have two
degrees of freedom each, the effect of these factors will be a combination of
two dummy coded variables, hence the “1" and “2" following the
recoded variable. The data matrix for
the multiple regression analysis is presented below:
The correlation matrix including all the dummy
coded variables is as follows:
‑ ‑ Correlation Coefficients
‑ ‑
|
DA DB1 DB2 DAB1 DAB2
DA
1.0000 .0000 .0000 .0000 .0000
DB1
.0000 1.0000 .0000 .0000 .0000
DB2
.0000 .0000 1.0000 .0000 .0000
DAB1
.0000 .0000 .0000 1.0000 .0000
DAB2
.0000 .0000 .0000 .0000 1.0000
|
Note that the independent variables are
uncorrelated with each other, the definition of orthogonality or
independence. The implication of
independence is that the order of entering the variables into the regression
model will not make a difference.
/DEPENDENT x
/METHOD=TEST (DA) (DB1,DB2)
(DAB1,DAB2).
|
The REGRESSION command which does the two
factor ANOVA with the dummy coded variables is presented below. The “/METHOD=TEST” subcommand will
sequentially test each of the combinations of variables in the parentheses
following the word “TEST”. In this
case, DA corresponds to the main effect of A, DB1 and DB2 to the main effect of
B, and DAB1 and DAB2 to the interaction effect. The result of this regression analysis is presented below.
* * * * M U L
T I P L E R E G R E S S I O N * * * *
|
Hypothesis Tests
Sum
of
DF Squares Rsq Chg F Sig F Source
1 460.05556 .36562 24.35588 .0003
DA
2 456.44444 .36275 12.08235 .0013
DB1 DB2
2 115.11111 .09148 3.04706 .0851
DAB1 DAB2
5 1031.61111 10.92294
.0004 Regression
12 226.66667 Residual
17 1258.27778 Total
‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑
Multiple R .90546
R Square .81986
Adjusted R Square .74480
Standard Error 4.34613
|
Note that the DF, Sum of Squares, F, and Sig
F are identical to those of the two MANOVA analyses.
Conclusion
It at first
appears that dummy coding is a great deal of trouble for very little gain. The results of the analysis are identical to
those of the simpler MANOVA procedures.
There are at least three reasons for viewing ANOVA as a special case of
multiple regression.
The first
reason is to achieve a better understanding of how the MANOVA procedure
works. When one uses the MANOVA
program, the little guy inside the computer dummy codes the independent
variable(s) and then applies a general linear model or regression analysis to
the results. In a like manner, a
greater understanding of other multivariate procedures, such a discriminant
analysis, is gained if one understands dummy coding.
The second
reason for the understanding of dummy coding and the use of multiple regression
to perform an ANOVA is that considerable flexibility is gained in the type of
analysis that may be done. If the
independent variables in an analysis include both interval and categorical
variables, both may be used in the same analysis. The categorical variables are dummy coded and entered either
before, with, or after the interval variables.
The dummy coded variables may be combined into different “effects” and
entered into the regression equation in different orders.
The final and
perhaps overriding reason for including a discussion of the relationship
between multiple regression and ANOVA in this book is that it is difficult to
explain the nature of the difficulty of unequal cell sizes in ANOVA without
recourse to multiple regression. That
is the subject of the next chapter.
Chapter
14
Unequal Cell
Frequencies
Up to this point all computational procedures
and theoretical development have been done for balanced designs. Balanced designs have equal or proportional
cell frequencies in each cell.
Employing unbalanced designs has profound effects on both the interpretation
and theoretical development of ANOVA.
Since an unbalanced design is often employed in real life situations, a
careful examination of unbalanced designs is warranted.
In the following, a review of a balanced
two-way between subjects ANOVA will first be presented. Following that, an unbalanced design of the
same form will be analyzed using various options available from SPSS. Since I entered the problem in a discussion
list (ED-STAT), the solutions resulting from the resulting discussion will be presented
last.
Equal Cell Frequency - Independence
of Effects
In a balanced design the computational
procedures to estimate the terms in the score model produce independent, or
uncorrelated, effects. This result of
independent effects is that the total
of the sum of squares for all effects adds up to the total SS. It makes no difference which effect is
computed first, second, or last, since the size of one effect has no
relationship to the size of any other effect
This can be seen very nicely in a source
table from a balanced A X B design.
Tests of
Significance for X using UNIQUE sums of squares
|
Source SS DF MS F Sig of F
WITHIN
CELLS 139.36 12
11.61
A .36
1 .36 .03 .863
B 123.08
2 61.54 5.30 .022
A BY B 108.73
2 54.36 4.68 .031
Model 232.17 5
46.43 4.00 .036
Total 271.53 17
|
Note that the SS for the Model is equal to SSA
+ SSB + SSAB.
Here no matter what order the A, B, or A BY B terms are entered into the
model, the result would be the same.
Unequal Cell Frequency - Dependent
Effects
When the AXB design is unbalanced, it makes a
great deal of difference which order the effects are entered into the
model. In order for the following to
make sense, the reader must understand that SPSS has two options for dealing
with unbalanced designs. These options
are selected by including either the subcommand "/METHOD = UNIQUE" or
"/METHOD = SEQUENTIAL" to the MANOVA command. The SEQUENTIAL procedure enters the effects
into the model in the order that they are listed in the "/DESIGN"
subcommand. For example, if the default
"/DESIGN = A B A BY B" was used, the A main effect would be entered
first, the B main effect second, and the interaction A BY B would be last.
The use of
"/DESIGN = A BY B B A" and "/METHOD = SEQUENTIAL"
would result in the interaction term being entered first, the B main effect
second, and the A main effect last. As
shall be seen, the order of effects is critical and greatly effects the results
of the analysis.
The "/METHOD = UNIQUE" subcommand
is the default in SPSS. That is, it is
assumed if no "/METHOD = " subcommand is used. This procedure presents the results as if
every term in the model had been entered last. The following is an article sent to the ED-STAT discussion group
on April 28, 1993.
In trying to understand how SPSS handles
unbalanced A X B designs, I generated a simulated data set. The following is the summary table using the
UNIQUE (default) option.
Tests of Significance for X using UNIQUE sums of squares
|
Source of
Variation SS DF MS F Sig of F
WITHIN+RESIDUAL
29961.11 76 394.23
FACTORA
927.96 1 927.96 2.35 .129
FACTORB
2500.39 2 1250.20 3.17 .048
FACTORA
BY FACTORB 3504.32 2 1752.16 4.44 .015
(Model)
9391.52 5 1878.30 4.76 .001
(Total)
39352.63 81 485.83
|
Note that the Sum of Squares for the three
effects sum to 6932.67, somewhat less than the Sum of Squares for the model.
Tests of Significance for X using SEQUENTIAL Sums of Squares
|
Source of Variation SS DF MS F
Sig of F
WITHIN+RESIDUAL 29961.11
76 394.23
FACTORA 797.17
1 797.17 2.02 .159
FACTORB 5090.03
2 2545.01 6.46 .003
FACTORA BY FACTORB 3504.32 2 1752.16 4.44 .015
(Model) 9391.52
5 1878.30 4.76 .001
(Total) 39352.63
81 485.83
|
The same data analysis using the SEQUENTIAL
method produced the following table.
In this case the SS for the factors add up to
the model SS and the interaction SS is the same as in the UNIQUE method. No problem.
The SS for FACTORA, entered into the equation first, is smaller than
that using the UNIQUE method. Listing
the effects in a different order on the DESIGN produces a different SS when the
SEQUENTIAL method is used.
Tests of Significance for X using SEQUENTIAL Sums of Squares
|
Source of Variation SS DF MS F
Sig of F
WITHIN+RESIDUAL 29961.11
76 394.23
FACTORB 5747.55
2 2873.77 7.29 .001
FACTORA 139.65
1 139.65 .35 .553
FACTORA BY FACTORB 3504.32 2 1752.16 4.44
.015
WITHIN+RESIDUAL 29961.11
76 394.23
FACTORA BY FACTORB 4780.86 2 2390.43 6.06 .004
FACTORA 2110.27
1 2110.27 5.35 .023
FACTORB 2500.39
2 1250.20 3.17 .048
WITHIN+RESIDUAL 29961.11
76 394.23
FACTORA BY FACTORB 4780.86 2 2390.43 6.06 .004
FACTORB 3682.70
2 1841.35 4.67 .012
FACTORA 927.96
1 927.96 2.35 .129
|
In all cases the final SS corresponds to that
using the UNIQUE method. How can the SS for an effect be smaller than the
UNIQUE SS for that effect, especially if that effect is entered into the model
first? To quote a student "I just
don't understand."
To review what occurred. If the "/METHOD = SEQUENTIAL"
subcommand was specified, the sum of the SS's for the effects was equal to the
SS for the model. The problem was that
the SS for each effect was different depending upon where it was listed in the
"/DESIGN" subcommand. The
size of an effect depended upon which effects were already in the model. An effect could be significant or not
depending upon where it was listed on the "/DESIGN" subcommand. In addition, it can be seen that the
"/METHOD = UNIQUE" produced results as if each term had been listed
last on the "/DESIGN" subcommand.
It all seems quite arbitrary.
Solutions for Dealing with Dependent
Effects
The article solicited many comments. Two of them are presented below:
From: Neil W. Henry
<nhenry@cabell.vcu.edu>
You are not going to like this answer: "That's life." Or another: "Don't use the word
'effect' so much." It carries
connotations that interfere with following what the numbers are doing. And 'unique effect' is even more dangerous. If you like correlations and regression, as
I do, more than ANOVA, consider the following example.
Y = X + W (exactly)
Var(X) = Var(Y) = Var(W) = 1.
Cor(X,W) = ‑.5 .
It follows that
Cor(Y,X) = Cor(Y,W) = .5 .
Consequently, W "when entered into the
model first", _explains [only] 25% of the variance in Y_ . But, since together
X and W "explain" 100% of the variance in Y, W's UNIQUE contribution
to the variance is 75% (i.e. its contribution when added to the model after X
is
already in).
Moral 1: Ain't negative numbers fun?
Moral 2: Don't let other people get your
kicks for you; or push their favorite jargon on you either.
*************************************************
`o^o' * Neil W. Henry
(nhenry@cabell.vcu.edu) *
‑<:>‑
* Virginia Commonwealth University *
_/ \_ * Richmond VA 23284‑2014 *
*(804)367‑1301 (math sciences, 2079
Oliver) *
* 7‑6650
(academic computing, B30 Cabell) *
*************************************************
From:
nichols@spss.com (David Nichols)
To:
dws148f@smsvma.bitnet
Subject: Re:
Unbalanced ANOVA
From a
computational perspective, you can think of UNIQUE sums of squares as the
difference between the total model sums of squares accounted for by a model
including all terms and one including all terms except for the one at issue.
Thus each term in the model is "adjusted" for all other terms, or
entered last in a regression. This will produce sums of squares that do not
generally add up to the total (they will in cases where there are the same
number of cases in each cell of the design). The SEQUENTIAL approach fits an
ordered set of terms one at a time, and this will produce a set of sums of
squares that add up to the total. Since the last term entered is after all
other terms in the model, it will always be the same as with UNIQUE sums of
squares.
In the general
case of data in which there are unequal numbers of cases in the cells of the
design (assuming at least one in each cell‑‑the case of empty cells
becomes substantially more complex), the UNIQUE sums of squares test the same
hypotheses that are tested in balanced designs. In the balanced case, both
types of sums of squares test the same hypotheses and thus reduce to the same
results. In unbalanced designs the statistical hypotheses tested by SEQUENTIAL
sums of squares test hypotheses that depend on the number of observations in
the cells, which is generally not considered a positive characteristic. This is
why UNIQUE sums of squares are the default.
As to the
question of how the sums of squares for a factor can be higher when it is
entered after another factor than when it is entered alone, probably the
easiest way to understand it is to think in terms of part correlations and
multiple regression. Recall that ANOVA and regression involve an identical
mathematical/statistical model. You can easily verify the fact that the sums of
squares accounted for by a term in a regression model at a particular stage is
given by the squared part correlation of that regressor with the dependent
variable, multiplied by the mean corrected total sums of squares. Thus if the
squared part correlation between X and Y with the linear effects of Z taken out
of X is higher than the simple squared correlation of X with Y, the sums of
squares of X fitted after Z will be greater than the sums of squares fitted
before Z (at least in the case where these are the only two predictors).
Another way to
look at this (related to the first) and one that may be more helpful, is to try
to picture it in geometric terms. The part correlation of X with Y, removing
the linear effects of Z from X is equivalent to regressing X on Z, taking the
residuals, and then regressing Y on those residuals. If you look at this in
terms of vectors, you can see that the residual vector can in general be
pointing in any direction and thus may be more nearly coincident with the Y
vector than was the original X vector.
It all depends
on the intercorrelations among the variables. There's an excellent description
of this from a geometrical perspective in the February 1993 _American
Statistician_, Vol. 47, No. 1, beginning on p. 26.
UNEQUAL
CELL SIZES FROM A MULTIPLE REGRESSION VIEWPOINT
A B X DA1 DB1 DB2 DAB1 DAB2
|
1 1 23
1 1 1 1 1
1 1 27
1 1 1 1 1
1 1 35
1 1 1 1 1
1 1 25
1 1 1 1 1
1 2 56
1 1 ‑1 1 ‑1
1 2 45
1 1 ‑1 1 ‑1
1 2 62
1 1 ‑1 1 ‑1
1 3 24
1 ‑2 0
‑2 0
1 3 21
1 ‑2 0
‑2 0
1 3 34
1 ‑2 0
‑2 0
1 3 28
1 ‑2 0
‑2 0
1 3 33
1 ‑2 0
‑2 0
1 3 21
1 ‑2 0
‑2 0
1 3 14
1 ‑2 0
‑2 0
1 3 26
1 ‑2 0
‑2 0
1 3 22
1 ‑2 0
‑2 0
1 3 20
1 ‑2 0
‑2 0
1 3 31
1 ‑2 0
‑2 0
1 3 38
1 ‑2 0
‑2 0
1 3 30
1 ‑2 0
‑2 0
1 3 25
1 ‑2 0
‑2 0
2 1 36
‑1 1 1
‑1 ‑1
2 1 41
‑1 1 1
‑1 ‑1
2 2 43
‑1 1 ‑1
‑1 1
2 2 48
‑1 1 ‑1
‑1 1
2 2 35
‑1 1 ‑1
‑1 1
2 2 41
‑1 1 ‑1
‑1 1
2 2 39
‑1 1 ‑1
‑1 1
2 2 44
‑1 1 ‑1
‑1 1
2 2 51
‑1 1 ‑1
‑1 1
2 3 18
‑1 ‑2 0
2 0
2 3 21
‑1 ‑2 0
2 0
|
An example data
set with unequal cell sizes will be analyzed first with MANOVA using both “/METHODS=UNIQUE”
and “/METHODS=SEQUENTIAL” subcommands.
The results of these analyses will correspond directly to those
discussed earlier in this chapter. The
same data set will be dummy coded and analyzed using a multiple regression
procedure. The results of the two
methods of MANOVA will be duplicated.
The example
data set is presented below. Note that
the cells have unequal n’s and the variables have been dummy coded using the
same recoding system as the two factor ANOVA in the last chapter.
The results of
the MANOVA using the “/METHOD=UNIQUE” subcommand is as follows:
A n a l y s I s o
f V a r I a n c e
|
Tests of
Significance for X using UNIQUE sums of squares
Source of
Variation SS DF
MS F Sig of F
WITHIN CELLS 975.02 26 37.50
A 27.63 1
27.63 .74 .399
B 2604.75 2
1302.38 34.73 .000
A BY
B 425.82 2
212.91 5.68 .009
(Model)
3255.94 5 651.19 17.36 .000
(Total)
4230.97 31 136.48
R‑Squared
= .770
Adjusted R‑Squared
= .725
|
A similar
analysis done using the “/METHOD=SEQUENTIAL” subcommand yields the following
table.
A n a l y s I s o f
V a r I a n c e
|
Tests of
Significance for X using SEQUENTIAL Sums of Squares
Source of
Variation SS DF MS F Sig of F
WITHIN
CELLS 975.02 26
37.50
A 398.82 1
398.82 10.63 .003
B 2431.30 2
1215.65 32.42 .000
A BY
B 425.82 2
212.91 5.68 .009
(Model)
3255.94 5 651.19 17.36 .000
(Total)
4230.97 31 136.48
R‑Squared
= .770
Adjusted R‑Squared
= .725
|
The results are
as expected considering the preceding discussion of the effect of unequal n’s
on the ANOVA. Both the cell means for
the groups and the marginal means from the MANOVA command are presented below.
Var Value Mean Std Dev Cases
|
A
1 30.47 11.79
21
B 1
27.50 5.25 4
B 2
54.33 8.62 3
B 3
26.21 6.51 14
A
2 37.90 10.25
11
B 1
38.50 3.53 2
B 2
43.00 5.38 7
B 3
19.50 2.12 2
|
Combined Observed Means for A
|
A
1 WGT. 30.47619
UNWGT. 36.01587
2 WGT. 37.90909
UNWGT. 33.66667
Combined
Observed Means for B
B
1 WGT. 31.16667
UNWGT. 33.00000
2 WGT. 46.40000
UNWGT. 48.66667
3 WGT. 25.37500
UNWGT. 22.85714
|
Note that the
weighted means are the means obtained using the MEANS command and each marginal
mean is computed by dividing the sum of scores by the number of scores. The unweighted marginal means are computed
by finding the mean of the cell means.
For example, the unweighted marginal mean for a1 would be the
mean of 27.50, 54.33, and 26.21.
REGRESSION
ANALYSIS OF UNEQUAL N ANOVA
The regression
analysis uses the dummy codes of the data.
With unequal cell sizes, the dummy coded variables will not be
orthogonal or uncorrelated. The greater
the difference in cell sizes, the greater the correlations between the dummy
codes. The correlation matrix for the
dummy codes in the example data is presented below.
‑ ‑ Correlation Coefficients
‑ ‑
|
DA1 DB1 DB2 DAB1 DAB2
DA1
1.0000 ‑.4606** .3427 ‑.1910 ‑.2835
DB1
‑.4606** 1.0000 ‑.1796 .5069** .2750
DB2
.3427 ‑.1796 1.0000 .0910 ‑.0823
DAB1
‑.1910 .5069** .0910 1.0000 .0296
DAB2
‑.2835 .2750 ‑.0823 .0296 1.0000
|
The implication
of these correlations is that the different dummy coded variables will share
variance in predicting the dependent variable.
The order of entry of a variable into the regression equation will make
a difference.
The REGRESSION
command appearing below places uses the subcommand “/METHOD=TEST(DA1) (DB1
DB2)(DAB1 DAB2).” The result of this
analysis directly corresponds to the “/METHOD=UNIQUE” subcommand in
MANOVA. The REGRESSION command and
output are presented below.
/DEPENDENT x
/METHOD=TEST (da1) (db1 db2) (dab1 dab2) .
* * *
* M U L T I P L E R E G R E S S I O N * * * *
Hypothesis Tests
Sum of
DF Squares Rsq Chg F Sig F
Source
1 27.63040 .00653 .73679 .3985
DA1
2 2604.75272 .61564 34.72919 .0000
DB1 DB2
2 425.82185 .10064 5.67749 .0090
DAB1 DAB2
5 3255.94494
17.36462 .0000 Regression
26 975.02381 Residual
31 4230.96875 Total
|
To generate the
ANOVA table of the “METHOD=SEQUENTIAL” subcommand in MANOVA, the “/METHOD=TEST”
subcommand is modified as follows in the REGRESSION command. The resultant output follows.
/DEPENDENT x
/METHOD=test (da1)
/METHOD=test (db1 db2)
/METHOD=test (dab1 dab2)
Variable(s) Entered on Step Number 1..
DA1
Hypothesis Tests
Sum of
DF Squares Rsq Chg F Sig F
Source
1 398.82156 .09426 3.12218 .0874
DA1
1 398.82156 3.12218
.0874 Regression
30 3832.14719 Residual
31 4230.96875 Total
Block Number
2. Method: Test
DB1 DB2
Hypothesis Tests
Sum of
DF Squares Rsq Chg F Sig F
Source
2 2431.30153 .57464 24.29834 .0000
DB1 DB2
3 2830.12309 18.85610
.0000 Regression
28 1400.84566 Residual
31 4230.96875 Total
Block Number
3. Method: Test
DAB1 DAB2
Hypothesis Tests
Sum of
DF Squares Rsq Chg F Sig F
Source
2 425.82185 .10064 5.67749 .0090
DAB1 DAB2
5 3255.94494 17.36462
.0000 Regression
26 975.02381 Residual
31 4230.96875 Total
|
Note that the
order of variables entered into the equation is similar to the default order of
entry in the MANOVA. In the case with
the multiple “/METHOD=TEST” commands the REGRESSION analysis tests the
hypothesis that the addition of the variable or variables predicts a
significant amount of the variance that remains after the preceding variables
have been entered into the equation.
RECOMMENDATIONS
The first
recommendation is to avoid multi-factor ANOVA designs with unequal cell
sizes. This recommendation is rather
glib and does not present a realistic alternative in most situations. Unequal cell sizes are a fact of life in
most real-life experimental designs.
The second
recommendation is to use the default “/METHOD=UNIQUE” when faced with unequal
cell sizes. The anomalous results that
this method sometimes generates can be more easily accepted if the user has an
understanding of how the procedure works.
A colleague presented a table of cell means which common sense would
suggest that the two-factor interaction should be significant. It was not, however, because the cell sizes
were grossly unequal. The colleague had
a difficult time accepting a statistical result that went against common sense.
The final
recommendation is to approach ANOVA using a multiple regression approach. Even if the results are similar to that
using MANOVA, at least the user has more control over the analysis and perhaps
a better understanding of the results.
Chapter
15
Subjects
Crossed With Treatments S X A
The Design
In S X A designs each subject
sees each level of factor A.
These designs are sometimes called repeated measures designs
because there must necessarily be more than one score per subject. Another name for the design is within
subjects design, because all effects are within rather than between
subjects.
With respect to purpose and function, the S
X A design is like the A design.
With respect to the score model,
it is much closer to the A X B design.
RUN NAME LOMBARD EFFECT DATA FOR
S X A EXAMPLE
|
DATA LIST
FILE='SXA DAT A'/SUBJECT 1‑3 SOFT 4‑6 MEDIUM 7‑9 LOUD 10‑12
NONE 13‑15.
LIST.
CORRELATION SOFT
TO NONE.
MANOVA SOFT
MEDIUM LOUD NONE
/WSFACTORS LOMBARD(4)
/PRINT = CELLINFO(MEANS)
/WSDESIGN
/DESIGN.
|
An example of an experiment that employs the S
X A design follows. Suppose a
researcher was interested in choral singing behavior. A phenomenon noted by many choral directors, called the Lombard
effect, is that people tend to sing louder when the people around them are
singing louder. The researcher wishes
to test the extent that this occurs.
The researcher has twelve subjects each sing
the Star Spangled Banner four times, measuring the loudness of their singing
each time. The subjects have on
headphones and are singing along with other voices, played soft, medium, and
loud. A forth condition has no
accompaniment. The subjects were
instructed to try to sing at the same level each time. The order of the loudness of the
accompanying voices is randomized across subjects. An apparatus measures the average decibels of the singing for the
entire song.
1 92
93 93 87
2 81
89 89 79
3 88
93 89 81
4 93
99 99 87
5 82
83 83 79
6 88
85 87 90
7 86
92 89 82
8 82
87 84 81
9 82
88 84 82
10 85
88 88 79
11 79
82 82 76
12 88
92 93 84
|
The Data
The data are arranged with each subject's
data as a row. The first column in the
data file is the subject number.
Subject numbers are not necessary
to run the analysis, but may come in very handy in locating incorrectly entered
data or missing subjects. The next four
columns of data are average decibels for the none, soft, medium and loud
accompaniment conditions.
SPSS commands
As in all previous analyses, the DATA LIST
command reads in the data from the data file and the LIST command gives a
picture of how the computer understands the data. The CORRELATION command gives a picture of how the repeated
measures are correlated.
After the keyword MANOVA, the names of the
four variables used for the measures under the different treatment levels are
listed. There is no BY listed because
there are no between subjects factors.
The subcommand WSFACTORS = is followed by a name for the factor and the
number of levels of that factor in parentheses. In this example the factor name selected was LOMBARD and the
number of levels of this factor was four.
The number of levels must correspond to the number of variables listed
after the MANOVA command.
As in the previous MANOVA commands, the PRINT
= CELLINFO(MEANS) will give an optional table of the means for the four
variables. The analysis will also go on
without the WSDESIGN and DESIGN subcommands, but putting them in avoids a
warning message.
The Correlation Matrix
‑ ‑
Correlation Coefficients ‑
‑
|
SOFT MEDIUM LOUD NONE
SOFT
1.0000 .7951** .8647**
.8071**
MEDIUM
.7951** 1.0000 .9143** .4824
LOUD
.8647** .9143** 1.0000 .5988*
NONE
.8071** .4824 .5988*
1.0000
* ‑
Signif.LE.05 ** ‑ Signif.LE.01 (2‑tailed)
|
The correlation matrix showing the linear
relationship between the within subjects variables is presented below.
The Table of Means
The following table is an example of the
output of the optional PRINT command.
Note that the results of the experiment are as predicted, the louder the
accompaniment, the louder the subjects sang, except for a slight reversal from
Medium to Loud. The output has been
reduced in size.
Graphs
Cell
Means and Standard Deviations
|
Mean Std. Dev.
N 95 percent Conf. Interval
‑
‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑
‑ ‑ ‑ ‑
Variable .. SOFT
For
entire sample 85.500 4.442
12 82.678 88.322
‑
‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑
‑ ‑ ‑ ‑
Variable .. MEDIUM
For
entire sample 89.250 4.827
12 86.183 92.317
‑
‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑
‑ ‑ ‑ ‑
Variable .. LOUD
For
entire sample 88.333
4.924 12 85.205 91.462
‑
‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑
‑ ‑ ‑ ‑
Variable .. NONE
For
entire sample 82.250 4.070
12 79.664 84.836
|
A simple graph is produced from this
data. Because the graph contains so
little information, a table is usually used to present the results.
The ANOVA Table
The results produced by a within subjects
design has several additional statistics besides the ANOVA table. Each section of the results will now be
analyzed in turn.
Interpretation of Output
The first section of the results is presented
below:
‑ ‑ ‑ ‑ ‑
‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑ ‑
|
Tests of Between‑Subjects Effects.
Tests of Significance for T1 using UNIQUE
sums of squares
Source of Variation SS DF MS F
Sig of F
WITHIN CELLS 746.17 11 67.83
CONSTANT 357765.33
1 357765.33 5274.18
.000
|
* * * * * * * * * * * * * * * * * A N A L Y S I S O F
V A R I A N C E ‑‑ DESIGN 1 * * * * * * * * * *
|
Tests involving 'LOMBARD' Within‑Subject
Effect.
Mauchly sphericity test, W = .26143
Chi‑square approx. = 13.04327 with 5 D. F.
Significance = .023
Greenhouse‑Geisser Epsilon = .53263
Huynh‑Feldt Epsilon = .60890
Lower‑bound Epsilon = .33333
|
Since there are no between subjects effects,
other than the constant (m), in an S X A design, the information
presented above is not very useful. The
analysis says that the grand mean is significantly different from zero.
The second section of the results is
presented below.
The Mauchly sphericity test is an hypothesis
test of the additional assumptions necessary in a within subjects design. When the significance of this statistic is
below a set level (.05 is standard) then the assumptions necessary to do a
within subjects ANOVA have been violated.
The three Epsilons that follow are corrections to the degrees of freedom
to correct for the violations. Both the
numerator and denominator degrees of freedom of the F test that follows must be
multiplied by one of these values and the significance of the F ratio
revaluated. The Greenhouse-Geisser
Epsilon if fairly conservative, especially for small sample. The Huynh-Feldt Epsilon is an attempt at
correction of the conservatism (Norusis, 1990).
Multivariate Tests of Significance (S = 1,
M = 1/2, N = 3 1/2)
Test
Name Value Exact F Hypoth. DF Error
DF Sig. of F
Pillais .72457 7.89193
3.00 9.00 .007
Hotellings 2.63064 7.89193
3.00 9.00 .007
Wilks .27543 7.89193 3.00
9.00 .007
Roys .72457
Note..
F statistics are exact.
|
Multivariate tests of significance are
sometimes recommended for testing within subjects effects because the
assumptions necessary to perform the test are less restrictive. Four different multivariate tests are performed
and appear as follows:
Note that in this case all of the
multivariate tests give the same Exact F and significance level. These hypothesis tests provide evidence for
the reality of effects. The
interpretation of the effect is the same as the interpretation for a
significant main effect of A in design A.
Tests involving 'LOMBARD' Within‑Subject
Effect.
|
AVERAGED
Tests of Significance using UNIQUE sums of squares
Source of Variation SS DF MS F
Sig of F
WITHIN CELLS 176.00
33 5.33
LOMBARD 358.50
3 119.50 22.41
.000
|
The final table is a univariate test of the A
effect. Note that here the Lombard (A)
main effect is statistically significant.
Interpretation of this significant main effect proceeds identically to
the multivariate test for a significant main effect of A.
In any case the results of this study seem
clear, the louder the accompaniment, the louder the person sang. No accompaniment gave the quietest
singing. There may have been a ceiling
effect as moderate and loud accompaniment gave almost identical singing volume.
Additional Assumptions for
Univariate S X A Designs
To avoid distorting the decision-making
process, it is necessary to add an assumption when taking a univariate approach
to testing for the main effect of A in S X A designs. The assumption is that if an infinite number
of subjects were run in the experiment and scores were collected on all levels
of the within subjects treatments, then the correlations between the different
treatments would all be equal. In
addition, the variances would all be equal.
The sample correlation matrix between the treatment effects was
found in the example SPSS program.
Imagine a very large (infinite) data set and a similar correlation
matrix. The assumption underlying the
univariate repeated measures ANOVA is that all the values in the matrix would
be the same.
Norusis (1990) acknowledges difficulties with
the Mauchly Sphericity Test. For small
samples the test of the preceding assumption is not very powerful and for large
samples it may be too powerful. The
major problem here is that it the goal of the analysis is to accept the null
hypothesis that the correlations are not different.
Deciding to use the univariate or
multivariate approach is often a matter of preference. Winer (1962 p. 283) writes: "It is, however, difficult to formulate
a set of general rules by which one type of analysis will be preferred to the
other. The univariate approach pools
information from repeated measures - this kind of pooling usually carries with
it a set of homogeneity assumptions."
SS, MS, and Expected Mean Squares
(EMS)
SS and MS
As in the preceding designs, the SS for any
effect can be found by squaring the score model terms and summing the
squares. The mean squares may be obtained
by dividing the SS by the appropriate degrees of freedom. The F-ratio is then found by dividing the MS
for that term by an error MS. It is not
directly apparent whether the MS for the A effect (MSA)
should be tested by dividing by the MS for Subjects (MSS) or the MS
for error (MSE). For that
reason a new concept, expected mean squares, will be introduced.
Expected Mean Squares
If an experiment could be run with an
infinite number of subjects in all conditions, then the score model would not
need to be estimated, but would be known.
In that case, hypothesis testing for term effects would not be needed
because it would be known whether the terms in the model were zero or not. Since it is impossible and not very
cost-efficient to run an infinite number of subjects in an experiment,
hypothesis testing is standard operating procedure in science.
In an abstract and mathematical sense the
question of what would the world look like if there were and infinite number of
subjects in each group makes sense. In
fact it is the foundation of hypothesis testing theory. If there were an infinite number of subjects
in each group, the score model terms could be found for each score. The procedure for finding the terms is
conceptually similar, but not directly analogous, to the procedure which has
been used in the preceding chapters. It
is not directly possible to solve an infinite number of simultaneous linear
equations for an infinite number of variables.
A mathematical solution, however, is available. At this point the student is asked to believe.
If the real terms for the score model could
be found, the real variances for all the terms in the model could also be
found. There would be a variance for
each term in the score model. For
example,
Factor Model
Term Variance
A aa σA2
S ss σS2
E eas σE2
In real life, only a sample of subjects
(preferably random) is selected to be included in the experiment rather than an
infinite number of subjects. The terms
in the score model are estimated and the mean squares for each term are
computed. Basically, the means squares
for each term are variances of those estimated values.
Moving back again to the conceptual and
abstract arena, visualize an infinite number of similar experiments being
carried out. The score model terms and
associated mean squares are estimated for each experiment. Each time they are different, but seem to
cluster around one value or another. A
mathematician can find the
(theoretical) mean of the model terms and means squares using a technique
called expected value. The expected
mean square (EMS) is the expected value of the infinite number of observed mean
squares.
Using techniques that will not be covered in
this text, it is possible to solve for the EMS in terms of real variances of
model terms. The results show certain
patterns which have been placed in an algorithm so that given any standard
model, the expected mean squares may be found (Lee, 1975). I have written a computer program that uses
the algorithm presented in Lee so that students may find expected mean squares
for many standard ANOVA designs.
Instructions on how to use the program may be found in Homer &
Tessie.
Using the program the following results were found for design S X A.
Factor Model
Term Variance Mean Square EMS
A aa σA2
MSA σE2 + σAS2 + SσA2
S ss σS2 MSS σE2 + AσS2
E eas σE2 MSE σE2 + σAS2
The rule for selecting an error term to test
the effect of a factor is simple: the
error term (denominator) in the F-ratio is the mean square of the term with an
expected mean square containing all the terms in the expected mean square of
the term to be tested plus one - an error term - σE2.
From the above table it can be seen that MSE is the
appropriate error term. Also note that
there is no error term that could be used to test the subjects main effect.
The output from the program described in Homer
& Tessie differs somewhat from the Expected Mean Squares given above,
because of limitations of the screen output within the programming
language. The correspondence is as
follows:
Actual Expected Mean Squares Program Expected Mean Squares
σE2 + σAS2 + SσA2 σE + σAS + SσA
σE2 + AσS2 σE + AσS
σE2 + σAS2 σE + σAS
Note that the terms are not squared and terms
which follow the σ are
not subscripted. With a little effort,
the two presentation modes may be understood as equivalent.
Chapter
16
Subjects
Crossed With Two Treatments -
S X A X B
The Design
In S X A X B designs
each subject sees each level of all possible combinations of
levels of factors A and B. These designs are a more complex example of repeated measures
or within subjects designs. With respect to purpose and function, the S
X A X B design is like the A X B design.
An example of an experiment that employs an S
X A X B design is a variation of the S X A
experiment described in the previous chapter.
That study investigated the Lombard effect by having subjects sing the
national anthem along with a choral group played through earphones. In this variation of the study, the three
levels of factor A were three different locations in the song. Factor B investigated the effects of
instructions to the subjects. On the
first attempt, subjects were not given any instructions about how loudly to
sing, while on the second attempt subjects were instructed to try to sing at the same level throughout the song.
The Data
The data are arranged with each subject's
data as a row. As before, the first
column in the data file is the subject number. Following this are the six measures taken for each subject. Note that the variables are named as L1P1 to
L2P3 for the loudness and place levels for each subject. The MANOVA subcommand does not care what
these variables are named, the variable names could just as well have been V1
to V6. In this case the variable names
were selected to make life more organized for the data analyzer. The GENDER and EXPER variables are not
needed for this analysis, but will be needed in following chapters.
SUB GND EXP I1P1 I1P2 I1P3 I2P1
I2P2 I2P3
|
2
1 3 93 93 93
87 88 89
3
2 2 81 89 89
79 83 79
4
1 2 88 93 89
81 77 76
5
1 1 93 99 99
87 93 95
6
2 1 82 83 83
79 79 79
7
2 2 88 85 87
90 84 83
8
1 3 86 92 89
82 84 86
9
1 2 82 87 84
81 81 79
10 2
2 82 88 84 82
83 83
11 1
1 85 88 88 79
77 83
12 1
3 79 82 82 76
79 82
13 1
1 88 92 93 84
84 88
14 1
3 82 83 83 78
82 80
15 2
1 81 85 86 78
80 80
16 1
1 85 85 87 81
84 87
17 1
3 91 88 85 87
77 78
18 2
3 78 79 79 71
73 78
19 1
3 89 95 102 91
99 98
20 1
2 85 84 86 85
79 78
21 1
3 90 94 93 88
89 90
22 2
3 86 88 89 83
86 87
23 1
2 89 90 90 85
85 87
24 2
2 85 86 87 75
81 85
25 1
1 83 83 78 72
74 73
26 1
1 79 80 80 79
77 78
27 2
1 80 81 81 79
77 78
28 1
2 77 82 84 71
72 76
|
SPSS commands
As in all previous analyses, the DATA LIST
command reads in the data from the data file.
The VALUE LABELS command will make interpretation of output easier. As before, the LIST command gives a picture
of how the computer understands the data and the CORRELATION command gives a
picture of how the repeated measures are correlated.
After the keyword MANOVA, the names of the
six variables used for the measures under the different treatment levels are
listed. Because there are no between
subjects factors, the keyword BY is not used.
The subcommand WSFACTORS = is followed by a name for both the A
and B factors with the number of levels of that factor in parentheses
following the factor name. The slowest
moving variable is listed first and the fastest moving variable is listed last. In this example the factor names selected
were INSTRUCT and PLACE, the former reflecting whether or not instructions were
given to the subject to control for the Lombard effect and the latter
reflecting the position in the song.
The order of the variables following the MANOVA command determines the
order of factor names following the WSFACTORS subcommand. In this case, because the three position
measures under the no instruction condition appeared first and the second three
were under voice control instructions, the order of variables on the WSFACTORS
subcommand had to be INSTRUCT and then PLACE.
Changing the order of these factors would give very different
results. The product of the number of
levels of all factors listed after WSFACTORS must correspond to the number of
variables listed after the MANOVA command.
In this case there are two times three or six variables.
As in the previous MANOVA commands, the PRINT
= CELLINFO(MEANS) will give an optional table of the means for the six
variables. The analysis will also
proceed without the WSDESIGN and DESIGN subcommands, but putting them in avoids
a warning message.
The Correlation Matrix
The correlation matrix showing the linear
relationship between the within subjects variables is presented below. Note that the different measures are highly
correlated, indicating that the experiment gains considerable power by using a
within rather than between subjects design.
‑ ‑ Correlation Coefficients
‑ ‑
|
I1P1 I1P2 I1P3 I2P1
I2P2 I2P3
I1P1
1.0000 .8270** .7578**
.8091** .6620** .6519**
I1P2
.8270** 1.0000 .9020** .6882** .7627** .7323**
I1P3
.7578** .9020** 1.0000 .7260** .8716** .8696**
I2P1
.8091** .6882** .7260** 1.0000 .7667** .6606**
I2P2
.6620** .7627** .8716**
.7667** 1.0000 .9224**
I2P3
.6519** .7323** .8696**
.6606** .9224** 1.0000
0* ‑
Signif. LE .05 ** ‑ Signif.
LE .01 (2‑tailed)
|
The Table of Means
Cell Means and Standard Deviations
|
Mean Std. Dev. N 95 percent
Conf. Interval
Variable .. I1P1
For entire sample 84.704
4.513 27 82.918 86.489
Variable .. I1P2
For entire sample 87.185 5.054
27 85.186 89.184
Variable .. I1P3
For entire sample 87.037 5.626
27 84.811 89.263
Variable .. I2P1
For entire sample 81.111 5.423
27 78.966 83.256
Variable .. I2P2
For entire sample 81.741 6.010
27 79.363 84.118
Variable .. I2P3
For entire sample 82.778
5.976 27 80.414 85.142
|
The following table is an example of the
output of the optional PRINT subcommand under the MANOVA command. Note that the means must be collapsed over
variables to find effects. For example,
in order to find the size of the INSTRUCT factor, the mean of the first three
means would have to be compared with the mean of the last three means. The POSITION main effect could likewise be
found by comparing the means of I1P1 and I2P1, I1P2 and I2P2, and I1P3 and
I2P3.
Graphs
A graph of the interaction of A and B is generated from the
data if the interaction effect is found to be significant. The following presents a graph of the above
means.
The ANOVA Table
* * * * * * * * A N A L Y S I
S O F V A R I A N C E * * * * * *
|
Tests
involving 'INSTRUCT' Within‑Subject Effect.
Tests of
Significance for T2 using UNIQUE sums of squares
Source
of Variation SS DF MS F Sig of F
WITHIN
CELLS 280.27 26
10.78
INSTRUCT
795.56 1 795.56 73.80 .000
|
Like the S X A design, the
results produced by a within subjects design has several additional statistics
besides the ANOVA table. Each section
of the results will now be analyzed in turn.
Interpretation of Output
The first section of the results is presented
below:
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
|
Tests of Between‑Subjects Effects.
Tests of Significance for T1 using UNIQUE
sums of squares
Source
of Variation SS DF
MS F Sig of F
WITHIN CELLS 3753.11
26 144.35
CONSTANT 1145593.39
1 1145593.4 7936.20
.000
|
Since there are no between subjects effects,
other than the constant (m), in an S X A X B design, the
information presented above is not very useful. The analysis says that the grand mean is significantly different
from zero.
Because the INSTRUCT main effect has only two
levels, univariate and multivariate tests of significance do not differ. In this case the SPSS program does only the
univariate test with the following result.
Note that the INSTRUCT main effect is significant.
The third section of the results is presented
below.
* *
* * * * A N A L Y S I S O F V A R I A N C E * * * * * * *
|
Tests involving 'PLACE' Within‑Subject Effect.
Mauchly sphericity test, W = .51990
Chi‑square approx. = 16.35277 with 2 D. F.
Significance = .000
Greenhouse‑Geisser Epsilon = .67563
Huynh‑Feldt Epsilon = .69951
Lower‑bound Epsilon = .50000
AVERAGED Tests of
Significance that follow multivariate tests are equivalent
to univariate or split‑plot
or mixed‑model approach to repeated measures.
Epsilons may be used to
adjust d.f. for the AVERAGED results.
|
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
|
EFFECT
.. PLACE
Multivariate
Tests of Significance (S = 1, M = 0, N = 11 1/2)
Test
Name Value Exact F Hypoth. DF Error DF
Sig. of F
Pillais .23891
3.92384 2.00 25.00
.033
Hotellings .31391 3.92384 2.00
25.00 .033
Wilks .76109
3.92384 2.00 25.00
.033
Roys .23891
Note.. F statistics are exact.
|
As in the S X A design, the
Mauchly sphericity test is an hypothesis test of the additional assumptions
necessary in a within subjects design when the factors have more than two
levels. Because the PLACE factor has
more than two level, these assumptions are tested by SPSS. Note that in this case the significance of
the Mauchly sphericity test seriously questions the assumptions necessary to
test the PLACE main effect.
Multivariate tests of significance are
sometimes recommended for testing within subjects effects because the
assumptions necessary to perform the test are less restrictive. Four different multivariate tests are
performed and appear as follows:
Note that in this case all of the
multivariate tests give the same Exact F and significance level. These hypothesis tests provide evidence for
the reality of effects. The
interpretation of the effect is the same as the interpretation for a
significant main effect of A in design A.
The final table is a univariate test of the A
effect. Note that here the Lombard (A)
main effect is statistically significant.
Interpretation of this significant main effect proceeds identically to
the multivariate test for a significant main effect of A.
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
|
Tests involving 'PLACE' Within‑Subject
Effect.
AVERAGED
Tests of Significance using UNIQUE sums of squares
Source of Variation SS
DF MS F
Sig of F
WITHIN CELLS 448.89
52 8.63
PLACE 119.11
2 59.56 6.90
.002
|
In any case the results of this study seem
clear, if the accompaniment was louder at some position in the song, the person
sang louder.
Just as in the A X B design,
the S X A X B design has an A by B
interaction. This term is tested by
both multivariate and univariate analyses in SPSS. In addition, the sphericity test is done to test the assumptions
necessary to test this term. The
following presents the combined analysis.
Note the Mauchly sphericity test was not significant and both the
multivariate and univariate tests of the INSTRUCT BY PLACE interaction were
significant. The interaction was
probably mostly due to position 2 in the song where the instructions to control
loudness were relatively more effective.
* * * * A N A L Y S I S
O F V A R I A N C E * * * *
|
Tests involving 'INSTRUCT BY PLACE' Within‑Subject
Effect.
Mauchly sphericity test, W = .87792
Chi‑square approx. = 3.25506 with 2 D. F.
Significance = .196
Greenhouse‑Geisser Epsilon = .89120
Huynh‑Feldt Epsilon = .95230
Lower‑bound Epsilon = .50000
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
EFFECT .. INSTRUCT BY PLACE
Multivariate Tests of Significance (S = 1,
M = 0, N = 11 1/2)
Test
Name Value Exact F Hypoth. DF Error DF Sig. of F
Pillais .28144 4.89584 2.00
25.00 .016
Hotellings .39167 4.89584 2.00
25.00 .016
Wilks .71856
4.89584 2.00 25.00 .016
Roys .28144
Note.. F statistics are exact.
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
Tests involving 'INSTRUCT BY PLACE' Within‑Subject
Effect.
Source
of Variation SS DF
MS F Sig of F
WITHIN CELLS 166.91 52 3.21
INSTRUCT BY PLACE 23.75 2 11.88 3.70
.031
|
Additional Assumptions for
Univariate S X A X B Designs
To avoid distorting the decision-making
process, it is necessary to add assumptions when taking a univariate approach
to testing for the main effects and interaction effects in an S X A X B design. The assumptions identical to testing the
main effect of A in an S X A design, namely that if an
infinite number of subjects were run in the experiment and scores were
collected on all levels of the within subjects treatments, then the
correlations between the different treatments or combination of treatments
would all be equal. repeated measures
ANOVA is that all the values in the matrix would be the same.
SS, MS, and Expected Mean Squares
(EMS)
SS and MS
As in the preceding designs, the SS for any
effect can be found by squaring the score model terms and summing the
squares. The mean squares may be
obtained by dividing the SS by the appropriate degrees of freedom. The F-ratio is then found by dividing the MS
for that term by an error MS. As in the
S X A design it is not directly apparent whether the MS for the A
effect (MSA) should be tested by dividing by the MS for Subjects (MSS),
the MS for A by Subjects (MSAS), or the MS for error (MSE). Expected mean squares may be used to find
the correct error term.
Expected Mean Squares
The expected mean squares can be found by
using the program described in Homer & Tessie. The results of running the program for design A X B X S has been previously
presented in a text box. Modifying the
order of the terms yields the following:
Factor Model
Term Variance
A aa σA2
S ss σS2
E eas σE2
In real life, only a sample of subjects (preferably
random) is selected to be included in the experiment rather than an infinite
number of subjects. The terms in the
score model are estimated and the mean squares for each term are computed. Basically, the means squares for each term
are variances of those estimated values.
Moving back again to the conceptual and
abstract arena, visualize an infinite number of similar experiments being
carried out. The score model terms and
associated mean squares are estimated for each experiment. Each time they are different, but seem to
cluster around one value or another. A
mathematician can find the
(theoretical) mean of the model terms and means squares using a technique
called expected value. The expected
mean square (EMS) is the expected value of the infinite number of observed mean
squares.
Using techniques that will not be covered in
this text, it is possible to solve for the EMS in terms of real variances of
model terms. The results show certain
patterns which have been placed in an algorithm so that given any standard
model, the expected mean squares may be found (Lee, 1975).
The following results were found for
design S X A X B.
A‑ σE + BσAS + SBσA
AS‑ σE + BσAS
B‑ σE + AσBS + SAσB
BS‑ σE + AσBS
AB‑ σE + σABS + SσAB
ABS‑ σE + σABS
S‑ σE + ABσS
The rule for selecting an error term to test
the effect of a factor can now be recalled.
The error term (denominator) in the F-ratio is the mean square of the
term with an expected mean square containing all the terms in the expected mean
square of the term to be tested plus one - an error term - σE2. From the above table it can be seen that MSAS
is the appropriate error term for the A main effect. In a like manner, the MSBS is
used for B and MSABS is used for AB. As before, there is no error term that
could be used to test the subjects main effect.
The WITHIN CELLS term in the SPSS output
corresponds to the appropriate error term. For example, the MSABS term corresponds to the WITHIN
CELLS term in the test of the AB interaction effect.
Chapter
17
Mixed Designs
- S ( A ) X B
The Design
In S ( A ) X B designs each subject sees each
level of B and appears under one
and only one level of A. These
designs are called mixed designs because they contain both within and between
subjects factors. With respect to
purpose and function, the S ( A ) X B design is like the A
X B or S X A X B designs.
An example of an experiment that employs an S ( A ) X B design is a
variation of the S X A X B experiment described in the
previous chapter. That study
investigated the Lombard effect by having subjects sing the national anthem
along with a choral group played through earphones. In this variation of the study, the three levels of factor B
were three different locations in the song.
Factor A investigated the effects of the subjects previous choral
experience. In this blocking factor,
subjects either had no choral experience, some experience, or a great deal of
experience. The astute student might
realize that this data is identical to the first half the data presented in the
last chapter.
The Data
The data are arranged with each subject's
data as a row. As before, the first
column in the data file is the subject number. Following this are the three measures taken for each
subject. Note that the variables are
named I1P1 to I1P3 for the three places in the song that readings were taken
for each subject. In this analysis only
the condition when the subjects are given no instructions about how loudly to
sing is used.
SUB GND EXP I1P1 I1P2 I1P3
|
2
1 3 93 93 93
3
2 2 81 89 89
4
1 2 88 93 89
5
1 1 93 99 99
6
2 1 82 83 83
7
2 2 88 85 87
8
1 3 86 92 89
9
1 2 82 87 84
10 2
2 82 88 84
11 1
1 85 88 88
12 1
3 79 82 82
13 1
1 88 92 93
14 1
3 82 83 83
15 2
1 81 85 86
16 1
1 85 85 87
17 1
3 91 88 85
18 2
3 78 79 79
19 1
3 89 95 102
20 1
2 85 84 86
21 1
3 90 94 93
22 2
3 86 88 89
23 1
2 89 90 90
24 2
2 85 86 87
25 1
1 83 83 78
26 1
1 79 80 80
27 2
1 80 81 81
28 1
2 77 82 84
|
SPSS commands
After the keyword MANOVA, the names of the
three variables used for the within subjects factor are listed. Following the keyword BY, the between
subjects factors are listed, followed by the beginning and ending levels in
parentheses. Here the between subjects
factor is EXPER with factor levels starting at 1 and ending at 3.
The subcommand WSFACTORS = is followed by a
name for the B factor with the number of levels of that factor in
parentheses following the factor name.
This is done in a manner identical to the design specification if a S
X B design had been employed. In
this case the factor is labeled PLACE and has three levels.
As in the previous MANOVA examples, the PRINT
= CELLINFO(MEANS) will give an optional table of the means for the three place
variables at the three levels of the EXPER variable. As before, the analysis will go on without the WSDESIGN and
DESIGN subcommands, but putting them in avoids a warning message.
RUN NAME ANOVA EXAMPLE MIXED DESIGN ANALYSIS
|
DATA LIST FILE='ANOVA4 DAT A'/SUBJECT GENDER EXPER 1‑6
I1P1 I1P2 I1P3 7‑15.
VALUE LABELS GENDER 1 'MALE' 2 'FEMALE'/
EXPER 1 'NONE' 2 'SOME' 3 'LOTS'.
LIST.
MANOVA I1P1 TO I1P3 BY EXPER (1,3)
/WSFACTORS PLACE (3)
/WSDESIGN
/PRINT CELLINFO(MEANS)
/DESIGN.
|
The Table of Means
Cell Means and Standard Deviations
|
Variable .. I1P1
FACTOR CODE
Mean Std. Dev 95 percent Conf. Interval
EXPER NONE
84.000 4.387 9
80.627 87.373
EXPER SOME
84.111 3.951 9
81.074 87.148
EXPER LOTS
86.000 5.339 9
81.896 90.104
For entire sample 84.704 4.513
27 82.918 86.489
Variable .. I1P2
EXPER NONE
86.222 6.016 9
81.598 90.847
EXPER SOME
87.111 3.333 9
84.549 89.673
EXPER LOTS
88.222 5.783 9
83.777 92.668
For entire sample 87.185 5.054
27 85.186 89.184
Variable .. I1P3
EXPER NONE
86.111 6.679 9
80.977 91.245
EXPER
SOME 86.667 2.345 9 84.864 88.469
EXPER LOTS
88.333 7.053 9
82.912 93.755
For entire sample 87.037 5.626
27 84.811 89.263
|
The following table is an example of the output
of the optional PRINT subcommand under the MANOVA command for a mixed
design.
Graphs
A graph of the interaction of A and B is generated from the
data if the interaction effect is found to be significant. Here the Song Position by Choral Experience
interaction is not significant, but the graph is presented anyway.
The ANOVA Table
The complexity of the results of the S ( A ) X B design falls between the A X B and S X A X B designs. As before, each section of the results will
now be analyzed.
Interpretation of Output
The first section of the results is presented
below:
* * * * A N A L Y S I S O F
V A R I A N C E * * * *
|
Tests of Between‑Subjects Effects.
Tests of Significance for T1 using UNIQUE
sums of squares
Source
of Variation SS DF
MS F Sig of F
WITHIN CELLS 1714.37 24 71.43
EXPER 62.91
2 31.46 .44
.649
|
The result of the analysis of the
between-subjects effect, here the main effect of A, is presented first.
In the above example, the main effect of EXPER (Choral Experience) was
not significant.
The Mauchly sphericity test is for the within
subjects factor ( PLACE ) is presented next.
In this case this significance test is again significant and the tests
of the within subjects factors must proceed with some consideration about
whether the assumptions have been violated.
Multivariate tests of significance are
sometimes recommended for testing within subjects
* * * * * * * A N A L Y S I S
O F V A R I A N C E * * * * * *
|
Tests involving 'PLACE' Within‑Subject
Effect.
Mauchly sphericity test, W = .75133
Chi‑square approx. = 6.57602 with 2 D. F.
Significance = .037
Greenhouse‑Geisser Epsilon = .80085
Huynh‑Feldt Epsilon = .92074
Lower‑bound Epsilon = .50000
AVERAGED Tests of Significance that follow
multivariate tests are
equivalent to univariate or split‑plot
or mixed‑model approach to
repeated measures.
Epsilons may be used to adjust d.f. for the
AVERAGED results.
|
The multivariate and univariate tests for the
B and A X B effects are presented next. Note that all of the multivariate tests give
the same Exact F and significance level. The univariate tests, while providing a slightly different
significance level, make the same decisions as the multivariate tests.
* * * * A N A L Y S I S O F V A R I A N C E *
* * *
|
EFFECT .. EXPER BY PLACE
Multivariate
Tests of Significance (S = 2, M = ‑1/2, N = 10 1/2)
Test
Name Value Approx. F Hypoth.
DF Error DF sig. of F
Pillais
.02538 .15421 4.00 48.00 .960
Hotellings .02598 .14292 4.00 44.00 .965
Wilks
.97465 .14860 4.00 46.00 .963
Roys
.02436
Note.. F statistic for WILK'S Lambda is
exact.
* *
* * A N A L Y S I S O F V A R I A N C E * * * *
EFFECT .. PLACE
Multivariate
Tests of Significance (S = 1, M = 0, N = 10 1/2)
Test
Name Value Exact F Hypoth. DF Error DF
Sig. of F
Pillais .44336 9.15950 2.00
23.00 .001
Hotellings .79648 9.15950 2.00
23.00 .001
Wilks .55664
9.15950 2.00 23.00 .001
Roys .44336
Note.. F statistics are exact.
* *
* * A N A L Y S I S O F V A R I A N C E * * * *
Tests involving 'PLACE' Within‑Subject
Effect.
AVERAGED
Tests of Significance using UNIQUE sums of squares
Source of Variation SS
DF MS F
Sig of F
WITHIN CELL 237.41
48 4.95
PLACE 104.62
2 52.31 10.58
.000
EXPER BY PLACE 1.98 4 .49 .10 .982
|
The results of the study seem clear. At positions in the national anthem where
the choir sang louder, the person singing along sang louder. Experience in choral singing had no
significant effect as either a main effect or interaction.
Expected Mean Squares (EMS)
The expected mean squares can be found by
using the program described in Appendix B.
The results of running the
program for design S ( A ) X B has been previously
presented in a text box. Modifying the
order of the terms yields the following:
A‑ σE + BσS + SBσA
S-A σE + BσS
B‑ σE + σBS + SAσB
AB‑ σE + σBS + SσAB
BS‑A σE + σBS
The rule for selecting an error term to test
the effect of a factor can now be recalled.
The error term (denominator) in the F-ratio is the mean square of the
term with an expected mean square containing all the terms in the expected mean
square of the term to be tested plus the error term - σE2. From the above table it is seen that MSBS
is the appropriate error term for the B main effect and AB
interaction effect. The WITHIN CELLS
term in the SPSS output corresponds to the appropriate error term. For example, the MSABS term
corresponds to the WITHIN CELLS term in the test of the B and AB
interaction effect.
Chapter
18
Three Factor
ANOVA
This chapter will focus on four designs which
serve the same function, to test the effects of three factors
simultaneously. The designs which will
be studied include:
S X A X B X C
S ( A ) X B X C
S ( A X B ) X C
A X B X C
Since the naming of the factors is arbitrary,
these designs include all possible three factor designs. In a departure from the last few chapters,
the similarities of these designs will first be studied, followed by the
differences. The advantages and disadvantages
of each will be then be presented.
Effects
The function of the four designs given above
is to test for the reality of three kinds of effects, main, two-way interaction, and three-way interaction. Although the first two have been described
in detail in earlier chapters, the different forms of the effects will be
discussed. The three-way interaction
will be discussed in detail.
A study of effects begins with a table of
means. This table might be constructed
by averaging over subjects in any number of ways, depending upon the
design. An example of a table of means
follows.
c1
c2
b1 b2 b3 b1 b2
b3
┌────┬────┬────┐ ┌────┬────┬────┐
a1 │
5 │
5 │
5 │ 5 │
6 │
7 │
8 │ 7
├────┼────┼────┤
├────┼────┼────┤
a2 │
7 │
7 │
7 │ 7 │
6 │
5 │
4 │ 5
└────┴────┴────┘
└────┴────┴────┘
6 6 6 6 6 6 6 6
Main Effects
Main effects are found in a manner analogous
to finding main effects in a two factor design, except that the data must be
collapsed over two other effects rather than one. In the case of a three factor experiment, there will be three
main effects, one for each factor, A, B, and C
.For example, in order to find the main
effect of factor A, one must find the mean of each level of A,
collapsing over levels of B and C. In the above example _a.. = ( _a11 + _a12 + ... +
_ABC) / BC. Where a=1, B=3,
and C=2, _1.. = ( _111
+ _112 + _121 + _122 + _131 + _132
) / 6 = ( 5 + 5 + 5 + 6 + 7 + 8 ) / 6 = 6. Likewise, _2.. = 6 and there would be no main effect
of A, because these values are similar.
In a like manner, _.1. = _.2.
= _.3. = 6 and there would
be no main effect of B. From the
table above it can be seen that _..1 = _..2 = 6 and
there would be no main effect of C.
Thus, this table is an example of a three factor experiment where no
main effects would be found.
Two-Way Interactions
Each combination of two factors produces a
two-way interaction by collapsing over the third factor. The three two-way interactions are
interpreted just like the single two-way interaction would be in an A X B
design.
By collapsing over the C factor, the AB
interaction yields the following table and graph. Note that an AB interaction is present because the simple
main effect of B does changes over levels of A, in one instance
increasing with B and the other decreasing. This table also clearly illustrates the lack of an A or B
main effect.
b1 b2 b3
┌────┬────┬────┐
a1 │ 5.5│ 6 │ 6.5│ 6
├────┼────┼────┤
a2 │ 6.5│ 6 │ 5.5│ 6
└────┴────┴────┘
6 6 6 6
By collapsing over the B factor, the AC
interaction produces the following table and graph. The cells in the table
reproduce the numbers which appeared as row means in the full table. In this case there is an AC
interaction present.
c1 c2
┌────┬────┐
a1 │
5 │
7 │ 6
├────┼────┤
a2 │
7 │
5 │ 6
└────┴────┘
6 6 6
By collapsing over the A factor, the BC
table and graph are produced. The
numbers in the graph appear as row means on the separate tables in the original
data. In this case the interaction is
absent.
b1 b2 b3
┌────┬────┬────┐
c1 │
6 │
6 │
6 │ 6
├────┼────┼────┤
c2 │
6 │
6 │
6 │ 6
└────┴────┴────┘
6 6 6 6
Three-Way Interaction
The three-way interaction, ABC, is a
change in the simple two-way interaction over levels of the third factor. A simple two-way interaction is a two-way
interaction at a single level of a third factor. For example, going back to the original table of means in this
example, the simple interaction effect of AB at c1 would be
given in the means in the left-hand boxes.
The same simple interaction at c2 would be given in the
right-hand boxes.
A change in the simple two-way interaction
refers a change in the relationship of the lines. If in both simple two-way interactions the lines were parallel,
no matter what the orientation, there would be no three-way interaction. Similarity, if the lines in the simple
two-way interactions intersected at the same angle, again no matter what the
orientation, there would be no three-way interaction.
In the case of the example data, graphed
below, the orientation of the lines comprising the simple interactions changes
from parallel to non-parallel from one graph to the other. In this case a three-way interaction would
exist. It may or may not be significant
depending upon the size of the error term.
Additional Examples
All Effects Significant
The following table of means was constructed
such that all effects would be significant, given that the error terms were
small relative to the size of the effects.
c1
c2
b1 b2 b3 b1 b2
b3
┌────┬────┬────┐ ┌────┬────┬────┐
a1 │
4 │
5 │
6 │ 5 │
5 │
5 │
5 │ 5
├────┼────┼────┤
├────┼────┼────┤
a2 │
7 │
8 │
9 │ 8 │
6 │
5 │
4 │ 5
└────┴────┴────┘
└────┴────┴────┘
5.5 6.5 7.5 6.5
The A X B interaction in table
and graph form follow:
b1 b2 b3
┌────┬────┬────┐
a1 │ 4.5│ 5 │ 5.5│ 5
├────┼────┼────┤
a2 │ 6.5│ 6.5│ 6.5│ 6.5
└────┴────┴────┘
5.5 5.75 6 5.75
The same is now done for the A X C interaction:
c1 c2
┌────┬────┐
a1 │
5 │
5 │ 5
├────┼────┤
a2 │
8 │
5 │ 6.5
└────┴────┘
6.5 5 5.75
In a similar fashion the table and graph for
the B X C interaction:
b1 b2 b3
┌────┬────┬────┐
c1 │ 5.5│ 6.5│ 7.5│ 6.5
├────┼────┼────┤
c2 │ 5.5│ 5 │ 4.5│ 5
└────┴────┴────┘
5.5 5.75 6 5.75
Finally the graph of the three-way
interaction is given.
Example 3 - B, AC, and BC
Selecting a somewhat arbitrary combination of
effects, one could ask what table of means could produce a combination of
effects such that B, AC, and BC would possibly be
significant and all other effects would not be significant. The following tables are one solution.
c1
c2
b1 b2 b3 b1 b2
b3
┌────┬────┬────┐ ┌────┬────┬────┐
a1 │
4 │
5 │
6 │ 5 │
7 │
7 │
7 │ 7
├────┼────┼────┤
├────┼────┼────┤
a2 │
6 │
7 │
8 │ 7 │
5 │
5 │
5 │ 5
└────┴────┴────┘
└────┴────┴────┘
5 6 7 6 6
6 6 6
The presence of a B main effect and
the lack of an A main effect and AB interaction is seen in the
following table and graph.
b1 b2 b3
┌────┬────┬────┐
a1 │ 5.5│ 6 │ 6.5│ 6
├────┼────┼────┤
a2 │ 5.5│ 6 │ 6.5│ 6
└────┴────┴────┘
5.5 6 6.5 6
The AC interaction is seen in the
following.
c1 c2
┌────┬────┐
a1 │
5 │
7 │ 6
├────┼────┤
a2 │
7 │
5 │ 6
└────┴────┘
6 6 6
The BC interaction is seen in the
following.
b1 b2 b3
┌────┬────┬────┐
c1 │
5 │
6 │
7 │ 6
├────┼────┼────┤
c2 │
6 │
6 │
6 │ 6
└────┴────┴────┘
5.5 6 6.5 6
The ABC three-way interaction is not
significant because the simple interaction of AB does not change over levels of
C. In this case the lines are parallel
in both cases.
Two More Examples
Because a three-way interaction does not
always appear as intuitive to students, two additional examples three-way
interactions are now given. In the
first case, the three-way interaction is not significant because the
relationships between the lines in the simple interactions do not change. In the second example, only the three-way
interaction is significant.
c1
c2
b1 b2 b3 b1 b2
b3
┌────┬────┬────┐ ┌────┬────┬────┐
a1 │
5 │
5 │
5 │ 5 │
7 │
7 │
7 │ 7
├────┼────┼────┤
├────┼────┼────┤
a2 │
5 │
7 │
9 │ 7 │
3 │
5 │
7 │ 5
└────┴────┴────┘
└────┴────┴────┘
5 6 7 6 5 6 7 6
The reader should verify that in the above
example there might be a significant main effect of B, an AB interaction,
and an AC interaction, but no other effects could be significant.
c1
c2
b1 b2 b3 b1 b2
b3
┌────┬────┬────┐ ┌────┬────┬────┐
a1 │
4 │
6 │
8 │ 6 │
8 │
6 │
4 │ 6
├────┼────┼────┤
├────┼────┼────┤
a2 │
8 │
6 │
4 │ 6 │
4 │
6 │
8 │ 6
└────┴────┴────┘
└────┴────┴────┘
6 6 6 6 6 6 6 6
In the above example, only the three-way
interaction could be significant. There
could be no other significant effects.
Expected Mean Squares
As describe previously, each term in the
score model has expected mean squares terms which determine which mean square
term is used as an error term to test the significance of an effect. The same four designs are used to illustrate
expected mean squares, but the terms are rearranged to show how the various
terms in each model are tested.
A X B X C
A‑ όE + BCόA
B‑ όE + ACόB
C‑ όE + ABόC
AB‑ όE + CόAB
AC‑ όE + BόAC
BC‑ όE + AόBC
ABC‑
όE + όABC
S-ABC σE
S ( A X B ) X C
A‑ όE + CόS + SBCόA
B‑ όE + CόS + SACόB
AB‑ όE + CόS + SCόAB
S‑AB όE + CόS
C‑ όE + όCS + SABόC
AC‑ όE + όCS + SBόAC
BC‑ όE + όCS + SAόBC
ABC‑ όE + όCS + SόABC
CS‑AB όE + όCS
S ( A ) X B X C
A‑ όE + BCόS + SBCόA
S‑A όE + BCόS
B‑ όE + CόBS + SACόB
AB‑ όE + CόBS + SCόAB
BS‑A όE + CόBS
C‑ όE + BόCS + SABόC
AC‑ όE + BόCS + SBόAC
CS‑A όE + BόCS
BC‑ όE + όBCS + SAόBC
ABC‑
όE + όBCS + SόABC
BCS‑A
όE + όBCS
S
X A X B X C
A‑ όE + BCόAS + SBCόA
AS‑ όE + BCόAS
B‑ όE + ACόBS + SACόB
BS‑ όE + ACόBS
C‑ όE + ABόCS + SABόC
CS‑ όE + ABόCS
S‑ όE + ABCόS
AB‑ όE + CόABS + SCόAB
ABS‑
όE + CόABS
AC‑ όE + BόACS + SBόAC
ACS‑ όE + BόACS
BC‑ όE + AόBCS + SAόBC
BCS‑
όE + AόBCS
ABC‑
όE + όABCS + SόABC
ABCS‑
όE + όABCS
Tests of Significance
Error Terms
The error term used to test an effect will
differ depending upon the design of the experiment. As can be seen from the above EMS for the various designs, if a
term to be tested has any factor which is crossed with subjects, then the
interaction with subjects is used as an error term. For example, in the S(AXB)XC design,
the AB term is tested using the S-AB or subjects mean square,
while the ABC term is tested using the CS-AB mean square.
SPSS Output
The greater the number of factors that are
crossed with subjects (S), the more output the MANOVA package within
SPSS will provide. For every factor
crossed with subjects the program does both a test of the assumptions (Mauchly
Sphericity Test) and three multivariate tests of significance in addition to
the univariate tests of significance.
The output from an SXAXBXC design can be
somewhat forbidding. An example of an
experiment employing each of the functional three-factor designs will now be
presented in addition to the data file, MANOVA commands, and example output.
Examples
S ( A X B X C)
S ( A X B ) X C
S ( A ) X B X C
In its full form, the experiment designed to
test the Lombard effect in choral singers neatly fulfill this design. Factor A was choral singing
experience - None, Some, or Lots. Each
subject sang the national anthem twice, Factor B, the first time with no
instructions and the second with instructions to try to control vocal
intensity. Singing loudness (in
decibels) was measured at three different points (Factor C) in the song.
The data has been presented in its full form
in a previous chapter (Design SXAXB), so it is not necessary
to present it again. The MANOVA
commands necessary to do the analysis is given below. Note that the only difference between this set of commands and
the set of commands do the SXAXB analysis the BY EXPER (1,3) on the
first line following the MANOVA command.
MANOVA I1P1 TO I2P3 BY EXPER (1,3)
|
/WSFACTORS INSTRUCT (2)
PLACE (3)
/WSDESIGN
/PRINT CELLINFO(MEANS)
/DESIGN.
|
The means and standard deviations are
presented first.
Cell Means and Standard Deviations
|
Variable .. I1P1
FACTOR CODE Mean Std.
Dev. N 95 percent Conf. Interval
EXPER
NONE 84.000 4.387 9
80.627 87.373
EXPER
SOME 84.111 3.951 9
81.074 87.148
EXPER
LOTS 86.000 5.339 9
81.896 90.104
For entire sample 84.704 4.513
27 82.918 86.489
Variable .. I1P2
EXPER
NONE 86.222 6.016 9
81.598 90.847
EXPER
SOME 87.111 3.333 9
84.549 89.673
EXPER
LOTS 88.222 5.783 9
83.777 92.668
For entire sample 87.185 5.054
27 85.186 89.184
Variable .. I1P3
EXPER
NONE 86.111 6.679 9
80.977 91.245
EXPER
SOME 86.667 2.345 9
84.864 88.469
EXPER
LOTS 88.333
7.053 9 82.912 93.755
For entire sample 87.037 5.626
27 84.811 89.263
Variable .. I2P1
EXPER
NONE 79.778 4.147 9
76.590 82.965
EXPER
SOME 81.000 5.635 9 76.669 85.331
EXPER
LOTS 82.556 6.502 9
77.558 87.554
For entire sample 81.111 5.423
27 78.966 83.256
Variable .. I2P2
EXPER
NONE 80.556 5.725 9 76.155 84.956
EXPER
SOME 80.556 4.065 9
77.431 83.681
EXPER
LOTS 84.111 7.656 9
78.226 89.996
For entire sample 81.741 6.010
27 79.363 84.118
Variable .. I2P3
EXPER
NONE 82.333 6.671 9
77.206 87.461
EXPER
SOME 80.667 3.969 9
77.616 83.717
EXPER
LOTS 85.333 6.576 9
80.278 90.388
For entire sample 82.778 5.976
27 80.414 85.142
|
The results of the hypothesis tests from the
preceding set of commands is presented below in its full form.
* * * * * * * * * A N A L Y S I S O F V A R I A N C E * *
* * * * * * * *
Tests of Between‑Subjects Effects.
Tests of Significance for T1 using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
3527.19 24 146.97
EXPER
225.93 2 112.96 .77 .475
* * * * * * * * * A N A L Y S I
S O F V A R I A N C E * * * * * * * * * *
Tests involving 'INSTRUCT' Within‑Subject Effect.
Tests of Significance for T2 using UNIQUE sums of squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
260.37 24 10.85
INSTRUCT
795.56 1 795.56
73.33 .000
EXPER BY INSTRUCT
19.90 2 9.95 .92 .413
Tests involving 'PLACE' Within‑Subject Effect.
Mauchly sphericity test, W =
.51268
Chi‑square approx. = 15.36643 with 2 D. F.
Significance = .000
Greenhouse‑Geisser Epsilon = .67235
Huynh‑Feldt Epsilon = .75715
Lower‑bound Epsilon = .50000
AVERAGED Tests of Significance that follow multivariate tests are
equivalent to
univariate or split‑plot or mixed‑model approach to
repeated measures.
Epsilons may be used to adjust d.f. for the AVERAGED results.
* * * * * * * * * A N A L Y S I S O F V A R I A N C
E * * * * * * *
EFFECT .. EXPER BY PLACE
Multivariate Tests of Significance (S = 2, M = ‑1/2, N = 10
1/2)
Test Name
Value Approx. F Hypoth. DF Error DF Sig. of F
Pillais
.06553 .40650 4.00 48.00
.803
Hotellings
.06957 .38266 4.00 44.00
.820
Wilks
.93472 .39481 4.00 46.00
.811
Roys
.06147
Note.. F statistic for WILK'S Lambda is exact.
* * * * * * * * * A N A L Y S I
S O F V A R I A N C E * * * * * * * * * *
EFFECT .. PLACE
Multivariate Tests of Significance (S = 1, M = 0, N = 10 1/2)
Test Name
Value Exact F Hypoth. DF Error DF
Sig. of F
Pillais
.24300 3.69164 2.00 23.00
.041
Hotellings
.32101 3.69164 2.00 23.00
.041
Wilks
.75700 3.69164 2.00 23.00
.041
Roys
.24300
Note.. F statistics are exact.
* * * * * * * * * A N A L Y S I S O F V A R I A N C E * *
* * * *
Tests involving 'PLACE' Within‑Subject Effect.
AVERAGED Tests of Significance for MEAS.1 using UNIQUE sums of
squares
Source of Variation
SS DF MS F
Sig of F
WITHIN CELLS
437.04 48 9.10
PLACE
119.11 2 59.56 6.54 .003
EXPER BY PLACE
11.85 4 2.96 .33 .860
* * * * * * * * * * A N A L Y S I S O F V A R I A N C
E * * * * *
Tests involving 'INSTRUCT BY PLACE' Within‑Subject Effect.
Mauchly sphericity test, W =
.91778
Chi‑square approx. = 1.97333 with 2 D. F.
Significance = .373
Greenhouse‑Geisser Epsilon = .92403
Huynh‑Feldt Epsilon = 1.00000
Lower‑bound Epsilon = .50000
AVERAGED Tests of Significance that follow multivariate tests are
equivalent to
univariate or split‑plot or mixed‑model approach to
repeated measures.
Epsilons may be used to adjust d.f. for the AVERAGED results.
* * * * * * * * A N A L Y S I
S O F V A R I A N C E * * * *
* * * * *
EFFECT .. EXPER BY INSTRUCT BY PLACE
Multivariate Tests of Significance (S = 2, M = ‑1/2, N = 10
1/2)
Test Name
Value Approx. F Hypoth. DF Error DF
Sig. of F
Pillais
.17727 1.16704 4.00 48.00
.337
Hotellings .20882 1.14852
4.00 44.00 .346
Wilks
.82521 1.15950 4.00 46.00
.341
Roys
.16200
Note.. F statistic for WILK'S Lambda is exact.
* * * * * * * * A N A L Y S I
S O F V A R I A N C E * * * * * * *
EFFECT .. INSTRUCT BY PLACE
Multivariate Tests of Significance (S = 1, M = 0, N = 10 1/2)
Test Name
Value Exact F Hypoth. DF Error DF
Sig. of F
Pillais
.29476 4.80658 2.00 23.00
.018
Hotellings
.41796 4.80658 2.00 23.00
.018
Wilks
.70524 4.80658 2.00
23.00 .018
Roys
.29476
Note.. F statistics are exact.
* * * * * * * * A N A L Y S I S
O F V A R I A N C E * * * * * *
*
Tests involving 'INSTRUCT BY PLACE' Within‑Subject Effect.
AVERAGED Tests of Significance for MEAS.1 using UNIQUE sums of
squares
Source of Variation
SS DF MS F Sig of F
WITHIN CELLS
148.30 48 3.09
INSTRUCT BY PLACE 23.75
2 11.88 3.84 .028
EXPER BY INSTRUCT BY
18.62 4 4.65 1.51 .215
PLACE
From the preceding analysis the results are
reasonably clear. Main effects were
found for the INSTRUCT and PLACE factors.
When instructed, the singers sang with less intensity. In different places in the song the singers
also sang with different intensity levels over both levels of instructions.
The significant interactions of EXPER BY
PLACE and INSTRUCT BY PLACE are best seen in the three-way interaction, even
though the three-way interaction was not significant. The graph of the EXPER BY INSTRUCT BY PLACE interaction is
presented below.
From the above graphs it can be seen that the
more experienced the singer, the more loudly he or she sang. In addition, experienced singers sang
relatively more loudly than inexperienced singers at song positions 2 and
3. Also, the instructions to sing at an
even level seemed to work best at position 2, combined over all groups.
For an explanation of what all this means, it
is perhaps best to refer to the original author (Tonkinson,1990, p. 25)
The original problem was: Is the Lombard effect, to a significant
degree, and unconscious response in choral singing at different levels of
experience and training, and can it be consciously avoided. Most of the choral singers in this study,
regardless of experience, tended to succumb to a Lombard effect when faced with
increasing loss of auditory feedback.
They were, however, able to control the level of vocal intensity with
some brief instructions. It appears
that simple training in awareness would be enough for a member of an amateur
choir to begin regulating the intensity of their voice in a healthy manner.
BIBLIOGRAPHY
CMS User's Manual. (1992). SMSU.
Electronic Mail User's Guide. (1992). SMSU.
Hays, W. L. (1981). Statistics (3rd ed.).
New York: Holt, Rinehart and Winston.
Kerlinger, F. N. (1973). Foundations of Behavioral Research (2nd
Edition). New York: Holt, Rinehart, and Winston.
Letcher, John S. Jr., Marshall, John K.,
Oliver, James C. III, and Salvesen, Nils.
(1987) Stars & Stripes. Scientific American, 257(2), 34-40.
McGuire, L. C. Adults' Recall and Retention of Medical Information: What Does the Patient Really Remember? Dissertation Abstracts, 1993.
Michell, J. (1986). Measurement Scales and Statistics: A Clash of Paradigms. Psychological
Bulletin, 100-3, 398-407.
Norusis, Marija. (1990). The SPSS Guide to Data Analysis for
Release 4. Chicago: SPSS, Inc.
Reed, Valerie. The Student's Knowledge of United States Geography. A project done for PSY 200. Southwest Missouri State University, 1983.
SPSS Reference Guide.
(1990). Chicago: SPSS, Inc.
Stevens, S. S. (1951). Mathematics, measurement and
psychophysics. In S. S. Stevens(Ed.), Handbook
of experimental psychology (pp. 1-49).
New York: Wiley.
Tonkinson, S. E. The Lombard Effect in Choral Singing. Dissertation Abstracts, 1990.
Torgerson, W. S. (1967). Theory and Methods of Scaling. New York:
Wiley.
Tufte, Edward R. (1983). The Visual Display of Quantitative
Information. Cheshire, Conn: Graphics Press.
Winer, B. J. (1971). Statistical Principles in Experimental
Design. New York: McGraw-Hill.
dummy
codes......................... 232
MANOVA command............. 166
multiple regression.......... 219,
222
orthogonal transformations...... 214
SPSS
A X B.................................. 173
t-test...................................... 170
Table
of Means...................... 194
theory-driven.......................... 171
/DESIGN................ 166,
226, 228
/METHOD
SEQUENTIA....................... 226
SEQUENTIAL............ 226,
228
UNIQUE............................. 226
/METHOD = SEQUENTIAL. 226
/METHOD = UNIQUE.......... 226
/METHOD=SEQUENTIAL... 233
/METHOD=TEST.................. 234
/METHOD=UNIQUE............ 232
/METHODS=UNIQUE.......... 231
A X B..................................... 172
Analysis of variance................. 214
ANOVA 132,
138, 158, 163, 164, 199, 214, 216, 217, 219, 229
single-factor.......................... 203
summary table....................... 176
ANOVA table 151,
152, 193, 195, 203‑205, 207, 241
Assumption
within-subjects designs.......... 244
Assumptions
A X B X C........................... 258
AVOVA table......................... 195
B WITHIN A.......................... 195
B(A)............................... 193,
207
B(A) ...................................... 141
Balanced designs............. 225, 230
Between groups....................... 173
Between subjects..... 142, 173, 261
Between subjects effects.......... 242
Between subjects factors......... 149
Blocking factor................ 136, 140
Bolded capital letters............... 133
Bulimia.................................... 146
BY.......................................... 166
Carry-over effects................... 140
categorical............................... 224
Causal inferences..................... 136
Cell means............... 175,
190, 233
Central Limit Theorem...... 154‑156
Choir....................................... 145
Collapsing............................... 177
Complex experiments.............. 149
constant term........................... 217
Contrast
difference.............................. 210
non-orthogonal............. 199, 208
orthogonal.................... 198,
200
polynomial............................ 211
simple................................... 210
Contrasts................................. 197
CORRELATION.................... 239
correlation matrix 217, 222, 234,
240, 244, 251
Correlations............................. 258
Counter variable...................... 168
Critical value............................ 152
Crossed.................................. 139
Data file 165,
173, 215, 239, 249, 261
Data files................................. 165
DATA LIST............ 173,
239, 250
Data matrix...................... 148,
222
Data table................................ 190
data-driven.............................. 171
default..................................... 227
/METHOD........................... 227
degrees of freedom 151, 206, 222, 243,
258
orthogonal contrasts.............. 204
Dependent measure................. 147
Dependent variable.......... 133, 165
continuous............................ 214
DESIGN 174,
193, 240, 251, 262
unbalanced........................... 226
Designs
unbalanced........................... 225
Df........................................... 151
Difference contrasts................. 210
discrete variables..................... 215
discriminant analysis................. 223
Dot notation.................... 168,
190
Dots........................................ 189
dummy codes.......................... 234
dummy coding 214,
217, 222, 223
Duncan’s Multiple Range test... 170
E............................................. 137
ED-STAT....................... 225,
227
Effects............. 132,
152, 197, 268
EMS............... 246,
247, 259, 279
Error term 247,
260, 267, 276, 279
Essay...................................... 172
Exact F................................... 265
Example
non-orthogonal contrasts....... 208
Expected mean square 246, 259, 260, 267
Expected mean squares 245, 247, 258,
267, 276
Experiment-wise error rate....... 151
Experimental design 132, 143, 145
experimentwide error rate........ 170
F-distribution................... 159, 162
F-ratio 151,
157‑160, 162, 167, 245, 247, 258,
260, 267
Factor..................... 132,
214, 268
fixed..................................... 137
random................................. 138
Factor name............................ 166
Factors.................................... 133
crossed................................. 139
Fcrit................................ 160,
164
Fixed....................................... 137
FOBS..................................... 164
G............................................ 135
Gender.................................... 136
general linear model................. 223
Geometrical perspective........... 231
Geometrical perspective .......... 231
Grand mean..................... 175,
254
Grand mean ............................ 190
Graph...................................... 264
Graphical presentation............. 153
Greenhouse-Geisser Epsilon.... 243
Group factor............................ 135
Hierarchical design................... 141
Hierarchical designs................. 192
Hierarchical designs ................ 141
Homer & Tessie...................... 258
Huynh-Feldt Epsilon................ 243
hypothesis test......................... 219
Hypothesis testing............ 132, 158
Hypothesis testing theory......... 246
independence.................. 157,
222
Independent effects.................. 225
Independent variable 132, 165, 214, 223
independent variables............... 222
Infinite..................................... 259
Interaction 178,
206, 226, 237, 253, 264, 272, 279, 285
simple................................... 275
simple two-way.................... 271
three-way 268,
271, 273, 275, 285
two-way............................... 269
Interaction effect...... 222, 260, 267
Interaction term....................... 226
Interactions
three-way............................. 271
interval.................................... 224
Interval scale........................... 211
Large standard deviation.......... 188
Levels of a factor..................... 133
Linear effects........................... 231
Linear trend..................... 211,
212
LIST............................... 166,
250
Lombard effect 144, 238, 249, 261,
280
Main effect 206,
222, 226, 243, 265, 267
simple........................... 178,
207
Main effects 177,
255, 269, 285
nested................................... 194
MANOVA 132,
137, 174, 175, 202, 212, 226, 232, 233, 236, 239, 250, 252, 262, 263, 279, 280
/METHOD=TEST................ 236
contrasts............................... 209
METHOD=SEQUENTIAL.. 235
MANOVA command.............. 164
Marginal means....... 175, 177, 233
matrix
transpose.............................. 217
Mauchly sphericity test 243, 245, 255, 256, 265, 279
Mean
cell....................................... 175
grand.................................... 175
Marginal............................... 175
Mean Square................... 151,
247
Mean Square Between............ 161
Mean Square Within................ 161
Mean Squares Between........... 157
MEAN SQUARES WITHIN.. 155
means.............................. 175,
214
unweighted........................... 233
weighted............................... 233
MEANS command.................. 164
Means squares........................ 246
Measures per subject............... 148
Mixed design........................... 263
Model............................. 162,
229
adjusted................................ 230
MS................................. 151,
258
MSB ...................................... 161
MSW...................................... 161
Multiple regression 197, 214, 215,
217, 223, 231
ANOVA.............................. 237
Multiple t-tests......................... 150
Multivariate............................. 254
Multivariate ANOVA.............. 165
Multivariate approach.............. 245
Multivariate test....................... 256
Multivariate tests.............. 243, 265
Multivariate tests of significance 255, 265, 279
Negative numbers.................... 229
Nested.................... 140,
147, 192
Nested main effect................... 195
Nested main effects................. 194
Newman-Keuls....................... 170
Nichols, David......................... 230
Non-orthogonal contrasts 199, 209
Non-parallel............................ 271
Non-significant........................ 203
Nonsignificant effects............... 152
Notational system............ 133, 169
Number of subjects................. 148
orthogonal............................... 234
orthogonal contrast.......... 198, 217
Orthogonal contrasts 199, 200, 213, 215‑217,
219, 220
sum of squares...................... 204
orthogonal transformations....... 214
Parallel.................................... 178
Parameter................................ 156
Parameters...... 153,
154, 159, 178
Parsimonious........................... 200
Part correlations...................... 231
Polynomial Contrasts
example................................ 212
post-hoc.................................. 170
Pre-existing ............................ 135
pre-planned contrasts.............. 170
PRINT.................................... 252
PRINT = CELLINFO(MEANS) 240, 251, 262
PRINT CELLINFO(MEANS) 174
PRINT=CELLINFO(MEANS 166
Probability models................... 153
Probed recall........................... 143
Quadratic trend............... 211, 212
Random factor......................... 138
Ratio....................................... 243
Real effect............................... 177
Real effects.............. 152,
160, 161
Reality Therapy ...................... 150
recoding.................................. 217
Regression....................... 229,
234
/METHOD=TEST 234, 235
METHOD=TEST................. 234
Repeated measures.......... 238, 249
Repeated measures designs...... 140
residual vector......................... 231
Reverse Helmert contrast......... 210
S............................................. 137
S ( A )
X B............................ 261
S X A..................................... 238
S X A X B.............................. 249
S(A)........................................ 142
Sample statistics.............. 153, 155
Sampling distribution 153‑155, 161
Sampling distribution of the mean 154
Score model 225,
238, 245, 246, 258, 276
SEQUENTIAL....... 226,
227, 230
Set of contrasts........................ 198
Set of orthogonal contrasts....... 204
polynomial trends.................. 211
Sheffe’.................................... 170
Sig.......................................... 151
Sig of F........................... 152,
177
Significant................ 177,
203, 272
Significant effects..................... 152
Significantly different................ 164
Simple effects.......................... 195
Simple main effect............ 178, 269
Simple two-way interaction...... 271
Small letters............................. 133
Solitaire................................... 133
Source............................ 151,
177
Sphericity test.................. 243, 256
SPSS 164,
165, 193, 201, 202, 213, 215, 225, 227,
254, 260, 267, 279
MANOVA........... 219,
220, 223
REGRESSION............ 218,
222
SS........... 151,
204, 228, 245, 258
non-orthogonal contrasts....... 209
orthogonal contrasts.............. 204
WITHIN.............................. 208
Standard error of the mean...... 156
Star Spangled Banner.............. 238
Straight line.............................. 211
Subject numbers...................... 239
Subjects.......................... 132,
137
Subjects main effect................. 260
Subscript................................. 168
Subscripted............................. 248
Subscripts....................... 133,
189
Sum of Squares............... 151, 227
non-orthogonal contrasts....... 208
total - balanced designs......... 225
Summary table................. 174,
177
Summation...................... 168,
189
Summation sign........................ 189
Sums of squares.............. 230, 231
Swimsuit issue......................... 146
T............................................. 135
T-distribution........................... 159
T-test.............................. 159,
163
Table of means 175, 202, 268, 272,
274
Tables of Means...................... 193
Theoretical probability distribution 159
Three-way interaction...... 268, 276
Treatment factors..................... 134
Treatments...................... 132,
134
Trend...................................... 211
Trials factor............................. 135
Two-tailed test........................ 163
Two-way interactions.............. 269
Type I error............................. 151
Unbalanced............................. 227
Unbalanced design................... 225
Unbalanced designs 225, 227, 230
uncorrelated............ 217,
222, 234
unequal cell sizes.............. 224, 237
Unequal numbers..................... 230
UNIQUE......................... 226‑230
Unit factor............................... 137
Univariate........................ 254,
265
Univariate approach................. 245
VALUE LABELS........... 174,
250
vectors.................................... 231
Vocal intensity......................... 145
WARNING............................ 166
Warning message..................... 262
WITHIN................................. 193
WITHIN CELLS............ 260,
267
Within method......................... 155
Within subjects 140, 238, 249, 261,
265
Within subjects design
assumptions.......................... 243
Within subjects designs............ 238
Within subjects factor.............. 262
Within subjects factors............. 149
WSDESIGN........... 240,
251, 262
WSFACTORS....................... 251
WSFACTORS = 239,
250, 262
X............................................ 139
X-axis..................................... 176
