SPSS Library: Advanced issues in understanding and using SPSS MANOVA

Once the ONEWAY basis matrices have been specified, the overall basis matrix is
constructed from Kronecker products of the column partitions of the ONEWAY matrices:

BASIS AB = [ A  x B  | A  x B  | A  x B  | A  x B  ] 
              1    1    2    1    1    2    2    2 
 
where A  and B  represent the ith column partitions of the A and B basis 
       i      i 
matrices. The result is the overall basis matrix printed by MANOVA: 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Between-Subjects Design Matrix (Reduced Model) 
  1-A  2-B 
 Factor                                    PARAMETER 
 
    1   2              1         2         3         4         5         6 
 
    1   1        1.00000   1.00000   1.00000    .00000   1.00000    .00000 
    1   2        1.00000   1.00000    .00000   1.00000    .00000   1.00000 
    1   3        1.00000   1.00000  -1.00000  -1.00000  -1.00000  -1.00000 
    2   1        1.00000  -1.00000   1.00000    .00000  -1.00000    .00000 
    2   2        1.00000  -1.00000    .00000   1.00000    .00000  -1.00000 
    2   3        1.00000  -1.00000  -1.00000  -1.00000   1.00000   1.00000 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

The resulting overall contrast matrix could be derived from the formula given earlier,
or constructed from Kronecker products of the row partitions of the A and B contrast
matrices (in a manner similar to that shown for the overall basis construction). The
resulting overall contrast matrix is:

             _                 _ 
            |  1  1  1  1  1  1 | 
        1   |  1  1  1 -1 -1 -1 | 
        -   |  2 -1 -1  2 -1 -1 | 
        6   | -1  2 -1 -1  2 -1 | 
            |  2 -1 -1 -2  1  1 | 
            |_-1  2 -1  1 -2  1_| 

Factoring out the divisor of 6 which applies to all terms allows us to see the nature
of the contrasts more easily. The CONSTANT term is simply the unweighted grand mean:

             __    __    __    __    __    __            _   _ 
CONSTANT = ( AB  + AB  + AB  + AB  + AB  + AB  ) / 6 = ( A + A ) / 2 = 
               11    12    13    21    22    23           1   2 
             _   _   _ 
           ( B + B + B ) / 3 = 6 
              1   2   3 
The A contrast measures the deviation of level 1 of A from the grand mean: 
      __    __    __    __    __    __          
A = ( AB  + AB  + AB  - AB  - AB  - AB  ) / 6 = 
 1      11    12    13    21    22    23        
      _   _         _   __ 
    ( A - A ) / 2 = A - AB = -1   
       1   2         1 
 
The B contrasts measure the deviations of levels 1 and 2 of B from the grand mean: 
       __    __    __     __    __    __          
B = ( 2AB  - AB  - AB  + 2AB  - AB  - AB  ) / 6 =  
 1       11    12    13     21    22    23       
       _   _   _           _   __ 
    ( 2B - B - B ) / 3 = ( B - AB ) = 1 
        1   2   3           1 
       __     __    __    __     __    __         
B = (- AB  + 2AB  - AB  - AB  + 2AB  - AB  ) / 6 = 
 2       11     12    13    21     22    23         
       _    _   _         _   __ 
    (- B + 2B - B ) / 3 = B - AB = -.5 
        1    2   3         2 

The interaction contrasts are specified by the main effect contrasts. The first
contrast tests whether the deviation of level 1 of factor A is the same at level 1 of
factor B as at levels 2 and 3 of factor B, which is equivalent to testing whether the
deviation of level 1 of factor B is the same at both levels of factor A. The last contrast
tests whether the deviation of level 1 of factor A is the same at level 2 of B as at
levels 1 and 3, which is equivalent to testing whether the deviation of level 2 of factor
B is the same at both levels of factor A. In comparing these formulas with the output
given below, note that the parameter numbering in MANOVA is hard coded and cannot be
changed by the user.

        __    __    __     __    __    __ 
AB = ( 2AB  - AB  - AB  - 2AB  + AB  + AB  ) / 6 = 
  1       11    12    13     21    22    23 
           __    _       __    _       __    _ 
     [ 2 ( AB  - B ) - ( AB  - B ) - ( AB  - B ) ] / 3 =  
             11   1        12   2        13   3 
         __    _       __    _ 
     [ ( AB  - A ) - ( AB  - A ) ] / 2 = 0 , 
           11   1        21   2 
        __     __    __    __     __    __ 
AB = (- AB  + 2AB  - AB  + AB  - 2AB  + AB  ) / 6 = 
  2       11     12    13    21     22    23 
           __    _       __    _       __    _ 
     [ 2 ( AB  - B ) - ( AB  - B ) - ( AB  - B ) ] / 3 =  
             12   2        11   1        13   3 
         __    _       __    _ 
     [ ( AB  - A ) - ( AB  - A ) ] / 2 = .5 . 
           12   1        22   2 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Estimates for Y1 
 --- Individual univariate .9500 confidence intervals 
 CONSTANT 
 
  Parameter     Coeff.  Std. Err.    t-Value     Sig. t Lower -95%  CL- Upper 
 
        1   6.00000000     .49786   12.05161     .00000    4.92444    7.07556 
 A 
 
  Parameter     Coeff.  Std. Err.    t-Value     Sig. t Lower -95%  CL- Upper 
 
        2   -1.0000000     .49786   -2.00860     .06582   -2.07556     .07556 
 B 
 
  Parameter     Coeff.  Std. Err.    t-Value     Sig. t Lower -95%  CL- Upper 
 
        3   1.00000000     .72575    1.37789     .19149    -.56788    2.56788 
        4   -.50000000     .70408    -.71015     .49016   -2.02107    1.02107 
 A BY B 
 
  Parameter     Coeff.  Std. Err.    t-Value     Sig. t Lower -95%  CL- Upper 
 
        5   .000000000     .72575     .00000    1.00000   -1.56788    1.56788 
        6   .500000000     .70408     .71015     .49016   -1.02107    2.02107 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

Now that you have seen how MANOVA handles default contrasts, you might want to apply
the formulas listed earlier that relate the contrast and basis matrices to other types of
contrasts you use in your own analyses. This is only one of a number of topics on which
the SPSS MATRIX procedure can be an invaluable aid in understanding the functioning of
MANOVA and other SPSS procedures.

Appendix: Data 
 
A B Y1 
 
1 1  8 
1 1  5 
1 1  7 
1 1  4 
1 2  7 
1 2  2 
1 2  6 
1 3  4 
1 3  1 
1 3  7 
2 1  7 
2 1  9 
2 2  4 
2 2  6 
2 2  8 
2 3  7 
2 3  9 
2 3  6 
2 3  6 

TESTING SIMPLE EFFECTS IN MANOVA
David P. Nichols
Senior Support Statistician
SPSS, Inc.
From SPSS Keywords, May 1993

Factorial designs in analysis of variance and covariance, including designs with within
subjects factors, are very common in many fields of research. The SPSS MANOVA procedure
provides a powerful and flexible set of tools for performing most of the analyses that are
available under the general linear model framework. A very common problem is that of an
experiment in which interactions have been found and the researcher wants to explore the
data more carefully to determine what statements may be made about main effects or
interactions in the presence of the two-way or higher order interaction effects. Tests of
such simple main effects or simple interaction effects are generally easily handled in
MANOVA through the flexibility in model specification offered by the DESIGN and WSDESIGN
subcommands.

Two-way Between Subjects Models: Estimating Simple Main Effects 
                          

Let’s begin with the simplest case in which we might want to test for simple effects: a
two-way factorial design in which we have found an interaction effect. If the two factors
are both between subjects factors and have two and three levels respectively, we might
have the following syntax for the factorial analysis:

MANOVA Y BY A(1,2) B(1,3).

This one line of syntax will produce the full factorial analysis (MANOVA always does a
full factorial model by default), equivalent to specifying either

MANOVA Y BY A(1,2) B(1,3) 
 /DESIGN 
 
or 
 
MANOVA Y BY A(1,2) B(1,3) 
 /DESIGN=A, B, A BY B.

If the A by B interaction term is nonzero, the effects of the two factors A and B are
not the same across all levels of the other factor. That is, it is possible for A to have
a positive effect on the dependent variable at one level of B, no effect at another level
and a negative effect at a third level. Such a situation might lead to an overall main
effects test for A in which no evidence of any A effect was discovered. This is because
the effect of A is confounded with the A by B interaction effect. It is also possible that
A has a positive (or negative) effect at each level of B, but that this effect is stronger
at some levels of B than at others. In this case it does make sense to talk about an
overall positive (or negative) main effect for factor A, but discussion of the magnitude
of this effect must be conditioned on the particular levels of the B factor within which
these effects do not differ.

In each of these cases what is called for is to examine the effects of the A factor
separately within each level of the B factor. These effects are what are known as simple
main effects. Specification of such effects in MANOVA is simple, following a logical
algorithm applied to our model specifications on the DESIGN subcommand. The general
algorithm is as follows: To obtain the proper simple effects estimates and tests of one
factor at (within) each level of a second factor, replace the main effect of the factor of
interest and the two-way interaction involving these two factors with the simple effects
of the factor of interest within each level of the other factor. For our example, we would
replace the main effect of factor A and the A by B interaction with the simple effects of
factor A at each level of factor B:

MANOVA Y BY A(1,2) B(1,3) 
 /DESIGN=B, 
         A W B(1), A W B(2), A W B(3)

where the W operator is an acceptable shorthand for the WITHIN keyword.

Important notes to keep in mind here are the following: We have simply removed the main
effect of the A factor and the A by B interaction term from a full factorial specification
and have replaced them with a request for the simple effects of A within (separately for)
each level of factor B. The effect of this substitution is to repartition the same overall
model into different effects, but to maintain the same total model (total degrees of
freedom, total sums of squares accounted for, same predicted values and residuals, etc).
That is, we are estimating the same B main effect as in the original full factorial model,
and repartitioning the A main effect and the A by B interaction effect into the simple
main effects of A at each level of B.

This is important to note for two reasons. First, when working with data with unequal
numbers of observations in the cells of the design (generally referred to as unbalanced
data), the sums of squares for a particular effect such as A W B(1) will generally not be
the same when specified alone on the DESIGN subcommand as when specified as part of a
larger model, due to the intercorrelation among the factors in an unbalanced design. The
algorithm outlined here is designed to maintain the same overall model throughout the
testing of simple effects so that the simple effects estimated are logical followups to
the results of the overall full factorial analysis. Second, in releases beginning with
version 5.0 the default error term in MANOVA has been changed from WITHIN CELLS to
WITHIN+RESIDUAL. Thus even in balanced designs, the error term and degrees of freedom used
for testing simple effects would not be the same as in the original analysis unless the
same overall model was estimated or unless the user explicitly specified /ERROR=WITHIN on
the ERROR subcommand.

So far we have talked only in terms of the simple main effects of A at each level of B.
However, the implications of an interaction effect are completely symmetric. That is, to
say that the effects of factor A are different at different levels of factor B is
equivalent to saying that the effects of factor B are different at different levels of
factor A. Thus we would probably also want to test the simple main effects of B at each
level of A. To do this we would simply follow the same algorithm, reversing the role of
factor A and factor B. That is, we remove the main effect of factor B from the full
factorial specification, along with the A by B interaction and substitute the simple main
effects of B at each level of A. Our syntax would thus be:

MANOVA Y BY A(1,2) B(1,3) 
 /DESIGN=A, 
         B W A(1), B W A(2).

One important point to note is that since the A and B simple main effects each involve
a repartitioning of the interaction term, attempting to fit both sets of simple main
effects on one DESIGN subcommand would introduce redundant effects and should thus be
avoided. Estimation of both sets of simple main effects in one MANOVA run can be
accomplished simply by stacking two DESIGN subcommands:

MANOVA Y BY A(1,2) B(1,3) 
 /DESIGN=B, A W B(1), A W B(2), A W B(3) 
 /DESIGN=A, B W A(1), B W A(2).

General Between Subjects Models: Estimating Simple Main Effects

The algorithm outlined above generalizes immediately to cases of higher order designs.
Let’s illustrate with the case of a three-way design, with factors A, B and C. For the
sake of brevity we will assume that each factor has only two levels, since there is no
loss of generality in our discussion and this saves us from writing out more terms in our
DESIGN specifications.

If in a higher order design we wish to estimate simple main effects, the procedure is
exactly that outlined above, except that we would have other terms also listed on the
DESIGN subcommand. For example, in an A by B by C design in which we wanted to estimate
the simple effects of A at each level of B, we would perform the same replacement of main
and interaction effect terms as before, but would maintain the model specifications
involving the C factor. Thus our full factorial syntax

MANOVA Y BY A B C(1,2) 
 /DESIGN=A, B, C, A BY B, A BY C, B BY C, A BY B BY C

would become

MANOVA Y BY A B C(1,2) 
 /DESIGN=B, C, A BY C, B BY C, A BY B BY C, 
         A W B(1), A W B(2).

Since we are using UNIQUE or regression approach sums of squares, the order of effects
specified makes no difference, assuming that each cell of the design contains at least one
observation (designs involving empty cells are much more complicated and require careful
special handling).

Many statisticians might object to the foregoing simple effects tests because they are
being conducted in a model in which a higher order interaction is being estimated which
contains the effects in question. The logic behind this objection would be that first we
should test the three-way interaction. If this is significant we should then proceed to
test simple, simple main effects and/or simple interaction effects. If it is not
significant, remove the three-way interaction and re-estimate the model. That is,
follow-up tests on simple effects should not be performed until a final model has been
chosen. The algorithm outlined here is not affected by this approach. We would have first
re-estimated the model without a three-way interaction term, as

MANOVA Y BY A B C(1,2) 
 /DESIGN=A, B, C, A BY B, A BY C, B BY C

and the same substitutions would apply, resulting in

MANOVA Y BY A B C(1,2) 
 /DESIGN=B, C, A BY C, B BY C, 
         A W B(1), A W B(2).

Others would consider this approach somewhat rigid. That is, though an interaction
effect in a sample was not of sufficient magnitude to provide evidence at (say) the .05
alpha level of an interaction effect in the population, the assumption of no interaction
effect as opposed to a small one might be presumptuous. Thus another strategy would be to
fit the simple effects in the context of the overall factorial model, estimating them in
the presence of the estimated questionable interaction effects. Each user is responsible
for coming to her or his own conclusions as to what procedures should be followed in this
case; MANOVA can be made to analyze the data in either case.

If the three-way interaction had been significant in the above model, we would be faced
with a more complicated situation. That is, not only do the effects of factor A depend on
which level of factor B we consider, but they also depend on the level of factor C in
which our A by B designation of interest is found. The logical step at this point is to
examine the two-way interactions at each level of the third factor (such as A by B within
each level of C) to see if within each level of the third factor the main effects of the
other two factors are invariant. Generalization of the algorithm discussed in the two-way
case results in the A by B and A by B by C interactions being replaced by the simple
interaction effects of A by B at each level of C. Thus

MANOVA Y BY A B C(1,2) 
 /DESIGN=A, B, C, A BY B, A BY C, B BY C, A BY B BY C

becomes

MANOVA Y BY A B C(1,2) 
 /DESIGN=A, B, C, A BY C, B BY C, 
         A BY B W C(1), A BY B W C(2).

If the simple interaction effects are nonzero, the next step is to estimate the simple,
simple main effects of say, factor A at each level of the two-way breakdown of factors B
and C. The simple, simple effects of A at each level of factors B and C involve a
repartitioning of the A main effect, the A by B, A by C and A by B by C interactions:

MANOVA Y BY A B C(1,2) 
 /DESIGN=A, B, C, A BY B, A BY C, B BY C, A BY B BY C

becomes

MANOVA Y BY A B C(1,2) 
 /DESIGN=B, C, B BY C,  
         A W B(1) BY C(1), A W B(1) BY C(2), 
         A W B(2) BY C(1), A W B(2) BY C(2).

An equivalent specification would be

MANOVA Y BY A B C(1,2) 
 /DESIGN=B, C, B BY C,  
         A W B(1) W C(1), A W B(1) W C(2), 
         A W B(2) W C(1), A W B(2) W C(2).

As with the more simple two-way case, the factors here are perfectly symmetric, so we
could just as sensibly be using B or C in place of A. Also, the substitution rules used
here generalize to designs with any number of factors.

Models Involving Within Subjects Effects 
 
 

As most users are aware, MANOVA offers the capability of using the multivariate
approach to analyzing data involving within subjects (often involving repeated measures)
effects. The within subjects part of the model is specified separately from the between
subjects part, but in an analogous manner, via the WSDESIGN subcommand. Thus a two-way
completely within subjects design involving two two-level factors A and B could be
specified as:

MANOVA V1 TO V4 
 /WSFACTORS=A(2) B(2)

which would be the same as

MANOVA V1 TO V4 
 /WSFACTORS=A(2) B(2) 
 /WSDESIGN

or

MANOVA V1 TO V4 
 /WSFACTORS=A(2) B(2) 
 /WSDESIGN=A, B, A BY B.

The estimation of simple effects in completely within subjects designs requires no new
concepts; we simply apply the same rules to the WSDESIGN subcommand that we applied to the
DESIGN subcommand. So the simple effects of A at each level of B would be specified as

MANOVA V1 TO V4 
 /WSFACTORS=A(2) B(2) 
 /WSDESIGN=B, 
           A W B(1), A W B(2).

This is also true for the more complicated three-way and higher order cases.

Models Involving Between and Within Subjects Effects

Since the between and within subjects parts of the model are specified separately in
MANOVA, the case of a design involving both between and within subjects factors presents
some complications. The specifications for each part of the model are crossed by default.
That is, all between subjects factors are automatically crossed with all within subjects
factors. Since MANOVA will not allow the specification of between subjects factors on the
WSDESIGN subcommand or within subjects factors on the DESIGN subcommand, we need a way to
tell the procedure that we want to fit the effects of a factor of one type at each level
of one or more factors of the other type. Fortunately, there is a method for doing this,
and the algorithm involved is generally no more complex than the earlier one, and in many
cases it is even simpler.

Take the case of a two-way model involving one between subjects factor (call it A) and
a within subjects factor (TIME). The standard syntax for the full factorial model is

MANOVA V1 V2 BY A(1,2) 
 /WSFACTORS=TIME(2)

which is equivalent to specifying either TIME on the WSDESIGN or A on the DESIGN
subcommand, or both. If we want to estimate the simple effects of time for each level of
A, we use the MWITHIN keyword on the DESIGN subcommand, and replace the main effect of A
with MWITHIN A(1) and MWITHIN A(2):

MANOVA V1 V2 BY A(1,2) 
 /WSFACTORS=TIME(2) 
 /DESIGN=MWITHIN A(1), MWITHIN A(2).

MWITHIN stands for mean within, and it effectively turns the crossing of A and TIME
into the nesting of time within each level of A. This case requires some special caution
in reading the output, since what we are thinking of as a simple main effect, TIME at each
level of A, is listed on the output as an interaction effect. This analysis produces two
tables, the first of which contains the between subjects part of the analysis:

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * * 
 
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+RESIDUAL           60.64      17      3.57 
 MWITHIN A(1)             441.80       1    441.80    123.85      .000 
 MWITHIN A(2)             440.06       1    440.06    123.36      .000 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

What is actually being tested by the MWITHIN A(1) and MWITHIN A(2) terms here are the
null hypotheses that the average value across all time points (represented to within a
constant multiple by transformed variable T1) is zero within level 1 and level 2 of A,
respectively. These are in general not hypotheses in which we are usually interested. The
hypotheses of common interest are to be found in the within subjects section of the
output:

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * * 
 
Tests involving 'TIME' Within-Subject Effect. 
 
 Tests of Significance for T2 using UNIQUE sums of squares 
 Source of Variation          SS      DF        MS         F  Sig of F 
 
 WITHIN+RESIDUAL           78.64      17      4.63 
 MWITHIN A(1) BY TIME       9.80       1      9.80      2.12      .164 
 MWITHIN A(2) BY TIME      20.06       1     20.06      4.34      .053 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Transformed variable T2 represents a normalized difference variable comparing the two
TIME points. Thus a test of MWITHIN A(1) BY TIME represents a test of the null hypothesis
that the TIME differences are zero at level 1 of factor A, and MWITHIN A(2) BY TIME
corresponds to a similar test at level 2 of factor A.

The substitution rule in this case is even simpler than in cases in which all factors
are either between or within subjects in nature. That is, all we had to do was to remove
the A effect from the DESIGN subcommand and replace it with MWITHIN each level of A. The
same rule applies when we want to go the other way, to look at A differences at each TIME
point:

MANOVA V1 V2 BY A(1,2) 
 /WSFACTORS=TIME(2) 
 /WSDESIGN=MWITHIN TIME(1), MWITHIN TIME(2)

produces tests of corresponding null hypotheses with the roles of the two factors
reversed. However, in this case the tables are presented somewhat differently, as all four
hypothesis degrees of freedom in the analysis are defined as within subjects effects. In
each case we have a constant or intercept term, followed by the term of interest, labeled
essentially as an interaction term.

* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * * 
 
Tests involving 'MWITHIN TIME(1)' Within-Subject Effect. 
 
 Tests of Significance for T1 using UNIQUE sums of squares 
 Source of Variation          SS      DF        MS         F  Sig of F 
 
 WITHIN+RESIDUAL           65.29      17      3.84 
 MWITHIN TIME(1)          408.71       1    408.71    106.42      .000 
 A BY MWITHIN TIME(1)      10.82       1     10.82      2.82      .112 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 
* * * * * * A n a l y s i s   o f   V a r i a n c e -- design   1 * * * * * * 
 
Tests involving 'MWITHIN TIME(2)' Within-Subject Effect. 
 
 Tests of Significance for T2 using UNIQUE sums of squares 
 Source of Variation          SS      DF        MS         F  Sig of F 
 
 WITHIN+RESIDUAL           74.00      17      4.35 
 MWITHIN TIME(2)          473.68       1    473.68    108.82      .000 
 A BY MWITHIN TIME(2)      18.95       1     18.95      4.35      .052 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

In this case T1 is simply V1 and T2 is simply V2. That is, the transformation applied
to the dependent variables was an identity transformation. Thus the MWITHIN TIME(1) effect
tests the null hypothesis that the mean of V1 is zero, averaged across both levels of A,
and MWITHIN TIME(2) tests a similar hypothesis concerning V2. As before, these tests of
constant or intercept terms are not generally of interest. The terms labeled as
interactions, A BY MWITHIN TIME(1) and A BY MWITHIN TIME(2) are the effects we want, as
they test the null hypotheses that there are no population differences between the levels
of factor A for V1 and V2, respectively. The general substitution rule for designs with
both between and within subjects factors is to substitute MWITHIN each level of a
particular main or interaction effect for that factor or interaction and all effects
encompassed by that term. So if we wanted to estimate the effects of TIME at each level of
the breakdown of an A by B between subjects design, (the simple, simple main effects of
TIME within A by B) we would specify:

MANOVA V1 V2 BY A B(1,2) 
 /WSFACTORS=TIME(2) 
 /DESIGN=MWITHIN A(1) BY B(1), MWITHIN A(1) BY B(2), 
         MWITHIN A(2) BY B(1), MWITHIN A(2) BY B(2).

Thus MWITHIN A BY B specifications replace the A by B interaction and the A and B main
effects, which are encompassed within A by B. If we wanted to estimate TIME effects only
within the levels of A, we would specify:

MANOVA V1 V2 BY A B(1,2) 
 /WSFACTORS=TIME(2) 
 /DESIGN=B, A BY B, 
         MWITHIN A(1), MWITHIN A(2).

The same logic applies completely when testing simple effects of between subjects
factors at different levels of within subjects factors. The substitution algorithm here
can also, as in the case involving only between or within subjects factors, be extended to
as many factors as necessary.

As mentioned earlier, while it is possible in some designs to estimate more than one
set of simple effects at a time, it is safest to do them individually, as the results of
specifying redundant requests are often meaningless ANOVA tables. This is particularly
true with regard to use of the MWITHIN keyword in releases prior to verion 5.0 of SPSS. In
later releases only one term can be used with MWITHIN, but in earlier releases use of
redundant MWITHIN requests may produce output of questionable validity that some users
will not be able to properly interpret.

Finally, there is some disagreement in the ANOVA literature about the use of error
terms in designs involving both between and within subjects factors. Specifically, it is
sometimes claimed to be desirable to use a pooled error term when fitting A within each
TIME point, just as a pooled error term is used when fitting TIME effects within each A
level. However, the simple effects of A at each level of TIME are simply the A effects for
the original correlated dependent variables. They are therefore not independent and cannot
be pooled to obtain a test statistic with a proper F-distribution under the null
hypothesis. Therefore, in this situation MANOVA uses a separate error term at each level
of TIME, equivalent to a simple univariate analysis of variance on each dependent
variable.

NONORTHOGONAL CONTRASTS ON WSFACTORS IN MANOVA
David P. Nichols
Senior Support Statistician
SPSS, Inc.
From SPSS Keywords, December 1993

A substantial number of users have asked how to get SPSS MANOVA to produce nonorthogonal
contrasts in repeated measures, or within subjects, designs. The reason that nonorthogonal
contrasts (such as the default DEVIATION, or the popular SIMPLE, or some SPECIAL user
requested contrasts) are not available when using WSFACTORS is that the averaged tests of
significance require orthogonal contrasts, and the program has been structured to ensure
that this is the case when WSFACTORS is used (users working on version 5 and later of SPSS
should note that DEVIATION is no longer the default contrast type for WSFACTORS). MANOVA
thus transforms the original dependent variables Y(1) to Y(K) into transformed variables
labeled T1 to TK (if no renaming is done) which represent orthonormal linear combinations
of the original variables. The transformation matrix applied by MANOVA can be obtained by
specifying PRINT=TRANSFORM. Note that the transformation matrix has been transposed for
printing, so that the contrasts estimated by MANOVA are discerned by reading down the
columns.

Here is an example, obtained by specifying a simple repeated measures MANOVA with four
levels and no between subjects factors. The following syntax produces the output in Figure
1.

MANOVA Y1 TO Y4 
 /WSFACTORS=TIME(4) 
 /PRINT=TRANSFORM 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Orthonormalized Transformation Matrix (Transposed) 
 
                    T1         T2         T3         T4 
 
 Y1               .500       .707      -.408      -.289 
 Y2               .500       .000       .816      -.289 
 Y3               .500       .000       .000       .866 
 Y4               .500      -.707      -.408      -.289 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

To see what contrasts have been obtained, simply read down the columns of the
transformation matrix. Thus we have

T1 =  .500*Y1 + .500*Y2 + .500*Y3 + .500*Y4 
T2 =  .707*Y1 - .707*Y4 
T3 = -.408*Y1 + .816*Y2 - .408*Y4 
T4 = -.289*Y1 - .289*Y2 + .866*Y3 - .289*Y4

Three further points should be noted here. First, the coefficients of the linear
combinations used to form the transformed variables are scaled such that the
transformation vectors are of unit length (are normalized). This can be duplicated by
first specifying the form of the contrasts using integers, then dividing each coefficient
by the square root of the sum of the squared integer coefficients. For example,

T3 = ( -1*Y1 + 2*Y2 - 1*Y4 ) / SQRT[ (-1)**2 + 2**2 + (-1)**2 ]

Second, the first transformed variable (T1) is the constant term in the within subjects
model, a constant multiple of the mean of the original dependent variables. This will be
used to test between subjects effects if any are included in the model. Finally, note that
the contrasts generated here are not those that we asked for (since we did not specify any
contrasts, the default DEVIATION contrasts would be expected). An orthogonalization of a
set of nonorthogonal contrasts changes the nature of the comparisons being made. It is
thus very important when interpreting the univariate F-tests or the parameter estimates
and their t-statistics to look at the transformation matrix when transformed variables are
being used, so that the inferences being drawn are based on the contrasts actually
estimated. This is not the case with the multivariate tests. These are invariant to
transformation, which means that any set of linearly independent contrasts will produce
the same results. The averaged F-tests will be the same given any orthonormal set of
contrasts.

Now that we know why we can’t get the contrasts we want when running a design with
WSFACTORS, let’s see how to make MANOVA give us what we want. This is actually fairly
simple. All that we have to do is to get MANOVA to apply a nonorthogonal transformation
matrix to our dependent variables. This can be achieved through the use of the TRANSFORM
subcommand. What we do is to remove the WSFACTORS subcommand (and anything else such as
WSDESIGN or ANALYSIS(REPEATED) that refers to within subjects designs) and transform the
dependent variables ourselves. For our example, the following syntax produces the
transformation matrix given in Figure 2.

MANOVA Y1 TO Y4 
 /TRANSFORM=DEVIATION 
 /PRINT=TRANSFORM 
 /ANALYSIS=(T1/T2 T3 T4) 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Transformation Matrix (Transposed) 
 
                    T1         T2         T3         T4 
 
 Y1              1.000       .750      -.250      -.250 
 Y2              1.000      -.250       .750      -.250 
 Y3              1.000      -.250      -.250       .750 
 Y4              1.000      -.250      -.250      -.250 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Note that this transformation matrix has not been orthonormalized; it gives us the
deviation contrasts we requested. You might be wondering what the purpose of the ANALYSIS
subcommand is here. The analysis subcommand is used to separate the transformed variables
into effects so that the multivariate tests produced in this case are equivalent to those
in the run where WSFACTORS was used. This serves two purposes. First, it allows us to
check to make sure that we’re still fitting the same model. Second, it helps us to
identify the different effects on the output. In this case, we will have only effects
labeled "CONSTANT," since we don’t have any WSFACTORS as far as MANOVA is
concerned. MANOVA is simply doing a multivariate analysis on transformed variables. This
is the same thing as the WSFACTORS analysis, except that the labeling will not match for
the listed effects. In this case, we will look for the effects labeled CONSTANT with T2,
T3 and T4 as the variables used in the analysis. These correspond to the TIME effect from
the WSFACTORS run, as can be seen by comparing the multivariate tests, but the univariate
tests now represent the contrasts that we wanted to see (as would the parameter estimates
if we had printed them).

Often the design is more complex than a simple repeated measures analysis. Can this
method be extended to any WSFACTORS design? The answer is yes. If there are multiple
dependent variables to be transformed (as in a doubly multivariate repeated measures
design), each set can be transformed in the same manner. For example, if variables A and B
are each measured at 3 time points, resulting in A1, A2, A3, etc., the following MANOVA
statements could be used:

MANOVA A1 A2 A3 B1 B2 B3 
 /TRANSFORM(A1 A2 A3/B1 B2 B3)=SIMPLE 
 /PRINT=TRANSFORM 
 /ANALYSIS=(T1 T4/T2 T3 T5 T6)

The TRANSFORM subcommand tells MANOVA to apply the same transformation matrix to each
set of variables. The transformation matrix printed by MANOVA would then have a block
diagonal structure, with two 3 x 3 matrices on the main diagonal, and two 3 x 3 null
matrices off the main diagonal. The ANALYSIS subcommand separates the two constants, T1
and T4, from the TIME variables, T2 and T3 (for A), and T5 and T6 (for B).

Another complication that may arise is the inclusion of between subjects factors in an
analysis. The only real complication involved here is in the interpretation of the output.
Printing the transformation matrix always allows us to see what the transformed variables
represent, but there is also a way to identify specific effects without reference to the
transformation matrix. There are two keys to understanding the output from a MANOVA with a
TRANSFORM subcommand:

1) The output will be divided into two sections: those which report statistics and
tests for transformed variables T1, etc., which are the constants in the repeated measures
model, used for testing between subjects effects, and those which report statistics and
tests for the other transformed variables (T2, T3, etc.), which are the contrasts among
the dependent variables and measure the time or repeated measures effects.

2) Output indicating that transformed variable T1 has been used represents exactly the
effect stated in the output. Output indicating that transformed variables T2, etc. have
been used represents the interaction of whatever is listed on the output with the repeated
measures factor (such as time). In other words, an effect for CONSTANT using variates T2
and T3 is really the time effect, and an effect FACTOR1 using T2 and T3 is really the
FACTOR1 BY TIME interaction effect.

If between subjects effects have been specified, the CONSTANT term must be specified on
the DESIGN subcommand in order to get the TIME effects. Also, the effects can always be
identified by matching the multivariate results to those from the WSFACTORS approach as
long as the effects have been properly separated with an ANALYSIS subcommand.

An example might help to make these principles more concrete. The following MANOVA
commands produced the four sets of F-tests listed in Figure 3.

MANOVA Y1 TO Y4 BY A(1,2) 
 /WSFACTORS=TIME(4) 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#1--The A main effect 
 
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              36.45      17      2.14 
 A                          3.79       1      3.79      1.77      .201 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#2--The A BY TIME interaction effect 
 
 EFFECT .. A BY TIME 
 Multivariate Tests of Significance (S = 1, M = 1/2, N = 6 1/2) 
 
 Test Name         Value    Exact F Hypoth. DF   Error DF  Sig. of F 
 
 Pillais          .59919    7.47478       3.00      15.00       .003 
 Hotellings      1.49496    7.47478       3.00      15.00       .003 
 Wilks            .40081    7.47478       3.00      15.00       .003 
 Roys             .59919 
 Note.. F statistics are exact. 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#3--The TIME effect 
 
 EFFECT .. TIME 
 Multivariate Tests of Significance (S = 1, M = 1/2, N = 6 1/2) 
 
 Test Name         Value    Exact F Hypoth. DF   Error DF  Sig. of F 
 
 Pillais          .29487    2.09085       3.00      15.00       .144 
 Hotellings       .41817    2.09085       3.00      15.00       .144 
 Wilks            .70513    2.09085       3.00      15.00       .144 
 Roys             .29487 
 Note.. F statistics are exact. 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#4--The averaged F-tests for TIME and A BY TIME 
 
Tests involving 'TIME' Within-Subject Effect. 
 
 AVERAGED Tests of Significance for Y using UNIQUE sums of squares 
 Source of Variation          SS      DF        MS         F  Sig of F 
 
 WITHIN CELLS             231.32      51      4.54 
 TIME                      25.97       3      8.66      1.91      .140 
 A BY TIME                 30.55       3     10.18      2.25      .094 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

The second run used TRANSFORM to analyze the same data, producing the output in Figure
4.

MANOVA Y1 TO Y4 BY A(1,2) 
 /TRANSFORM=SIMPLE 
 /ANALYSIS=(T1/T2 T3 T4) 
 /DESIGN=CONSTANT, A 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Order of Variables for Analysis 
 
   Variates     Covariates 
 
    T1 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#1--The A main effect 
 
 Tests of Significance for T1 using UNIQUE sums of squares 
 Source of Variation          SS      DF        MS         F  Sig of F 
 
 WITHIN CELLS             145.79      17      8.58 
 CONSTANT                8360.21       1   8360.21    974.86      .000 
 A                         15.16       1     15.16      1.77      .201 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 Order of Variables for Analysis 
 
   Variates     Covariates 
 
    T2 
    T3 
    T4 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#2--The A BY TIME interaction effect 
 
 EFFECT .. A 
 Multivariate Tests of Significance (S = 1, M = 1/2, N = 6 1/2) 
 
 Test Name         Value    Exact F Hypoth. DF   Error DF  Sig. of F 
 
 Pillais          .59919    7.47478       3.00      15.00       .003 
 Hotellings      1.49496    7.47478       3.00      15.00       .003 
 Wilks            .40081    7.47478       3.00      15.00       .003 
 Roys             .59919 
 Note.. F statistics are exact. 
 
 EFFECT .. A (Cont.) 
 Univariate F-tests with (1,17) D. F. 
 
 Variable   Hypoth. SS   Error SS Hypoth. MS   Error MS          F  Sig. of F 
 
 T2           18.73743  135.78889   18.73743    7.98758    2.34582       .144 
 T3            9.58129  227.15556    9.58129   13.36209     .71705       .409 
 T4            2.24795  108.48889    2.24795    6.38170     .35225       .561 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
#3--The TIME effect 
 
 EFFECT .. CONSTANT 
 Multivariate Tests of Significance (S = 1, M = 1/2, N = 6 1/2) 
 
 Test Name         Value    Exact F Hypoth. DF   Error DF  Sig. of F 
 
 Pillais          .29487    2.09085       3.00      15.00       .144 
 Hotellings       .41817    2.09085       3.00      15.00       .144 
 Wilks            .70513    2.09085       3.00      15.00       .144 
 Roys             .29487 
 Note.. F statistics are exact. 
 
 EFFECT .. CONSTANT (Cont.) 
 Univariate F-tests with (1,17) D. F. 
 
 Variable   Hypoth. SS   Error SS Hypoth. MS   Error MS          F  Sig. of F 
 
 T2           23.15848  135.78889   23.15848    7.98758    2.89931       .107 
 T3            4.94971  227.15556    4.94971   13.36209     .37043       .551 
 T4           45.19532  108.48889   45.19532    6.38170    7.08202       .016 
 
 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The first table in each run is the test for the between subjects factor A. Note that
the F-values and associated significances are identical. The sums of squares differ by a
constant multiple due to the orthonormalization. The CONSTANT term in the TRANSFORM run is
indeed the constant, and is usually not of interest. The second and third tables in the
WSFACTORS run contain only multivariate tests for the A BY TIME and A factors,
respectively. The univariate tests here are not printed by default. The corresponding
tables in the TRANSFORM output are labeled A and CONSTANT, with the header above
indicating that variates T2, T3 and T4 are being analyzed. Note that the multivariate
tests are exactly the same as those for the WSFACTORS run. This tells us that we have
indeed fit the same model in both runs. The application of our rule for interpreting the
labeling in the TRANSFORM run tells us that the second table represents A BY TIME and that
the third table represents CONSTANT BY TIME, which is simply TIME. Since MANOVA is simply
running a multivariate analysis with transformed variables, as opposed to a WSFACTORS
analysis, univariate F-tests are printed by default. The univariate tests for TIME are
generally the major source of interest, as they are usually the reason for the TRANSFORM
run. The A BY TIME tests may be the tests of interest if interaction is present. Finally,
the WSFACTORS run presents the averaged F-tests, which are not available in the TRANSFORM
run (and which would not be valid, since we have not used orthogonal contrasts).

One further example setup might be helpful in order to clarify how we would proceed if
we had multiple within subject factors. This is probably the most complex and potentially
time consuming situation we will encounter when trying to get MANOVA to estimate
nonorthogonal contrasts in within subjects designs, since we must know the entire contrast
(transformation) matrix we want MANOVA to apply to our data.

In this case we must use a SPECIAL transformation, and spell out the entire
transformation matrix (or at least the entire matrix for each dependent variable; if there
are multiple dependent variables we can tell MANOVA to apply the same transformation to
each). Let’s look at a situation where we have a 2 x 3 WSDESIGN and we want to do SIMPLE
contrasts on each of our WSFACTORS. The standard syntax for the WSFACTORS run would be:

MANOVA V1 TO V6 
 /WSFACTORS=A(2) B(3)

The syntax for the TRANSFORM run would be:

MANOVA V1 TO V6 
 /TRANSFORM=SPECIAL(1  1  1  1  1  1 
                    1  1  1 -1 -1 -1 
                    1  0 -1  1  0 -1 
                    0  1 -1  0  1 -1 
                    1  0 -1 -1  0  1 
                    0  1 -1  0 -1  1) 
 /PRINT=TRANSFORM 
 /ANALYSIS=(T1/T2/T3 T4/T5 T6)

Note that the final two rows of the contrast matrix are simply coefficient by
coefficient multiples of rows two and three and two and four, respectively. Also, the
ANALYSIS subcommand here separates the effects into four groups: the CONSTANT and A
effects (each with one degree of freedom), and the B and A BY B interaction effect (with
two degrees of freedom). Once again, this separation allows us to compare the TRANSFORM
output with appropriate parts of the WSFACTORS output. Though this use of SPECIAL
transformations can be somewhat tedious if there are many WSFACTORS or some of these
factors have many levels, it is also very general and will allow us to obtain the desired
contrasts for designs of any size.

This page was adapted from a web page at the SPSS web page. 
We thank SPSS for their permission to adapt and distribute this page via our web site.