Date:         Fri, 26 Jul 1996 18:51:52 GMT
Reply-To:     David Nichols <nichols@SPSS.COM>
Sender:       "SPSSX(r) Discussion" <SPSSX-L@UGA.CC.UGA.EDU>
From:         David Nichols <nichols@SPSS.COM>
Organization: SPSS, Inc.
Subject:      Re: GLM and MANOVA (SPSSWIN 7.0) A bug?

In article <00001b82+00000c98@msn.com>, E Sandvik <Sand@msn.com> wrote:
>GLM and MANOVA  (SPSSWIN 7.0)
>
>I wondered if the following can really be true: I ran a repeated
>measures GLM with two data points, Time1 and Time2.  The mean at T1 =
>47, T2 = 42, N = 70.  GLM reports a signficant within-Ss F of 4.74*.
>I then added two personality covariates (non-varying).  Both of these
>are reported as non-significant with the within Ss factor (F=0.04,
>F=0.29), but now the within Ss factor is reported as NS as well,
>dropping from the previous 4.74* to F=0.001.  I repeated this
>analysis using the MANOVA procedure instead and the results are all
>the same EXCEPT the within-Ss F stays at 4.74* even with the NS
>covariates.  Same N of Ss on all analyses.
>
>Is this possible or a GLM bug?
 
The results are (I assume) correct, and the differences between GLM
and MANOVA explainable as follows. MANOVA treats covariates in a
somewhat unusual way, particularly with regard to within subjects
designs. It does not simply build a design matrix with the covariate
columns entered into it, as does GLM. Instead, MANOVA initially makes
no distinction between dependents and covariates, and estimates an
adjusted SSCP matrix among all continuous variables, adjusting for
the design matrix built based on the between subjects factors in the
model. For between subjects issues this generally isn't that relevant,
though it is if you try to use SEQUENTIAL sums of squares, as MANOVA
won't really do this unless you force covariates into the design
matrix by using ANALYSIS and putting them on the DESIGN. For analyses
with WSFACTORS, the results of the way MANOVA handles covariates are
of more common interest.
 
MANOVA applies the same transformation matrix to sets of covariates
as it does to sets of dependent variables. This is why in releases of
SPSS prior to 6.0, you had to make dummy copies of constant covariates
so that the number of covariates was equal to the number of implied
levels of WSFACTORS. In this case, on older versions you would have
to have created dummy copies of each of your two covariates and entered
four covariates. MANOVA would then create transformed variables: T1, a
normalized sum of the two dependents; T2, a normalized difference score
on the dependents; T3 and T5, normalized sums on the two covariate sets,
and T4 and T6, normalized differences on the two covariate sets. It
would then regress T1 on any between subjects design matrix, with T3
and T5 as covariates. That is, it would (in the full factorial case)
make a within cells SSCP matrix involving T1, T3 and T5, and compute
from that the regression of T1 on T3 and T5, having adjusted for any
between subjects factors. Next, it would do the same with T2, using
T4 and T6 as covariates. Now if the covariates were constant, T4 and
T6 would both be 0 for every case, and you'd see a note about them
being linearly dependent on preceding variables, and they would be
dropped from that part of the analysis. Thus, T2 winds up being
regressed on just the between subjects factors, with no covariates.
One way of looking at this is that since the covariates don't vary
over time, they don't play a role in the within subjects part of the
model. Another is to recognize that the model as fitted is assuming
that the adjustment to the means for the covariate is the same at
each time point, so it cancels out. (Note that the second analysis
produces the interactions between time and the between subjects
factor(s), with the constant in that model giving the time "main
effect," since we're using difference scores.)
 
On releases 6.0 and above, you can specify covariates as constant,
and we'll internally take care of things without making you make
dummy copies and having them thrown out of the within subjects
part of the analysis. MANOVA simply rescales the covariates to
match the metric for the transformed dependents, and uses them
only in the within subjects part of the model. It also uses a
different convention for naming the transformed covariates, adding
a T to the beginning of the name.
 
Only if the difference scores on the covariates are nonzero will
the transformed covariates remain in the within subjects part of
the model. In other words, only if the covariates vary over time.
For a discussion of the logic of this approach to repeated measures
with covariates, see, e.g., Winer's _Statistical Principles in
Experimental Design_.
 
This approach in the general case (more than two levels of the within
subjects factor) is only defined in the univariate or mixed models
approach. If you run a current version of MANOVA with varying
covariates and more than one transformed dependent variable in a
measure, you'll only get the averaged univariate tests. The general
multivariate approach analysis is not defined for this situation.
Older releases of SPSS (prior to release 5) gave questionable or
bogus multivariate and individual univariate results, and an
incorrect Mauchly sphericity test.
 
GLM in SPSS, like PROC GLM in SAS, does not allow for time varying
covariates. Covariates in these procedures are simply entered into
the between subjects design matrix without transformation, along
with the dummy variables representing factors. When you do this, and
regress transformed dependents on them, you get interactions between
covariates and within subjects factors, just as you get interactions
between between and within subjects factors; covariates are just
between subjects variables.
 
What MANOVA is giving you here is output that doesn't differ from
what you would get running the model without the covariates and
looking at your within subjects results. You can thus obtain the
same results from GLM by running the model a second time without
the covariates if you assume this is the proper model. Since both
of your covariate by within subjects factor interactions are not
significant, this may be a reasonable assumption.
 
To get the same results in MANOVA that you got in GLM (that is,
to have MANOVA also fit covariate by within subject factor
interactions), you would make MANOVA force the covariates into
the design matrix. The syntax for this depends on whether you're
using dummy covariates or constant covariates. For dummy ones:
 
MANOVA Y1 Y2 BY GROUP(1,2) WITH COV1 COV1DUM COV2 COV2DUM
 /WSFACTORS=TIME(2)
 /ANALYSIS=T1 T2
 /DESIGN=GROUP, T3, T5.
 
For covariates defined as constant in 6.0 and above,
 
MANOVA Y1 Y2 BY GROUP(1,2) WITH (COV1 COV2)
 /WSFACTORS=TIME(2)
 /ANALYSIS=T1 T2
 /DESIGN=GROUP, TCOV1, TCOV2.
 
Feel free to post or email me with follow up questions.
 
--
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David Nichols             Senior Support Statistician              SPSS, Inc.
Phone: (312) 329-3684     Internet:  nichols@spss.com     Fax: (312) 329-3668
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