== Multivariate output explained in SPSS repeated measures ==

In SPSS using the GLM:repeated measures procedure a multivariate analysis of 
variance is also presented. For factor(s) with more than two levels these results will not agree with univariate repeated measures.

Let's illustrate what is going on with a simple example.

Consider nine subjects who perform a task under each of three conditions. The scores are given in the table along with the
differences, D1 and D2, of C1-C2 and C1-C3 respectively.

||||<20% style="TEXT-ALIGN: center"> '''C1''' || '''C2''' || '''C3''' || '''D1''' || '''D2''' ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 1  || 2 || 1  || 0 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 1 || 2  || 1 || -1 || 0 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 3  || 2 || -1 || 0 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 3 || 2  || 3 || 1  || 0 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 3  || 4 || -1 || -2 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 1 || 4  || 5 || -3 || -4 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 3 || 5  || 3 || -2 || 0 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 6  || 4 || -4 || -2 ||
||||<20% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 1 || 4  || 5 || -3 || -4 ||

Now, the multivariate anova test of the overall mean difference tests if
the vector (D1 mean, D2 mean) equals (0,0) using the 2 by 2 (D1, D2) covariance matrix, V, consisting of the D1 and D2 variances and their ''covariance''. This gives a multivariate test statistic of 

N(D1 mean, D2 mean)^T ^V ^-1 ^(D1 mean, D2 mean)

where N is the sample size (=9 in above example) which follows a bivariate analogue of the univariate t distribution, Hotelling's T^2 ^ if there is no difference between the three conditions.

The univariate anova test, on the other hand, takes the score and fits a 
multiple regression with a series of indicator variables. The indicators for 
the first four subjects are coded in the table. 
Two of these indicators represent condition and eight represent 
subjects. The SS(condition), for example, in the Univariate repeated measures
anova represents the unexplained variance in scores which is not
explained by subject differences ie 31.037 - 19.407 = 11.63.  

||||<9% style="TEXT-ALIGN: center"> '''SCORE''' || '''INDC2''' || '''INDC3''' || '''INDS1''' || '''INDS2''' || '''INDS3''' || '''INDS4''' || '''INDS5''' || '''INDS6''' || '''INDS7''' || '''INDS8''' ||
||||<9% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 0  || 0 || 1  || 0 || 0 || 0 || 0 || 0 || 0 || 0 ||
||||<9% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 1 || 0  || 0 || 0  || 1 || 0 || 0 || 0 || 0 || 0 || 0 ||
||||<9% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 2 || 0  || 0 || 0  || 0 || 1 || 0 || 0 || 0 || 0 || 0 ||
||||<9% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center"> 3 || 0  || 0 || 0  || 0 || 0 || 1 || 0 || 0 || 0 || 0 ||

It can be seen from this that only the multivariate procedure assumes there
is a correlation of the change in score between conditions 1 and 2 with the
change in score between conditions 1 and 3 across the subjects.
Any discrepancy in results between univariate and multivariate results
will be due to the presence of such a correlation. The results may also differ due to the violation of the [[FAQ/MANOVA/manrm/sphericity|sphericity assumption.]]

Finally a piece of terminology: if 1) more than one variable representing a particular factor or measure is recorded at (2) multiple time points this set up is sometimes called a ''doubly'' multivariate design.

__Reference__

Tabachnick, B. G. and Fidell, L. S. (2007) Using multivariate statistics. Pearson Educational:Boston, USA. 

(Pages 316 to 323 of chapter 8 in the above book show the step-by-step computation of  multivariate F and T ratios for an interaction involving a between subjects factor (occupational group) with a within subjects factor (activity) and the two main effects based on matrices using differences of form D1 and D2 above).