# Multivariate pairwise comparisons after multivariate ANOVA

## Problem

I have just run a One-Way MANOVA with the the MANOVA or GLM procedure in SPSS. The multivariate tests for the group effect were significant. I would like to run multivariate pairwise comparisons as well as the usual univariate follow up tests. Is there a way to run these multivariate comparisons in SPSS?

## Resolving The Problem

The DISCRIMINANT procedure will run the multivariate pairwise comparisons if you add the FPAIR keyword to the /STATISTICS subcommand. FPAIR requests tests of the Mahalanobis distance between each pair of levels of the group variable. FPAIR is only available for stepwise models, where one predictor variable is added at each step until all have been added or no remaining predictors will improve the model. The dependent variables in the MANOVA are the predictor variables in DISCRIMINANT. You can work around the stepwise restriction of FPAIR by specifying variable inclusion criteria that will insure that all of the predictor variables are entered. Then just pay attention to the pairwise comparisons for the final step in the output.

The following example shows a MANOVA analysis with GLM, followed by the DISCRIMINANT run for the multivariate pairwise comparisons. There were 5 dependent variables in the MANOVA (DV1 to DV5) and 4 levels in the grouping variable (GROUP).

GLM dv1 to dv5 BY group
/METHOD = SSTYPE(3)
/INTERCEPT = INCLUDE
/CRITERIA = ALPHA(.05)
/DESIGN = group .

DISCRIMINANT GROUPS = group(1 4)
/VARIABLES= dv1 to dv5
/METHOD = WILKS
/FIN=0
/FOUT=0
/STATISTICS = FPAIR .

In the DISCRIMINANT command, /METHOD = WILKS requests a stepwise analysis with Wilks' lambda as the test to evaluate the model improvement of adding a variable at each step. /FIN=0 specifies the minimum partial F for a new predictor to be entered into the model. /FOUT specifies the minimum partial F for a predictor to be ineligible for removal from the model. With both FIN and FOUT set at 0, all predictors will eventually enter the model. The "Pairwise Comparisons" table in the DISCRIMINANT output will include a set of comparisons at each step. For the purpose of running multivariate posthoc comparisons to the MANOVA, you will probably only be interested in the comparisons at the final step, after all variables have been entered (step 5 in this example).

Note that FPAIR is not available in the graphic user interface (GUI), or menus and dialog boxes, for Discriminant Analysis, so you must run the procedure as a syntax command. You can build most of the command from the GUI
(Analyze->Classify->Discriminant), clicking Paste rather than OK and adding FPAIR to the /STATISTICS subcommand in the syntax window. The Wilks method and FIN and FOUT values are chosen in the Method dialog box, which is entered by clicking the Method button in the main Discriminant Analysis dialog box. The Method dialog box will not accept a value for FOUT that is not smaller than the value for FIN. (You can enter 0 for both, but you can't Continue out of the dialog with that setting.) So, you can enter an FIN of 1 and an FOUT of 0 in the Method dialog, then change the FIN to 0 in the syntax window.

For further discussion of this approach with an example, see:
Stevens, J. P. (2002). Applied Multivariate Statistics for the Social Sciences (4th Ed.). Mahwah NJ: Erlbaum. (Chapter 5, pp. 217-222).
Stevens notes that the significance levels, or p-values, that are printed for each pairwise comparison are Type I error probabilities for a single comparison. He suggests using a critical alpha (the maximum p-value for a pairwise difference to be considered significant) that is small enough for the overall error rate to be acceptably small. For example, four groups will yield six pairwise comparisons. If we require p-values of .01 or less for each comparison to be judged significant, then the overall Type I error rate for the pairwise comparisons will be approximately 0.06. (Stevens actually accepts a larger overall error rate in his discussion (0.15)). Stevens' 3rd Edition (1996) also includes this discussion. Stevens discusses the multivariate pairwise comparisons in terms of Hotelling T^2 tests, which are equivalent to the Mahalanobis tests in DISCRIMINANT.
(Also see Kotz., S. & N.L. Johnson (Eds.) (1983). Encyclopedia of Statistical Sciences: Vol. 3. New York: Wiley. (p. 670).

## Related Information

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Modified date:
16 April 2020

swg21477075