Doubly Multivariate Analysis of RM Designs



Doubly Multivariate Analysis of Repeated Measures Designs:Multiple Dependent VariablesAn analysis may be “doubly multivariate” in at least two different ways. First, a set of noncommensurate (not measured on the same scale) dependent variables may be administered at two or more different times. For example, I measure subjects’ blood pressure, heart rate, cholesterol level, and percent body fat. Subjects participate in a monthlong cardiac fitness program or in some placebo activity. I measure the four dependent variables just before the program starts, just after it ends, a month after it ends, and a year after it ends. I have a Group x Time mixed design with multiple dependent variables. I take the multivariate approach with respect to the Time variable (to avoid the sphericity assumption), and I have multiple dependent variables, so I have a doubly multivariate design. For each effect (Group x Time, Time, Group) I obtain a test on an optimally weighted combination of the four dependent variables (weighted to maximize that effect). If (and only if) the test from the doubly multivariate analysis (which simultaneously analyzes all four dependent variables) is significant, I then conduct tests of that effect on each dependent variable (one at a time). This procedure may provide some protection against inflation of alpha with multiple dependent variables. Suppose that only the time effect was significant. I would then conduct singly multivariate analyses on each dependent variable, ignoring the group and Group x Time tests in those analyses.A second sort of doubly multivariate design exists when two or more noncommensurate sets of commensurate dependent measures are obtained at one time. Experiment 1 of Karl’s dissertation, which we used as an example for the multivariate approach to the (A x S) and the A x (B x S) ANOVAs, will serve as an example. In addition to measuring how much time each subject spent in each of four differently scented tunnels (one set of commensurate variates), I measured how many visits each subject made to each tunnel (a second set) and each subjects’ latency to first entry of each tunnel (a third set).Obtain from my SPSS Data Page TUNNEL4b.sav. Bring it into SPSS. Copy the following syntax to the syntax editor and run it, or just look at my output document.manova v_clean to v_rat t_clean to t_rat L_clean to L_rat by nurs(1,3) / wsfactors = scent(4) / contrast(scent)=helmert / measure = visits time latency / print=transform signif(univ hypoth) error(sscp) / design .The order of the variates in the MANOVA statement must be: The variate representing the first dependent variable at level 1 of the withinsubjects factor; the variate representing the first dependent variable at level 2 of the withinsubjects factor; . . . variate for first dv at the last level of within factor; variate for second dv at level 1 of within factor; variate for second dv at level 2 of within factor; . . . variate for last dv at last level of the withinsubjects factor.Our dependent variables (named in the MEASURE command) are Visits, Time, and Latency. Our withinsubjects factor is scent, with four levels (clean, Mus, Peromyscus, and Rattus, in that order). First listed in the MANOVA statement are the four variates for the visits dependent variable, then the four for time, then latency. I asked that the transformation matrix, univariate tests, and error and hypothesis SSCP be printed.Look at the orthonormalized coefficients (coefficients are orthogonal and for each contrast the sum of the squared coefficients is one) SPSS used to compute the differencescores used in the analysis. T1, T5, and T9 compare the Visits, Time, and Latency variates to zero. T2, T6, and T10 compare Clean vs Scented (Mus, Pero, Rat) scores on Visits, Time, and Latency. T3, T7, and T11 compare Mus vs Other Rodent (Pero, Rat). T4, T8, and T12 compare Pero (a prey species) vs Rat (a predator species). These particular contrasts result from my having chosen the Helmert contrast. They actually make some sense. The default contrast is polynomial, which does not make any sense, but which would lead to the same omnibus statistics.The “Multivariate Tests of Significance” for the NURS (nursing groups) factor indicate an effect significant beyond the .01 level. The “Univariate Ftests” output there shows that the transformed variates T1, T5, and T9 were used to conduct the test of the null hypothesis that the NURS variable had no effect on the Visits, Time, and Latency variables. If you will look back at the A x (B x S) singly multivariate analysis we did earlier you will find that the F conducted on T5 (Time) is exactly the same as the betweensubjects test of NURS’ effect on Time. NURS’ significant multivariate effect is accompanied by significant univariate effects on Visits (T1) and Latency (T9), but not Time (T5). Were there no significant higherorder effects we might now go on to pairwise comparisons to explain these significant main effects of NURS.Ignore the “EFFECT..constant” output (which tests the null that all variates have means of zero) and move on to the Tests involving “SCENT” WIthin-Subject Effect. The main diagonal of the WITHIN CELLS SSCP matrix here contains the SSerror for the NURS BY SCENT Univariate F tests that follow. The main diagonal of the next matrix contains the SShypothesis for the those F tests.Multivariate Tests of Significance show a significant interaction. For an optimally weighted combination of the difference scores T2, T3, T4, T6, T7, T8, T10, T11, and T12, the NURS BY SCENT effect is significant.The Univariate F-tests show that the Nursing groups differed significantly on the contrast between Peromyscus and Rattus for visits (T4), Time (T8) and Latency (T12).For Effect .. SCENT, we are given the hypotheses SSCP matrix, whose main diagonal contains the SShypothesis for the Univariate F tests involving the scent variable. The SSerror were given earlier (same ones used for the interaction). The multivariate tests are significant, and the univariate contrasts show significant effects on T3 (Visits to Mus versus other rodent), T7 (Time, Mus versus other rodent), and T8 (Time, Peromyscus versus Rattus).All remaining output of interest is for averaged statistics.The AVERAGED WITHIN CELLS Sum-of-Squares and Cross-Products are constructed from the error SSCP matrix displayed earlier. Look at that earlier matrix. Sum the sums of squares for T2:T2, T3:T3, and T4:T4. You get 7.599 + 17.964 + 9.612 = 35.174 = the averaged error SS for Visits. Now sum T6:T2, T7:T3, and T8:T4. You get 97.943 + 126.81 + 52.511 = 277.264 = the cross-product for Time-Visits.The entries in the Adjusted Hypothesis Sum-of-Squares and Cross-Products matrix represent the same sort of sum of the entries in the hypothesis SSCP matrices for the interaction presented earlier.The AVERAGED Multivariate Tests of Significance really represent an analysis where the multiple dependent variables (visits, time, and latency) are evaluated simultaneously, but the within-subjects contrasts are based on averaged statistics (that is, those one would get with univariate repeated measures analyses, one on each of visits, time, and latency) rather than a multivariate treatment of the repeated measures dimension(s). Although I have seen both the fully multivariate and the averaged multivariate tests used, it seems to me that the fully multivariate tests are the better choice, except, perhaps, when sample sizes are very small (the averaged tests have more error df). In fact, our sample sizes for the current analysis are so small that our variancecovariance matrices are singular (we have more dummy variates, T1T12, than we have subjects per group). This can be a problem because it lowers power and prevents us from conducting Box’s M, but with equal sample sizes we should not be concerned with Box’s M (the MANOVA being quite robust under those conditions) and when we have rejected the null hypothesis power is not an issue.The Univariate F-tests given after the Averaged Multivariate Tests give us the same statistics that would obtain if we were to do univariate Nurs x Scent ANOVAs, one each on visits, time, and latency. In such ANOVAs, the interaction is significant for each dependent variables. A bit later we see that the main effect of scent is significant for visits and time, but not for latency.AddendumIsabelle asked “Since I have 2 groups (Group), several dependent variables (A, B, C, and D) and 4 measurement times (1, 2, 3, and 4) in my study, the data corresponds to your first description: a set of noncommensurate (not measured on the same scale) dependent variables administered at two or more different times. So I am asking you if you could maybe help me with it and describe me how to perform the analysis in SPSS.”This should do the trick:manova A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 D1 D2 D3 D4 by Group(1,2) / wsfactors = time(4) / contrast(time)=helmert / measure = A B C D / print=transform signif(univ hypoth) error(sscp) / design .Summary of MANOVA syntaxPre-Post MANOVAReturn to Wuensch’s Statistics Lessons PageCopyright 2016, Karl L. Wuensch - All rights reserved. ................
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