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Maren Kandulla wrote (excerpted):

I do have one request regarding your remark: > With 75 people and four groups, you will have unequal representations among > groups. I think that MANOVA does best with equal representation among > groups, just like factorial ANOVA does. At the least, you might need to > adjust the one-quarter to some group-size-weighted fraction in order to > get > the -lincom- estimate to match that by -summarize-.

I have a very unbalanced design. To be more precise than in my previous email where I "combined information", I have following n-distribution: 1. Cohort with 4 groups: 21, 14, 35, 17; total 87 and 2. Cohort with 3 groups: 35, 21, 19; total 75; cohorts are analysed seperately. In the Stata-Manual I found following information: > manova fits multivariate analysis-of-variance (MANOVA) and multivariate analysis-of-covariance (MANCOVA) models for balanced and unbalanced designs...

I therefore decided not to do any adjustment. Please, correct me if this was wrong!

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The remark was based on an analogy to weighted and unweighted means analysis for unbalanced factorial ANOVA. There is a controversy over the use of so-called SAS Type I and Type III SS. Among the reasons for preferring the former, it is said to have higher power than the latter does in the absence of an interaction, although I recall that the difference was not major in simulations with -anova , sequential- and -anova , partial- done some time ago.

A re-reading this morning of the relevant passages in R. A. Johnson & D. W. Wichern, _Applied Multivariate Statistical Analysis_ 4th Ed., (Englewood Cliffs, New Jersey: Prentice-Hall, 1998) doesn't suggest that unequal group size is a problem with MANOVA if you have only a single grouping factor.

If you're ever in doubt, you can always try -manovatest- with and without adjustment to see whether it makes any practical difference in the test for change across time. The simulation below is of a repeated-measures design that replicates group sizes of your Cohort 1. I've given it a between-group difference of one-half standard deviation in one group, and a linear trend across time of one-half standard deviation for all groups, i.e., parallel time-course (additive, no interaction). The covariance structure is intended to mimic something that you might see with a rather long-duration between observations, probably more extreme than what would be encountered in the typical short-term experimental study in the fields of biology, psychology, or medicine.

There is a difference in power as expected. When you do not adjust the elements of the test matrix for -manovatest- according to the respective group sizes, the power is 66%. When you do the adjustment, the power rises modestly to 73%. There is analogously a slight rise in power for main effects of time in repeated-measures ANOVA (using the Huynh-Feldt epsilon).

The important difference, however, is between MANOVA and repeated-measures ANOVA with the Huynh-Feldt degrees of freedom adjustment. The latter is more powerful with your sample sizes, at least for main effects. I've copied the results into a table below, because the simulation takes a while to run, and the do-file might be wrapped during e-mail processing.

-------------------------------------- Percent null hypothesis Term rejection -------------------------------------- Main effects of group MANOVA 29 ANOVA 60

Main effects of time MANOVA (with adj.) 73 MANOVA (no adj.) 66 ANOVA (sequential) 87 ANOVA (partial) 84

Group X Time MANOVA 4.7 ANOVA 5.5 --------------------------------------

You can re-run this ado-file substituting zeroes for the mean vector for -drawnorm- to see how well the Type I error rate is controlled by the Huynh-Feldt adjustment for the main effects of time with this covariance structure. The null hypothesis was true for the interaction of group and time in the do-file below, so the values in the table above give the Type I error rates for that term.

Joseph Coveney

clear set more off set seed `=date("2005-11-18", "ymd")' set obs 6 forvalues i = 1/6 { generate float a`i' = (`i' == _n) * 1 + /// (`i' != _n) * 0.5^abs(`i' - _n) local responses `responses' response`i' } mkmat a*, matrix(A) * capture program drop runem program define runem, rclass syntax , responses(namelist) corr(name) tempname S M sequential partial drawnorm `responses', means(0.0 0.1 0.2 0.3 0.4 0.5) /// corr(`corr') n(87) clear generate byte treatment = 0 foreach size in 21 14 35 { local group = `group' + `size' replace treatment = treatment + 1 if _n > `group' } macro drop _group forvalues i = 1/6 { replace response`i' = response`i' + 0.5 /// if treatment == 1 } generate byte row = _n manova `responses' = treatment matrix `S' = e(stat_m) return scalar mmain = `S'[1,5] matrix `M' = (1, -1, 0, 0, 0, 0 \ /// 0, 1, -1, 0, 0, 0 \ /// 0, 0, 1, -1, 0, 0 \ /// 0, 0, 0, 1, -1, 0 \ /// 0, 0, 0, 0, 1, -1) matrix `sequential' = (1, `=21/87', `=14/87', `=35/87', `=17/87') matrix `partial' = (1, 0.25, 0.25, 0.25, 0.25) manovatest , test(`sequential') ytransform(`M') matrix `S' = r(stat) return scalar mstime = `S'[1,5] manovatest , test(`partial') ytransform(`M') matrix `S' = r(stat) return scalar mptime = `S'[1,5] manovatest treatment, ytransform(`M') matrix `S' = r(stat) return scalar minter = `S'[1,5] quietly reshape long response, i(row) j(time) anova response treatment / row | treatment time treatment*time, /// category(treatment time row) repeated(time) sequential return scalar amain = Ftail(e(df_1), e(dfdenom_1), e(F_1)) return scalar astime = Ftail(`=e(hf1) * e(df_3)', /// `=e(hf1) * e(df_r)', e(F_3)) return scalar ainter = Ftail(`=e(hf1) * e(df_4)', /// `=e(hf1) * e(df_r)', e(F_4)) anova response treatment / row | treatment time treatment*time, /// category(treatment time row) repeated(time) return scalar aptime = Ftail(`=e(hf1) * e(df_3)', /// `=e(hf1) * e(df_r)', e(F_3)) end * simulate mmain = r(mmain) mstime = r(mstime) mptime = r(mptime) /// minter = r(minter) amain = r(amain) astime = r(astime) /// aptime = r(aptime) ainter = r(ainter), /// reps(3000) nodots: runem , responses(`responses') corr(A) foreach var of varlist _all { generate byte p_`var' = `var' < 0.05 } summarize p_mmain p_amain summarize p_mstime p_mptime p_astime p_aptime summarize p_minter p_ainter exit

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Posted at 11/17/2005 05:30:00 AM EST (UTC -0500) [Permanent Link]
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