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Re: st: time-effect in manova (anova with repeated measures )

From   Joseph Coveney <>
To   Statalist <>, Maren Kandulla <>
Subject   Re: st: time-effect in manova (anova with repeated measures )
Date   Thu, 17 Nov 2005 19:29:42 +0900

Maren Kandulla wrote (excerpted):

I do have one request regarding your remark:
> With 75 people and four groups, you will have unequal representations
> 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

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


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

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.

                       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

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))
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

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