# [R] Hotelling T-Squared vs Two-Factor Anova

Lucke, Joseph F Joseph.F.Lucke at uth.tmc.edu
Mon Apr 16 17:59:39 CEST 2007

```Sean

Both Bill V and Peter D are right regarding traditional
repeated-measures ANOVA that assumes equality of the variance-covariance
matrices across groups and compound symmetry of the covariance matrix.
However, there is a multivariate approach to repeated measures that does
not require the assumption of compound symmetry.  This approach is given
in the reference below.  The difference between multivariate repeated
measures and general MANOVA is that the former imposes a model matrix
(contrasts) on the measures to reflect the repeated measures design.
General MANOVA does not have this extra matrix.  The Hotelling T-squared
approach with a measures model matrix would be equivalent to
multivariate repeated measures.

In your case, you need to specify the measure model matrix for the 4
time points (polynomial, differences, Helmert), which usually cannot be
done in a standard T-square program.  However, you could compute the
model matrix results (contrasts) beforehand and submit them to the
T-square program.  Your taking the difference scores on the 1st factor
is doing just that.

Joe

@BOOK{Bock1975,
author = {Bock, R. D.},
title = {Multivariate statistical methods in behavioral research},
year = {1975},
publisher = {McGraw-Hill},
keywords = {regression; analysis of covariance; analysis of variance;
log-linear
analysis; matrices; distribution theory;},
}

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Sean Scanlan
Sent: Saturday, April 14, 2007 7:38 PM
To: r-help at stat.math.ethz.ch
Subject: [R] Hotelling T-Squared vs Two-Factor Anova

Hi,

I am a graduate student at Stanford University and I have a general
statistics question.  What exactly is the difference between doing a
two-factor repeated measures ANOVA and a Hotelling T-squared test for a
paired comparison of mean vectors?

Given:

Anova: repeated measures on both factors, 1st factor = two different
treatments, 2nd factor = 4 time points, where you are measuring the
blood pressure at each of the time points.

Hotelling T^2: You look at the difference in the 4x1 vector of blood
pressure measurements for the two different treatments, where the four
rows in the vector are the four time points.

I am mainly interested in the main effects of the two treatments.  Can
someone please explain if there would be a difference in the two methods
or any advantage in using one over the other?

Thanks,
Sean

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