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

[Peter Dalgaard]

Lucke, Joseph F wrote:
> 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.
>   
I mentioned in my reply that one might look at successive differences to 
remove the mean level, I think this is effectively the same general 
idea. It is actually implemented in anova.mlm(), which allows you to 
specify a transformation matrix, possibly using formula notation to 
compute the relevant model matrix/matrices.

There is, btw, an alternative approach which involves _conditioning_ on 
contrasts that at known to have mean zero (second order differences for 
a linear within-subjects trend, e.g.). This is not implemented 
systematically anywhere though, as far as I know.

> 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},
>   address = {New York},
>   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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