st: Re: "Repeated-measures" form of linear regression?

["Joseph Coveney" <jcoveney@bigplanet.com>]

Pietro Mazzoni wrote:

My question is not specific to Stata, but I am hoping that someone can at least
point me to the topic of statistical testing on which I can do more reading to
answer this question. I appreciate any suggestions.

My hypothesis, in a motor control experiment, is that there is a linear
relationship between a perturbation (say, force F) and a response (say, lateral
deviation D). I obtained data from several subjects who were tested at various
levels of force and whose response was a particular deviation for each force
pulse. I feel that it would be incorrect to perform a linear regression of D vs.
F on the pooled data, because this would ignore within-subject correlations. I
see this data as analogous to repeated-measures ANOVA, because multiple pairs of
(F, D) were collected within each subject.  So my intuition tells me I should
perform linear regression within each subject, and then somehow combine these
results across the group, but I don't know what procedure might be the analog of
rm-ANOVA for regression. One suggestion I got was to perform individual
regressions within each subject and then average the values of the slopes and
intercepts obtained to estimate slope and intercept f!
 or the group. But I am more interested in establishing whether there is a
significant relationship at the group level (R square, p value for the group)
than in determining the parameters of the regression, and I don't know how to
combine the R squares or p values obtained for individual subjects.

Where can I read about repeated-measures approaches for continuous
variables/regression?

----------------------------------------------------------------------------

Sounds like a time-varying-covariate repeated-measures ANOVA.  You can 
Google "growth curve model" to find out more.  

Michael Mitchell has already recommended a couple of SAS references, and I 
agree that they're both good.  You might find their reading a little 
heavy-going at a few points if you're not into the arithmetic.  

I would also agree with Michael's implication that -xtmixed- is the way to 
go with your analysis, except for your mention of "several subjects".  
SAS's PROC MIXED still has an edge over the competition when it comes to 
hypothesis testing in small-sample datasets.  Until -xtmixed- implements an
option corresponding to PROC MIXED's DDFM=KENWARDROGER, it's probably not 
the best approach in Stata.

I think that you can still use least-squares-based methods in Stata to test
your hypothesis of interest if your dataset is balanced.  One approach is 
the time-varying-covariate repeated-measures ANOVA approach.  I posted a 
worked example for a simple case from B. J. Winer's _Statistical Principles
in Experimental Design_ (the second example in the do-file) at
www.stata.com/statalist/archive/2004-01/msg00032.html (full reference in 
that post).  In order to test the hypothesis you mentioned, you'd work with 
the linear component of the orthogonal contrasts.  If you're worried about 
autocorrelation, then you can use -manova-, again setting up and testing 
the linear contrast of interest from the regression coefficients.

Regarding your colleague's suggestion, you can find out more by Googling 
"longitudinal data" AND "summary measures" (together).  If I had a 
small-sample dataset that had a small amount of imbalance, I would pursue 
this approach.

Joseph Coveney


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