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Re: st: RE: ancova?

From   Joseph Coveney <>
To   Statalist <>
Subject   Re: st: RE: ancova?
Date   Thu, 20 Feb 2003 10:01:25 +0900

Daichi Nozaki asked about using analysis of covariance (ANCOVA) for an experimental 
study examining differences in physiological responses elicited from 10 subjects each 
tested under each condition.  Each condition within each subject was also tested 
repeatedly.  The study design included a covariable for reducing unaccounted-for 
variation in the response; the response-covariate slope might vary by subject, but is not 
expected to vary with condition.

----------------------begin excerpt from posting--------------------------------

I recorded physiological responses for 3 different conditions from 10 subjects. In each 
condition, the data were obtained several times with an associated variable.

Namely, the data look like this.
subject condition response assocvar
1 1 0.1 0.2
1 1 0.3 0.5
1 1 0.5 0.8
1 2 0.4 0.3
1 2 0.7 0.6
1 2 1.0 0.9
1 3 0.8 0.3
2 ..............

I'd like to test the difference of physiological responses among 3 conditions after 
adjusting the data using the associated variable

Possibly, the slope of physiological response data to the associated variable is different 
from subject to subject, but is independent of the conditions within each subject.

----------------------end excerpt from posting----------------------------------

Al Feiveson mentioned the possibility of using -xtrchh- or -gllamm-, and indicated that 
ANCOVA would not be appropriate.

Since condition and covariate aren't expected to interact, the best way to approach this 
analysis, I believe, is first (Step 1) to regress the response values against the covariate 
disregarding the condition and replication.  Then (Step 2), using the residuals (-predict , 
residuals-) from Step 1 as the dependent variable in the next step, the analysis 
continues as a conventional three-way cross-over design with replicated measurements 
for each treatment.  The set-up for -anova- of the latter can be found in standard 
references, e.g., B. J. Winer, D. R. Brown & K. M. Michels, _Statistical Principles in 
Experimental Design_ (McGraw-Hill, 1991).

There are two approaches to consider for the first step (regression on the covariate):  
the first is to use -regress- with a subject-by-covariate interaction term; the second, as 
Al Feiveson mentions, is to use -gllamm- for a random-coefficients mixed-effects 
model.  The latter will provide a little "shrinkage" of the between-subjects variation in 
slopes compared to the former, but I'm not sure that it will make much practical 
difference which Daichi chooses.

For the second step, technically one degree of freedom will need to be manually 
deducted from the error mean square in order to account for the use of the covariate.  
Neglecting this, though, won't make much practical difference, since the degrees of 
freedom will be large.  The use of -anova- for the second step assumes that data are 
"balanced," i.e., equal numbers of replicates within equal numbers of conditions for 
each subject.  From Daichi's partial listing, the dataset might not be balanced in terms 
of the number of replicate determinations within each condition.  If so, and if the 
imbalance is at least unsystematic, then Daichi could use -xtreg, re- for a random-
effects regression instead of -anova-.  Daichi has only 10 subjects, and small-sample 
adjustments for mixed-effects models are still being worked out by the experts, but one 
approach in the interim is to run the model using -anova- and take the degrees of 
freedom from the ANOVA table for use in constructing tests after -xtreg, re-.

Joseph Coveney
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