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.
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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.
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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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