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From |
Joseph Coveney <jcoveney@bigplanet.com> |

To |
Statalist <statalist@hsphsun2.harvard.edu> |

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 * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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