Regression Models for Repeated Measurements

Abstract

In his Response to a Query, Aitkin (Biometrics 37, 831-832, December 1981) uses standard ANOVA methodology to analyze data from a trial in which rectal temperatures were measured on a group of ten subjects, under four combinations of ambient temperatures, with six equally-spaced observations taken for each temperature. I feel that the assumptions needed for the 'standard' ANOVA are almost certainly not satisfied for this, or comparable data sets. To simplify the discussion, I assume that the ambient temperatures were applied in random order. Under this condition, the tests for temperature and the test for the linear components of the time effect and interaction, do have the stated F distribution if modified slightly, whereas the other tests do not have an F distribution under Ho. Aitkin suggests a partition of the residual term to investigate the possibility that the slope varies between the subjects. The partition should be done even if the slope is not of interest, since it permits analysis of a more general model which includes Subject x Time and SubjectxTemperature interactions (Neter and Wasserman, 1974, p. 732). In addition, as indicated below, it allows evaluation of the temperature effect, even if serial correlation due to time is present. Although the necessary and sufficient condition for validity of the model does not require equal variances and covariances, it does imply that the covariances are equal if the variances are equal (Huynh and Feldt, 1970). Furthermore, Huynh (1978) states that 'it would be difficult to conceptualize a situation which would give rise to a Huynh-Feldt matrix'. I suggest that for all practical purposes the required assumption should be that of equal variances and equal covariances. This assumption is probably not satisfied for the analysis of time in this data set, since the correlation between the 20and 40-minute readings in rectal temperature is probably not the same as between 20 minutes and two hours. Since the ambient temperatures were applied in random order, it would be reasonable to assume that the correlations between temperatures are equal and thus MS (Temperature)/MS (Subject x Temperature) has an F distribution (see also Mendoza, Toothaker and Cain, 1976). Since the test for trend in the time effect is based on a linear combination of the within-individual observations, MS {Time(L)}/MS {SubjectxTime(L)} has an F distribution. Huynh (1978) cites various approaches for giving approximate critical values for the F ratio in cases where the Huynh-Feldt assumptions do not hold. If one can postulate that the serial correlation between observations u time units apart is pu, conservative critical values are obtained by multiplying the nominal degrees of freedom by 5(t+1)/(2t2+7) (Wallenstein and Fleiss, 1979).