The power to detect differences in average rates of change in longitudinal studies

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Abstract
With considerable current interest in longitudinal epidemiologic studies, little is available regarding sample size requirements. This paper considers a method for analysis of longitudinal data, where one compares the mean rates of change for two or more groups, and proposes a statistic for use in determining sample size requirements. One calculates individual rates of change with least squares estimates of slopes of individuals' responses regressed over time. The assumption of linear change over time, while clearly not applicable for some data, applies to many biological measurements, either as recorded or with some transformation. The variances of these estimated slopes have two components: within-individual variability based on measurement error and length and frequency of follow-up, and true between-individual slope variability. It is assumed that measurement error is the same for all subjects, so that the total variances differ due to differences in follow-up. The question addressed is: when can one use the usual ANOVA F statistic to compare group means of estimated slopes? Expected mean squares demonstrate that this F is appropriate when either each group has the same number of subjects, or when each subject has the same length and frequency of follow-up. A procedure for computing power and sample size is presented, where one can specify the maximum detectable difference in any two average slopes. Moment estimation and maximum likelihood estimation of variance components from prior data are discussed.