Power and sample size calculations for longitudinal studies comparing rates of change with a time-varying exposure

Existing study design formulas for longitudinal studies have assumed that the exposure is time-invariant. We derived sample size formulas for studies comparing rates of change by exposure when the exposure varies with time within a subject, focusing on observational studies where this variation is not controlled by the investigator. Two scenarios are considered, one assuming that the effect of exposure on the response is acute and the other assuming that it is cumulative. We show that accurate calculations can often be obtained by providing the intraclass correlation of exposure and the exposure prevalence at each time point. When comparing rates of change, studies with a time-varying exposure are, in general, less efficient than studies with a time-invariant one. We provide a public access program to perform the calculations described in the paper (http://www.hsph.harvard.edu/faculty/spiegelman/optitxs.html).