Power and sample size calculations for longitudinal studies estimating a main effect of a time-varying exposure

Existing study design formulas for longitudinal studies assume that the exposure is time invariant or that it varies in a manner that is controlled by design. However, in observational studies, the investigator does not control how exposure varies within subjects over time. Typically, a large number of exposure patterns are observed, with differences in the number of exposed periods per participant and with changes in the cross-sectional mean of exposure over time. This article provides formulas for study design calculations that incorporate these features for studies with a continuous outcome and a time-varying exposure, for cases where the effect of exposure on the response is assumed to be constant over time. We show that incorrectly using the formulas for time-invariant exposure can produce substantial overestimation of the required sample size. It is shown that the exposure mean, variance and intraclass correlation are the only additional parameters needed for exact solutions for the required sample size, if compound symmetry of residuals can be assumed, or to a good approximation if residuals follow a damped exponential correlation structure. The methods are applied to several examples. A publicly available programme to perform the calculations is provided.