Web-based Supplementary Materials for ’Power and Sample Size Calculations for Longitudinal Studies Comparing Rates of Change with a Time-Varying Exposure’ by X. Basagaña and D. Spiegelman Web Appendix A Equivalence of conditional likelihood and a model on differences

Verbeke et al.[1] proved this equivalence for the mixed effects model, where Σi = ZiDZi + σ wI. This model has the special feature that conditional on the random effects, the observations are independent. The DEX model does not follow this structure. The proof given here is for a general response covariance matrix, Σi , and thus extends their results. Suppose that we have subject-specific intercepts ai, which can be fixed or random, and assume that E (Yi) = ai1 + Xiγ, where 1 is a vector of ones, Xi a matrix of covariates and γ a vector of regression parameters. Assuming normality of Yi and V ar (Yi) = Σi, the probability density function has