Direct regression models for longitudinal rates of change

Comparing rates of growth, or rates of change, across covariate-defined subgroups is a primary objective for many longitudinal studies. In the special case of a linear trend over time, the interaction between a covariate and time will characterize differences in longitudinal rates of change. However, in the presence of a non-linear longitudinal trajectory, the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change. Therefore, we propose regression methodology for longitudinal data that allows a direct, structured comparison of rates across subgroups even in the presence of a non-linear trend over time. Our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time, but this assumption can be relaxed. Rates are compared relative to a generally specified time trend for which we discuss both parametric and non-parametric estimating approaches. We develop mixed model longitudinal methodology that explicitly characterizes subject-to-subject variation in rates, as well as a marginal estimating equation-based method. In addition, we detail a score test to detect violations of the proportionality assumption, and we allow time-varying rate effects as a natural generalization. Simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non-linear trend in time. We apply our method to a study of growth among infants born to HIV infected mothers and conclude with a discussion of possible extensions for our methods.