Methodology for Examining Differential Rates of Change for Longitudinal Data

A common objective for longitudinal studies is to characterize differences in the rate of growth, or rate of change of an outcome across covariate-defined groups. The statistical challenges and potential extensions of models for comparing rates of change are intriguing with a broad scope for improving scientific research. We present research and newly proposed methodology on various scientific and statistical application of models for comparing rates of change in longitudinal outcomes across groups. We first discuss a generalized approach to modeling rates of change through direct structuring of differences in rates of change. The regression methodology offers a direct and parsimonious comparison of rates of change across groups and allows flexibility for structuring the underlying time trend of the outcome. By directly structuring rates of change relative to general time trend, power for detecting differences in the rate of change is improved compared to an equivalent linear models approach when the outcomes time trajectory is non-linear. Secondly, methodology for differentiating rates of change is extended to modeling multivariate longitudinal data. A common or global difference in the rate of change between groups is measured across all outcomes while separately structuring the time trend and mean level group differences for each outcome. When the true difference in the rate of change is similar for each outcomes, the global rate parameter method improves the ability to distinguish between groups compared to estimating separate rate effects for each outcome. Finally, the direct modeling of rates of change is made more robust to model misspecification by developing a semi-parametric estimation approach. Non-parametric estimation of a smooth time trend function is incorporated with parametric estimation of differences in the rate of change. We describe methods for estimating the time trend non-parametrically based on penalized spline methodology. We illustrate the proposed methodology for longitudinal rates of change and its extensions using studies of growth in infant subjects. Models for comparing rates can also be applied to areas such as treatment trials and studies of environmental exposures. We conclude with a discussion of future areas of work and possible extensions for modeling rates of change using longitudinal data.