Cumulative regression function tests for regression models for longitudinal data

Abstract

The longitudinal regression model $Y_{i,j} = m(V_\tau_{i,j}^i) + \varepsilon_{i,j}$ where $Y_{i,j}$, is the $j$th measurement of the $i$th subject at random time $\tau_{i,j}$, $m$ is the regression function, $V_\tau_{i, j}$ is a predictable covariate process observed at time $\tau_{i,j}$ and $\varepsilon_{i,j}$ is noise, often provides an adequate framework for modeling and comparing groups of data. The proposed longitudinal regression model is based on marked point process theory, and allows a quite general dependency structure among the observations.

In this paper we find the asymptotic distribution of the cumulative regression function (CRF), and present a nonparametric test to compare the regression functions for two groups of longitudinal data. The proposed test, denoted the CRF test, is based on the cumulative regression function (CRF) and is the regression equivalent of the log-rank test in survival analysis. We show as a special case that the CRF test is valid for groups of independent identically distributed regression data. Apart from the CRF test, we also consider a maximal deviation statistic that may be used when the CRF test is inefficient.