Semiparametric and Nonparametric Regression Analysis of Longitudinal Data

TLDR

The paper addresses regression analysis of irregular, subject‑specific longitudinal measurements. It proposes least‑squares estimators for the constant and time‑varying regression coefficients, and a closed‑form estimator for their cumulative function. The authors introduce semiparametric and nonparametric models with unspecified stochastic structure, deriving n½‑consistent, asymptotically normal estimators and a Gaussian‑process‑convergent cumulative estimator. Simulations confirm accurate asymptotic approximations and high efficiency compared to parametric methods, and the approach is illustrated on HIV CD4 count data.

Abstract

This article deals with the regression analysis of repeated measurements taken at irregular and possibly subject-specific time points. The proposed semiparametric and nonparametric models postulate that the marginal distribution for the repeatedly measured response variable Y at time t is related to the vector of possibly time-varying covariates X through the equations E{Y(t)|| X(t) = α0(t) + β′0X(t) and E{Y(t)||X(t)} = α0(t)+ β′0(t)X(t), where α0(t) is an arbitrary function of t, β0 is a vector of constant regression coefficients, and β0(t) is a vector of time-varying regression coefficients. The stochastic structure of the process Y(·) is completely unspecified. We develop a class of least squares type estimators for β0, which is proven to be n½-consistent and asymptotically normal with simple variance estimators. Furthermore, we develop a closed-form estimator for a cumulative function of β0(t), which is shown to be n½-consistent and, on proper normalization, converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Extensive simulation studies demonstrate that the asymptotic approximations are accurate for moderate sample sizes and that the efficiencies of the proposed semiparametric estimators are high relative to their parametric counterparts. An illustration with longitudinal CD4 cell count data taken from an HIV/AIDS clinical trial is provided.