Joint partially linear model for longitudinal data with informative drop-outs

Abstract

In biomedical research, a steep rise or decline in longitudinal biomarkers may indicate latent disease progression, which may subsequently cause patients to drop out of the study. Ignoring the informative drop-out can cause bias in estimation of the longitudinal model. In such cases, a full parametric specification may be insufficient to capture the complicated pattern of the longitudinal biomarkers. For these types of longitudinal data with the issue of informative drop-outs, we develop a joint partially linear model, with an aim to find the trajectory of the longitudinal biomarker. Specifically, an arbitrary function of time along with linear fixed and random covariate effects is proposed in the model for the biomarker, while a flexible semiparametric transformation model is used to describe the drop-out mechanism. Advantages of this semiparametric joint modeling approach are the following: 1) it provides an easier interpretation, compared to standard nonparametric regression models, and 2) it is a natural way to control for common (observable and unobservable) prognostic factors that may affect both the longitudinal trajectory and the drop-out process. We describe a sieve maximum likelihood estimation procedure using the EM algorithm, where the Akaike information criterion (AIC) and Bayesian information criterion (BIC) are considered to select the number of knots. We show that the proposed estimators achieve desirable asymptotic properties through empirical process theory. The proposed methods are evaluated by simulation studies and applied to prostate cancer data.

Keywords: Joint models; Longitudinal data; Nonparametric maximum likelihood; Partially linear model; Random effects; Sieve maximum likelihood; Transformation models.

Figure 1

Simulation results for α(t) estimation when ϕ = 1.7. The solid curve indicates true α(t) = (t + 0.5)^−1.5 + (t/2 + 1)³ − 4. The dash-dotted (dotted) curves indicate the maximum, average, and minimum of the estimated α(t) over 1000 simulated data sets when the AIC-based (BIC-based) knot selection procedure was used. The AIC-estimates appeared to have larger variations in both tails than the BIC-estimates. This figure appears in color in the electronic version of this article.

Figure 2

Bayesian information criterion (BIC) plotted for different transformations H(x) = log(1 + ηx)/η and different numbers of interior knots (K_n).

Figure 3

Coefficient function of log PSA score, adjusted by T-stage, Gleason score, and age, under the best fit of transformation H(x) = log(1 + x) and 5 interior points. The solid (dashed) curve is an estimate from the joint (marginal) model. The circles and dots present the full history of all post-radiation PSA values for patients who dropped out informatively and non-informatively, respectively. The mark “×” on some circles indicates the last observation of PSA before the informative drop-out occurred.