Abstract: This paper proposes a nonparametric approach for jointly modelling longitudinal and survival data using functional principal components. The proposed model is data-adaptive in the sense that it does not require pre-specified functional forms for longitudinal trajectories and it automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error are represented by flexible basis functions, such as B-splines, and the model dimension is reduced by functional principal component analysis. The relationship between the longitudinal process and event history is assessed using Cox regression model. Although the proposed model inherits the flexibility of a nonparametric approach, the estimation procedure based on EM algorithm is intrinsically parametric, and thus is simple and easy to implement. The computation is more efficient by reducing the dimension of random coefficients, i.e., functional principal component scores. The reduction of dimension achieved from eigen-decomposition also makes the model particularly applicable for the sparse data often encountered in longitudinal studies. An iterative selection procedure based on the Akaike Information Criterion (AIC) is proposed to choose tuning parameters, such as the knots of spline basis and the number of principal components, so that an appropriate degree of smoothness can be assessed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data collected in a clinical trial for comparing the efficacy and safety of two antiretroviral drugs.
Key words and phrases: Cox regression, EM algorithm, functional principal components, longitudinal data, smoothing, survival.