Nonparametric estimation of conditional transition probabilities in a non-Markov illness-death model

Abstract

One important goal in multi-state modeling is the estimation of transition probabilities. In longitudinal medical studies these quantities are particularly of interest since they allow for long-term predictions of the process. In recent years significant contributions have been made regarding this topic. However, most of the approaches assume independent censoring and do not account for the influence of covariates. The goal of the paper is to introduce feasible estimation methods for the transition probabilities in an illness-death model conditionally on current or past covariate measures. All approaches are evaluated through a simulation study, leading to a comparison of two different estimators. The proposed methods are illustrated using a real colon cancer data set.

Appendix: Additional simulation results

In this section we give the additional simulation results for the two estimators (IPCW and LIN-based) using local linear weights instead of NW weights. The results were obtained using the dpik function which is available from the R KernSmooth package. See Tables 4, 5 and 6 below. Results for independent censoring were also obtained (not shown), leading to similar conclusions to those shown in Sect. 3 and in this Appendix.

Table 4 IMSE ($\times$1,000) of the estimated transition probabilities $\hat{p}_{11}(x;s,t)$ along 1,000 trials for different sample sizes. Estimates based on the local linear estimators

Table 5 IMSE ($\times$1,000) of the estimated transition probabilities $\hat{p}_{12}(x;s,t)$ along 1,000 trials for different sample sizes. Estimates based on the local linear estimators

Table 6 IMSE ($\times$1,000) of the estimated transition probabilities $\hat{p}_{22}(x;s,t)$ along 1,000 trials for different sample sizes. Estimates based on the local linear estimators