Bridging the gap between two-stage and joint models: The case of tumor growth inhibition and overall survival models
Francois Mercier (1), Danilo Alvares (2)
(1) Roche-Genentech, CH, (2) MRC Biostatistics Unit, UK
Objectives: Many clinical trials generate both longitudinal biomarkers and time-to-event data. We might be interested in their relationship, as in the case of tumor size and overall survival (OS) in oncology drug development. Many well-established methods exist for analyzing such data either sequentially (two-stage models) or simultaneously (joint models). Two-stage models (2stgM) have been condemned (i) for not acknowledging that biomarkers are endogenous covariable to OS [1], and (ii) for not propagating the uncertainty of the longitudinal biomarker sub-model into the survival sub-model [2]. On the other hand, joint models (JM), which properly circumvent both problems, have been criticized for being computationally expensive, and difficult to use in practice [3]. In this work, we explore a third approach, referred to as the novel two-stage (N2stgM) model. This strategy reduces the model complexity without compromising the parameter estimates' accuracy.
Methods: The three approaches (JM, 2stgM, and N2stgM) are formulated, and a Bayesian framework is considered for their implementation. The tumor size time dynamics were modeled using a bi-exponential model [4], with a parameter k_g capturing the tumor re-growth rate. The Weibull hazard model was expressed as a function of time, relevant baseline covariates, and k_g. A motivating dataset was taken from the control arm of the HORIZON III study (N = 577). In our analysis, two parameters were entered into the survival model to represent the sex (gamma_sex) and baseline LDH (gamma_LDH) effects. The goodness-of-fit is evaluated using an integrated leave-one-out (LOO) criterion. In addition, a simulation study is run to compare across models the gain in parameter estimation precision and accuracy, as well as computing time, when the sample size is reduced (N = 100, 250, or 400 patients data).
Results: The three approaches gave satisfying diagnostics plots for the longitudinal sub-model, but the results differed for survival. In particular, the mean survival sub-model parameter estimate for k_g was significantly lower in the 2stgM compared to N2stgM and JM (respectively, alpha_k_g = 1.79 vs. 3.36 vs. 4.06). The 95%CI of the N2stgM parameters, and the resulting posterior predictive survival curve, were largely overlapping with the ones of the JM (respectively, alpha_k_g 95%CI: [2.48, 4.36] vs. [3.09, 5.18] ) while the ones from the 2stgM were offset (alpha_k_g 95%CI: [1.17, 2.41]). The CPU time for the JM was about 2.5 times larger than the one for N2stgM. In the simulation study, the parameter estimates from the results were similar except with larger uncertainty in the parameter estimates due to a lower sample size.
Conclusions: We have introduced a new class of models based on conditional distribution which performs as well as JM, without being computationally expensive. This approach may further be extended to other cases where the conditional distribution of one process carries information about the other process.
References:
[1] Kalbfleisch JD, Prentice RL. The statistical analysis of failure time data. Chap. 6. pages 193-217. New Jersey, US: John Wiley & Sons. 2nd ed. 2002.
[2] Prentice RL. Covariate measurement errors and parameter estimation in a failure time regression model. Biometrika 1982, 69: 331-342.
[3] Ye W, Lin X, Taylor J. Semiparametric modeling of longitudinal measurements and time-to-event data - A two-stage regression calibration approach. Biometrics 2008, 64: 1238-1246.
[4] Stein W, Figg WD, Dahut W, et al. Tumor growth rates derived from data for patients in a clinical trial correlate strongly with patient survival: A novel strategy for evaluation of clinical trial data. The Oncologist 2008, 13: 1046-1054.