Statistical issues in survival analysis (longitudinal mediation)
July 2, 2025 The importance of evaluating a longitudinal biomarker in survival analysis for overall or diseasefree survival can be important. The authors have defined a new joint model for a longitudinal biomarker and a time-to-event endpoint, taking into account clustered data from a meta-analysis (or centers in a multicenter clinical trial) as well as the causal mediating effect of the biomarker as ascertained through its indirect and direct effects. They defined a partial treatment effect (PTE) as the ratio of the natural indirect effect to the total treatment effect. A surrogate marker would be validated if the PTE value is large enough. The PTE(t) is a function of time. They represented the joint model as two separate function for Mij2 and TijzMx^2. The Mij2 is written as function of theta and beta. The theta vector is a sum of the fixed effect and individual random effects associated with each component of fij(t) so if fij(t) = (1,t)’ then theta will give a random intercept and random slope model. The g(t) part is used to take into account potential interaction of time and treatment. The baseline hazard function is estimated by cubic M-splines. Also the beta parameters are the fixed treatment effect on the biomarker and time-toevent while the random effects are trial level effects taking into account the heterogeneity of the treatment effects on both outcomes across trials and are assumed to be jointly Gaussian. They then defined a phi parameter to be a vector of the parameters of the model and an estimate of it can be obtained by maximizing the penalized likelihood using the Levenberg-Marquardt algorithm. This approach is very similar to their already existing frailtypack library in R. They chose a Monte-Carlo approach for integrating over the trial-level random effects and a pseudo-adaptive Gauss-Hermite quadrature for integrating over the individual-level random effects. They had to make some assumptions for identifiability. The stable unit treatment value assumption (SUTVA) requires that each version of the treatment is well defined as they have stated and also that there is no interference between an individual treatment and another’s outcome. The first part is okay for clinical trials but the second part may not hold. Another assumption is the consistency one which for the subject ij, the biomarker process Mij equals the potential process associated with the treatment actually assigned to the patient. Usually this