A Causal Transformation Model for Time-to-Event Data Affected by Unobserved Confounding

Motivated by studies investigating causal effects in survival analysis, we propose a transformation model to quantify the impact of a binary treatment on a time-to-event outcome. The approach is based on a flexible linear transformation structural model that links a monotone function of the time-to-event with the propensity for treatment through a bivariate Gaussian distribution. The model equations are specified as functions of additive predictors, allowing the impacts of observed confounders to be accounted for flexibly. Furthermore, the effect of the instrumental variable may be regularized through a ridge penalty, while interactions between the treatment and modifier variables can be incorporated into the model to acknowledge potential variations in treatment effects across different subgroups. The baseline survival function is estimated in a flexible manner using monotonic P-splines, while unobserved confounding is captured through the dependence parameter of the bivariate Gaussian. The proposal naturally provides an intuitive causal measure of interest, the survival average treatment effect. Parameter estimation is achieved via a computationally efficient and stable penalized maximum likelihood estimation approach and intervals constructed using the related inferential results. We revisit a dataset from the Illinois Reemployment Bonus Experiment to estimate the causal effect of a cash bonus on unemployment duration, unveiling new insights. The modeling framework is incorporated into the R package GJRM, enabling researchers and practitioners to fit the proposed causal survival model and obtain easy-to-interpret numerical and visual summaries.