Favoring the hierarchical constraint in penalized survival models for randomized trials in precision medicine

The research of biomarker-treatment interactions is commonly investigated in randomized clinical trials (RCT) for improving medicine precision. The hierarchical interaction constraint states that an interaction should only be in a model if its main effects are also in the model. However, this constraint is not guaranteed in the standard penalized statistical approaches. We aimed to find a compromise for high-dimensional data between the need for sparse model selection and the need for the hierarchical constraint.

To favor the property of the hierarchical interaction constraint, we proposed to create groups composed of the biomarker main effect and its interaction with treatment and to perform the bi-level selection on these groups. We proposed two weighting approaches (Single Wald (SW) and likelihood ratio test (LRT)) for the adaptive lasso method. The selection performance of these two approaches is compared to alternative lasso extensions (adaptive lasso with ridge-based weights, composite Minimax Concave Penalty, group exponential lasso and Sparse Group Lasso) through a simulation study. A RCT (NSABP B-31) randomizing 1574 patients (431 events) with early breast cancer aiming to evaluate the effect of adjuvant trastuzumab on distant-recurrence free survival with expression data from 462 genes measured in the tumour will serve for illustration. The simulation study illustrates that the adaptive lasso LRT and SW, and the group exponential lasso favored the hierarchical interaction constraint. Overall, in the alternative scenarios, they had the best balance of false discovery and false negative rates for the main effects of the selected interactions. For NSABP B-31, 12 gene-treatment interactions were identified more than 20% by the different methods. Among them, the adaptive lasso (SW) approach offered the best trade-off between a high number of selected gene-treatment interactions and a high proportion of selection of both the gene-treatment interaction and its main effect.

Adaptive lasso with Single Wald and likelihood ratio test weighting and the group exponential lasso approaches outperformed their competitors in favoring the hierarchical constraint of the biomarker-treatment interaction. However, the performance of the methods tends to decrease in the presence of prognostic biomarkers.