Robust estimation of the expected survival probabilities from high-dimensional Cox models with biomarker-by-treatment interactions in randomized clinical trials

Abstract

Background: Thanks to the advances in genomics and targeted treatments, more and more prediction models based on biomarkers are being developed to predict potential benefit from treatments in a randomized clinical trial. Despite the methodological framework for the development and validation of prediction models in a high-dimensional setting is getting more and more established, no clear guidance exists yet on how to estimate expected survival probabilities in a penalized model with biomarker-by-treatment interactions.

Methods: Based on a parsimonious biomarker selection in a penalized high-dimensional Cox model (lasso or adaptive lasso), we propose a unified framework to: estimate internally the predictive accuracy metrics of the developed model (using double cross-validation); estimate the individual survival probabilities at a given timepoint; construct confidence intervals thereof (analytical or bootstrap); and visualize them graphically (pointwise or smoothed with spline). We compared these strategies through a simulation study covering scenarios with or without biomarker effects. We applied the strategies to a large randomized phase III clinical trial that evaluated the effect of adding trastuzumab to chemotherapy in 1574 early breast cancer patients, for which the expression of 462 genes was measured.

Results: In our simulations, penalized regression models using the adaptive lasso estimated the survival probability of new patients with low bias and standard error; bootstrapped confidence intervals had empirical coverage probability close to the nominal level across very different scenarios. The double cross-validation performed on the training data set closely mimicked the predictive accuracy of the selected models in external validation data. We also propose a useful visual representation of the expected survival probabilities using splines. In the breast cancer trial, the adaptive lasso penalty selected a prediction model with 4 clinical covariates, the main effects of 98 biomarkers and 24 biomarker-by-treatment interactions, but there was high variability of the expected survival probabilities, with very large confidence intervals.

Conclusion: Based on our simulations, we propose a unified framework for: developing a prediction model with biomarker-by-treatment interactions in a high-dimensional setting and validating it in absence of external data; accurately estimating the expected survival probability of future patients with associated confidence intervals; and graphically visualizing the developed prediction model. All the methods are implemented in the R package biospear, publicly available on the CRAN.

Keywords: Confidence intervals; Cox model; High-dimensional data; Penalized regression; Precision medicine; Prediction model; Prognostic biomarkers; Survival estimation; Treatment-effect modifiers.

Fig. 1

Schematic representation of the proposed framework. 1CV and 2CV: single and double cross-validation, ĥ ₀(t) baseline hazard at time t, $\widehat{\alpha },\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }}$: estimated regression parameters, ${\widehat{\phi }}_i={\displaystyle \sum _{j=1}^p}{\widehat{\beta }}_j{X}_{ij}$: prognostic score, ${\widehat{\eta }}_i={\displaystyle \sum _{j=1}^p}{\widehat{\gamma }}_j{X}_{ij}$: treatment-effect modifying score, $\widehat{\mathit{\pi }}$: entire linear predictor of the selected model, iBrier: integrated Brier score, C: Uno’s C-statistic, ΔC: different in arm-specific Uno’s C-statistics, Ŝ(t): estimated survival probability at time t

Fig. 2

Graphical illustration of the expected survival probability at a given timepoint against the treatment-effect modifying ore. Expected survival probability against the treatment-effect modifying score $\widehat{\eta }$ in the setting of: no prognostic biomarker identified and one (a) or multiple (b) treatment-effect modifiers; multiple prognostic markers and treatment-modifiers in the lowest (c) and higher (d) prognostic risk group. Dot: point estimate, vertical line: 95% pointwise confidence interval, solid curves: average smoothed splines for point estimates, dashed curves: average smoothed splines for confidence bounds. Graphical illustrations are coming from several selected models based on a simulated dataset from the scenario 6

Fig. 3

Arm-specific distant-recurrence free survival in the illustrated breast cancer trial. Vertical lines: 95% confidence interval at 5 years

Fig. 4

5-year distant-recurrence free survival against the treatment-effect modifying score of the effect of trastuzumab in early breast cancer. Graphical representation of the model showed in Table 4. DRFS: distant-recurrence free survival, dot: point estimate, solid curves: average smoothed splines for point estimates, dashed curves: average smoothed splines for confidence bounds