Statistical inference after variable selection in Cox models: A simulation study

Choosing relevant predictors is central to the analysis of biomedical time-to-event data. Classical frequentist inference, however, presumes that the set of covariates is fixed in advance and does not account for data-driven variable selection. As a consequence, naive post-selection inference may be biased and misleading. In right-censored survival settings, these issues may be further exacerbated by the additional uncertainty induced by censoring. We investigate several inference procedures applied after variable selection for the coefficients of the Lasso and its extension, the adaptive Lasso, in the context of the Cox model. The methods considered include sample splitting, exact post-selection inference, and the debiased Lasso. Their performance is examined in a neutral simulation study reflecting realistic covariate structures and censoring rates commonly encountered in biomedical applications. To complement the simulation results, we illustrate the practical behavior of these procedures in an applied example using a publicly available survival dataset.

💡 Research Summary

This paper addresses a critical gap in survival analysis: how to conduct valid statistical inference after data‑driven variable selection in Cox proportional‑hazards models. While Lasso‑type penalization (standard Lasso and adaptive Lasso) is routinely used to identify a sparse set of predictors in high‑dimensional biomedical data, the resulting coefficient estimates are biased and the usual Wald confidence intervals are no longer trustworthy because the selection step alters their sampling distribution. Moreover, right‑censoring adds another layer of uncertainty, making post‑selection inference even more challenging.

The authors evaluate three post‑selection inference strategies that have been proposed for linear models and adapt them to the Cox setting:

1. Sample Splitting – The data are randomly divided (here a 50/50 split). The first half is used to run the Lasso and select a sub‑model; the second half is then used to fit an unpenalized Cox model on the selected variables and construct standard Wald confidence intervals. Because inference uses data independent of the selection step, conditional coverage is guaranteed. However, the price is reduced statistical efficiency: confidence intervals are wider and power is lower, especially when the overall sample size is modest or censoring is heavy.

2. Exact Conditional Post‑Selection Inference (Exact PSI) – Building on the Lee et al. framework, this method conditions on the exact selection event (the active set chosen by the Lasso) and derives finite‑sample valid confidence intervals for the sub‑model coefficients. For the Cox model, the partial likelihood replaces the Gaussian likelihood, but the core conditioning principle remains. Implementation requires the penalty parameter λ to be fixed in advance (no data‑driven tuning), and randomized extensions that would allow cross‑validated λ are not yet available for Cox models. Consequently, while the method achieves nominal coverage, its practical applicability is limited by computational burden and the need for pre‑specified λ.

3. Debiased Lasso (also called Desparsified Lasso) – Originally developed for linear regression, this approach adds a one‑step correction to the Lasso estimator that removes the leading ℓ1‑penalty bias, restoring asymptotic normality. In the Cox context, the correction uses the score function of the partial likelihood and an estimator of the inverse Fisher information matrix that accounts for censoring. The resulting “debiased” estimator is asymptotically normal, allowing construction of Wald‑type confidence intervals without conditioning on the selection event. Hence coverage is unconditional (asymptotic) rather than conditional.

To compare these methods, the authors design a comprehensive simulation study that extends the framework of Kammer et al. (originally for Gaussian data) to right‑censored survival outcomes. Key design features include:

• Realistic covariate correlation structures (AR(1) and block dependence).
• Varying dimensionality (p = 100, 200) and sample sizes (n = 200, 500).
• Different censoring rates (≈20 %, 35 %, 50 %).
• Sparse true coefficient vectors (10–15 non‑zero β’s) with moderate (0.5) and strong (1.0) effect sizes.

Performance metrics are (i) selective coverage (the proportion of selected coefficients whose confidence intervals contain the true sub‑model parameter), (ii) average interval width, (iii) selective power (ability to detect truly non‑zero coefficients among those selected), and (iv) false‑positive selection rate.

Simulation Findings

• Sample Splitting consistently attains coverage close to the nominal 95 % but yields the widest intervals and the lowest power, especially under high censoring or small n. Its robustness stems from the independence between selection and inference data.
• Exact PSI achieves nominal coverage when λ is fixed, but the requirement to pre‑specify λ reduces flexibility; cross‑validated λ cannot be used without randomization, which is not yet implemented for Cox models. Computational time is substantially higher than the other methods.
• Debiased Lasso offers a favorable trade‑off: coverage is slightly below nominal (≈92–95 %) but intervals are markedly shorter than those from sample splitting, and selective power is the highest among the three. Coverage deteriorates modestly as censoring rises, reflecting the reliance on asymptotic approximations.

The authors also apply the three methods to a publicly available breast‑cancer survival dataset (e.g., METABRIC). After Lasso selection, they construct selective confidence intervals using each approach. Results mirror the simulation: the debiased Lasso provides the most concise intervals and identifies several biologically plausible predictors; Exact PSI yields intervals that are valid but limited to a smaller set of variables; sample splitting produces very wide intervals, reflecting its conservative nature.

Conclusions and Practical Recommendations

• No single method dominates across all scenarios. For small samples or heavy censoring, sample splitting is the safest choice because it guarantees conditional coverage, albeit at the cost of efficiency.
• When the sample size is moderate to large and censoring is not extreme, the debiased Lasso emerges as the most practical tool: it is computationally straightforward, does not require a pre‑specified λ, and delivers relatively tight intervals with acceptable coverage.
• Exact PSI is theoretically appealing for finite‑sample validity but its current implementation constraints (fixed λ, lack of randomization) limit its use in routine biomedical research.
• Future work should focus on developing randomized PSI for Cox models, improving high‑dimensional Fisher information estimation under censoring, and extending these ideas to settings with informative censoring or competing risks.

Overall, the study provides a thorough empirical benchmark that guides analysts in selecting an appropriate post‑selection inference method for Cox models, highlighting the trade‑offs between validity, efficiency, and computational feasibility.