Penalized variable selection in multi-parameter regression survival modeling

Abstract

Standard survival models such as the proportional hazards model contain a single regression component, corresponding to the scale of the hazard. In contrast, we consider the so-called "multi-parameter regression" approach whereby covariates enter the model through multiple distributional parameters simultaneously, for example, scale and shape parameters. This approach has previously been shown to achieve flexibility with relatively low model complexity. However, beyond a stepwise type selection method, variable selection methods are underdeveloped in the multi-parameter regression survival modeling setting. Therefore, we propose penalized multi-parameter regression estimation procedures using the following penalties: least absolute shrinkage and selection operator, smoothly clipped absolute deviation, and adaptive least absolute shrinkage and selection operator. We compare these procedures using extensive simulation studies and an application to data from an observational lung cancer study; the Weibull multi-parameter regression model is used throughout as a running example.

Keywords: Variable selection; Weibull; differential evolution algorithm; multi-parameter regression; penalized maximum likelihood.

Figure 1.

Kaplan–Meier curves (solid) for different treatment groups with model-based curves overlaid (dashed) for the Cox PH model (left) and Weibull MPR model (right). PH: proportional hazards; MPR: multi-parameter regression.

Figure 2.

The BIC function evaluated at different tuning parameter values for the Weibull MPR model with the one tuning parameter LASSO penalty for the lung cancer data analysed in Section 5. The equivalent plot for the two tuning parameter LASSO penalties can be found in the Supplemental Material. BIC: Bayesian information criterion; MPR: multi-parameter regression;LASSO: least absolute shrinkage and selection operator.

Figure 3.

(a) True zero coefficients correctly set to zero (C) and (b) mean squared error (MSE) by model, distributional parameter and sample size across 1000 replicates for the models with two tuning parameters.

Figure 4.

Coefficient estimates from the models with adaptive least absolute shrinkage and selection operator (ALASSO) penalties by sample size and across the 1000 replicates: (a) ${\beta }_1$ coefficient estimates and (b) ${\beta }_2$ coefficient estimates (the dashed line represents the true coefficient value).

Figure 5.

Boxplots of SEEs for ${\beta }_1$ along with SEs (dot). SEEs: estimated standard errors; SEs: standard errors.

Figure 6.

Weibull model check. Here $\hat{H}(t)$ , along with the 95% confidence intervals, come from the Kaplan–Meier estimator.