The practical utility of incorporating model selection uncertainty into prognostic models for survival data

Abstract

Predictions of disease outcome in prognostic factor models are usually based on one selected model. However, often several models fit the data equally well, but these models might differ substantially in terms of included explanatory variables and might lead to different predictions for individual patients. For survival data, we discuss two approaches to account for model selection uncertainty in two data examples, with the main emphasis on variable selection in a proportional hazard Cox model. The main aim of our investigation is to establish the ways in which either of the two approaches is useful in such prognostic models. The first approach is Bayesian model averaging (BMA) adapted for the proportional hazard model, termed ‘approx. BMA’ here. As a new approach, we propose a method which averages over a set of possible models using weights estimated from bootstrap resampling as proposed by Buckland et al., but in addition, we perform an initial screening of variables based on the inclusion frequency of each variable to reduce the set of variables and corresponding models. For some necessary parameters of the procedure, investigations concerning sensible choices are still required. The main objective of prognostic models is prediction, but the interpretation of single effects is also important and models should be general enough to ensure transportability to other clinical centres. In the data examples, we compare predictions of our new approach with approx. BMA, with ‘conventional’ predictions from one selected model and with predictions from the full model. Confidence intervals are compared in one example. Comparisons are based on the partial predictive score and the Brier score. We conclude that the two model averaging methods yield similar results and are especially useful when there is a high number of potential prognostic factors, most likely some of them without influence in a multivariable context. Although the method based on bootstrap resampling lacks formal justification and requires some ad hoc decisions, it has the additional positive effect of achieving model parsimony by reducing the number of explanatory variables and dealing with correlated variables in an automatic fashion.