Bayesian analysis of biomarker levels can predict time of recurrence of prostate cancer with strictly positive apparent Shannon information against an exponential attrition prior

Shariat et al previously investigated the possibility of predicting from clinical data (including Gleason grade and stage) and preoperative biomarkers, which of any pair of patients would suffer recurrence of prostate cancer first. We wished to establish the extent to which predictions of time of relapse from such a model could be improved upon using Bayesian methods. The same dataset was reanalysed with a Bayesian skew-Student mixture model. Predictions were made of which of any pair of patients would relapse first and of the time of relapse. The benefit of using these biomarkers relative to predictions made without them was measured by the apparent Shannon information, using as prior an exponential attrition model of relapse time independent of input variables. Using half the dataset for training and the other half for testing, predictions of relapse time from the strict Cox model gave $-\infty$ nepers of apparent Shannon information (it predicts that relapse can only occur at times when patients in the training set relapsed). Deliberately smoothed predictions from the Cox model gave -0.001 (-0.131 to +0.120) nepers, while the Bayesian model gave +0.109 (+0.021 to +0.192) nepers (mean, 2.5 to 97.5 centiles), being positive with posterior probability 0.993 and beating the blurred Cox model with posterior probability 0.927. These predictions from the Bayesian model thus outperform those of the Cox model, but the overall yield of predictive information leaves scope for improvement of the range of biomarkers in use. The Bayesian model presented here is the first such model for prostate cancer to consider the variation of relapse hazard with biomarker concentrations to be smooth, as is intuitive. It is also the first to be shown to provide more apparent Shannon information than the Cox model or to be shown to provide positive apparent information relative to an exponential prior.