Using Bayes' theorem to answer a practical heart valve question

The risk of thromboembolism (TE) risk in a heart valve patient who has lived perhaps for 15 years since implant without TE or thrombosis is unknown. As patients are heterogeneous with regard to embolic risk, these patients must have a lower than average risk; whether this risk reduction could be quantified was the aim of the present study.

If all patients had the same risk, the TE-free curve would be exponential, with a constant hazard (equal to the 'linearized' rate). With a mixture of risks in a population, the population hazard will be a decreasing function of time. By fitting a certain parametric function to the TE-free curve, the mixing distribution can be estimated. Subsequently, given an observation for a particular patient, e.g. zero emboli in 15 years, Bayes' theorem can be used to update the mixing distribution for these patients.

Using observed TE-free curves for Starr-Edwards valves, the mixing distribution for TE-free periods from one to 25 years was estimated. Because the risk distributions were skewed, the mean value was higher than the median. The mean (median) risk (% per year) fell from 4.5 (1.8) at implant to 1.7 (0.7) at 15 years for the aortic position, and from 7.0 (3.4) to 2.4 (1.2) for the mitral position. Thus, the average risk after 15 years was approximately 35-40% of the risk at implant for both the aortic and mitral positions, using either the mean or median risk.

The mean risk for patients after 15 TE-free years is approximately one-third of the risk of all patients at implant. However, there is a wide range, with some patients still having higher risks. Bayes' theorem is useful for deriving such answers. Linearized rates have limited value for describing TE risk in a heterogeneous population.