CORRInsights®: Can a Bayesian Belief Network Be Used to Estimate 1-year Survival in Patients With Bone Sarcomas?
Bayesian means Bayes’ Theorem, which relates the total probability of an event occurring to the conditional probability of it occurring, given that another event has happened. For example, sensitivity—a concept all clinicians know about—is technically defined as the conditional probability of a positive test result given that the patient has a disease. Positive predictive value is the corresponding probability of the patient having the disease given the positive test result. Bayes’ Theorem states that the relationship between the two is the ratio between the total probability of having the disease (the prevalence) and the total probability of having a positive result, (which can be calculated from sensitivity, specificity, and prevalence). A low prevalence explains why the positive predictive value of a test can be low even if the sensitivity is high.
The Belief refers to the Bayesian view of probability. In most of the statistical analyses published in journals, probability is assumed to describe the observable frequency of an event or set of events, and so this approach to probability is called “frequentist.” In Bayesian methods, probability is seen as an initial belief (called the prior), which is updated based on accumulation of evidence. This can be explained through the clinician's own experience. If a patient presents to the office with fevers and a draining sinus, the physician will estimate a “pre-test probability”, also known as the Bayesian prior, of the patient having an infection. This pre-test probability might be the overall prevalence of infection or it might be based on the initial assessment of the patient. After diagnostic tests (such as white blood cell counts or bone scans) are performed, a “post-test probability” can calculated based on the pre-test probability, the test results, and the known reliability of the tests. One of the arguments in favor of the Bayesian approach to statistics is that it more closely mirrors actual experience.
The Network element is shown in Figure 1 of the current study , where each block may be directly or indirectly associated with the outcome of 1-year survival. In a regression model, each of the blocks would directly affect 1-year survival, and the resulting diagram would have predictors distributed like spokes around a hub. A regression model may explicitly allow for interaction terms in which the effect of one predictor is dependent on the level of another predictor. For example, the odds ratio for age could have one value for men and another for women; this should be explicitly defined by the analyst. In contrast, the Bayesian belief network would express it as different probabilities of mortality conditional on age and gender groups, which may be a more flexible arrangement. The graphical network can be predetermined by the researcher, or can be determined by an algorithm as was done in this study.
Putting it all together, the Bayesian belief network estimates conditional probabilities at given nodes of the network. For a specific set of predictors, the estimated probability of a patient dying at 1 year is calculated from the combination of the conditional probabilities. Forsberg and colleagues [1, 2] suggest that the Bayesian belief network's representation of conditional probabilities is more robust to missing data than alternatives such as artificial neural networks or logistic regression.