Bayesian Odds Model
The odds form of Bayes' equation is useful as a general model upon which imputation-based methodology can be designed (see Figures 18.2 and 18.3). In the Bayesian causal/non-causal odds model, existing epidemiologic data determine a prior odds of product causation versus non-product causation that is available to the safety analyst at the outset of the procedure. It has been suggested that the prior odds can be approximated
Posterior odds = Prior odds x Summary likelihood ratio
Pr (d —► AE\b,C) Pr (D —B) Pr (C|D —*■ ae)
Pr (d-/*■ ae\b,c Pr (D "Aae| B) Pr (C| d -/*■ ae)
The best estimate for a particular case (C) of the ratio of posterior probabilities (posterior odds) that the AE (AE) was caused by (D —ME) or was not caused by (D AE) the evaluated product (D) can be calculated by multiplying the same ratio in the absence of case information (prior odds=prior probability of causality divided by the prior probability of non-causality) times a summary likelihood ratio. A summary likelihood ratio is the ratio of the probabilities that a case such as the one examined (C) would occur if the product was causal versus if the product was non-causal. The prior odds is conditional on baseline (B) information, while the posterior odds is conditional both on baseline and case (B,C) information. Reproduced from Clark et al. (2001), Epidemiologic Reviews 23(2), 191-210, Fig. 2, with permission from Oxford University Press.
Figure 18.2. The Bayesian odds model for evaluating the probability of causality versus non-causality in single product-AE cases.
The prior odds (lower limit) can be estimated as -
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