Likelihood ratios (LR) can be estimated as:
Dichotomous test for causality, positive result:
Dichotomous test for causality, negative result:
( Ie — -U Iu sensitivity 1 — specificity 1 — sensitivity specificity
Multiple (>2) independent dichotomous tests: LR x X LR 2 X . . .
*This prior odds estimation assumes that Jnc=Iu (see text for details). RD = incidence rate difference; Ic = product causal incidence rate; Inc= product non-causal incidence rate; Ie= product exposed incidence rate; Iu = product unexposed incidence rate; LR = likelihood ratio.
Figure 18.3. Estimating the parameters of the Bayesian odds model.
by dividing the incidence rate difference for the AE between product-exposed and product-unexposed groups by the product-unexposed incidence rate (Kramer, 1988a) (see Figure 18.3). If this quantity, also referred to as the relative excess rate, is used to estimate the prior odds, then the analyst must make a counterfactual assumption that the pro-duct-unexposed incidence rate is an acceptable approximation for the product-exposed, non-causal incidence rate, which may be incorrect (Greenland and Rothman, 1998). Since product exposure could accelerate the onset of AE occurrences that result from multifactorial (including product) causation, the product-unex-posed incidence rate could be greater than the product-exposed, non-causal incidence rate. For the purposes of this discussion, we note that the prior odds is a useful theoretical construct that expresses an initial expectation of relative causal/non-causal occurrence in product-exposed patients, and whose true value can be estimated as greater than or equal to the relative excess rate.
In the Bayesian odds paradigm, the prior odds is then multiplied by one or more likelihood ratios derived from case diagnostic data (see Figure 18.2). This result, called the posterior odds, represents a theoretical final view of relative product causality versus other causality after all relevant case information has been incorporated. Each likelihood ratio is the mathematical expression of a specific test result for product causality that was available for that case (a positive rechallenge test, for example). The likelihood ratio for a positive test result for a dichotomous test is equal to the test's sensitivity divided by 1 — specificity (Kramer, 1988b) (see Figure 18.3). The sensitivity (true positive rate) for a test is the probability of obtaining a positive test result in a population of causal cases, while the quantity 1 — specificity (false positive rate) of a test is the probability of obtaining a positive test result in a population of non-causal cases. The product of all likelihood ratios for all available tests for causality forms a single summary likelihood ratio for that AE occurrence. If no diagnostically applicable information is available for a particular AE occurrence, then the Bayesian odds model assumes a summary likelihood ratio of one.
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