1 = 0.11. You can safely think "probability"
whenever we use the term "odds" and vice versa, as we will be dealing with small probabilities. Upon reading Appendix D, which contains the simplest probabilistic formulation of Bayes rules, you will see why we use the odds-likelihood form in this chapter—it is simpler to learn Bayes rules using this form, and more illuminating.
5.2.2. Odds-Likelihood Form of Bayes Rule
Equation 2 is the odds-likelihood form of Bayes rules when we have only one finding.
This equation says that if we know the prior odds Odds(D) (read prevalence) of a disease D and we observe one finding f of that disease in an individual, we can compute the posterior odds (read probability) by multiplying the prior odds times the likelihood ratio of that finding for that disease.
Likelihood ratios are nothing more than an alternative way of expressing the sensitivity and specificity of a test or observation for a disease. In fact, the likelihood ratio is defined in terms of sensitivities and specificities. The likelihood ratio positive (positive means that the finding drooling saliva is known to be present) of drooling saliva for the disease FMD follows:
p(drooling saliva presentlFMD present) p(drooling saliva presentlFMD absent)
In Equation 3, the numerator is sensitivity, and the denominator is the false positive rate (equal to 1 — specificity).
Note that the beauty of the likelihood ratio LR+drooling saiiva|FMD is that it is a very direct measure of how well drooling saliva discriminates between animals with FMD and animals without FMD. If drooling of saliva occurs much more frequently in animals with FMD than animals without FMD, LR+drooling saliva|FMD will be a large number. For a finding that is pathognomic (meaning that the finding in itself is sufficient to diagnose a disease), the denominator will be zero and the LR+ will be infinity, meaning that no matter how small the prior odds are, the posterior odds will be infinity (which converts to a probability of 1.0, and means that the individual has the disease with certainty). If, on the other hand, drooling of saliva occurs with equal frequency in animals with FMD and animals without FMD, the numerator and the denominator will be equal and the LR+drooling saliva|FMD will be equal to one, which when multiplied times the prior odds of FMD will result in a posterior odds that is equal to the prior odds. This result makes sense. If a finding cannot discriminate between animals with FMD and without FMD, it contains no diagnostic information for the disease FMD, and should not increase or decrease our belief that the animal has FMD.
The likelihood ratio negative (negative means that the finding drooling saliva is known to be absent) follows:
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