_ p(drooling saliva absentlFMD present) drooimgsaiiva|FMD = p(drooling saliva absentlFMD absent)
In Eq. 4, the numerator is the false negative rate (equal to 1 - sensitivity), and the denominator is specificity. In medicine, the absence of a finding that we expect to see if the patient actually has the disease in question is useful information. There is even a term for it—significant negative finding. A significant negative finding, such as a negative laboratory test, helps to rule out a diagnosis. A likelihood ratio negative is always a number that is less than one (but greater or equal to zero) for findings that we expect to see more often in individuals with the disease than in individuals without the disease.
Equation 2 expresses the essence of Bayes rules. Diagnosticians use Bayes rules to update their prior belief in a diagnosis in light of new information. When a diagnostician has no information whatsoever about an individual, her belief that the individual has a disease should be the prevalence of the disease. If she makes an observation (whether positive or negative), she should update her belief in the diagnosis by multiplying the likelihood ratio for the test or observation for that disease times the prior odds of the disease (think "prevalence").
If we know nothing about a cow whatsoever, then our belief that the cow has FMD is simply the prevalence of FMD. If we subsequently observe that the cow is drooling saliva, Bayes rules instructs us to update our belief that the cow has FMD using the information in Table 13.1 and 13.2 using the following calculation:
Odds(FMD l drooling saliva)
= LRdrooling saliva lFMD * Odds(FMD)
= p(drooling saliva IFMD present) 0.001
p(drooling salivalFMD absent) l- G.GGl = 0.95 G.GGl = 0.05 x G.999 = 19 x 0.001 = 0.019
(If we observe that the cow is not drooling saliva, we would use the likelihood ratio negative in Bayes rules.)
If we subsequently observe that a second cow is sick, we can apply Bayes rules a second time. In effect, we are treating the posterior odds from the prior calculation as the new prior odds:
Odds(FMD I drooling saliva, more than one animal affected)
= LRmore than one animal affected I FMD * Md^FMD^roding sa
= p(more than one animal affected|FMD present) p(more than one animal affected|FMD absent)
In general, if we have N diagnostic facts about a cow (or person), the odds-likelihood form of Bayes rules has the following form:
Odds(Dlf1,f2,...,fn) = LRf |D x LR^ |D x...x LRf |D x Odds(D) (7)
The result of Eq. 6, Odds(FMDldrooling saliva, more than one animal affected) + 0.09, is the odds (think probability) that the cow has FMD given that we have observed drooling of saliva in this cow and at least one other cow. The probabilistic inference engine would repeat the same calculation for the disease MCD, using the prior odds of MCD and the likelihood ratios for the two findings for MCD. The result of this calculation (not shown) is 9.8 x 10-8 . Cows with MCD stagger, but rarely drool saliva.
Table 13.3 shows the differential diagnosis that our extremely simple diagnostic expert system would show to a user. The two diseases are sorted in order of posterior odds. Note that we converted the posterior odds back to posterior probabilities odds using the formula P - ^ + odds , which we obtained by solving Eq. 2 for p.
Note that the posterior odds (and probabilities) in this example are very low.The low posterior odds are also partly due to the low prior prevalence of FMD and MCD that we arbitrarily assign (note that zero prior probability is not acceptable for the likelihood ratio computation). Countries that are currently free from these diseases or within the final stage of a disease eradication program for the diseases use low prior prevalence for exotic diseases (e.g., FMD and MCD). The LR+s for these findings for the disease FMD are only 19 and 4.75, respectively. If we had a third finding, such as blisters on feet, the LR+ for this finding would be a third multiplier in the equation, possibly increasing the posterior odds for one or both diagnoses. If we had a positive result from a highly sensitive and specific test for FMD (very high LR+), the posterior odds for FMD might be quite high. The key question for biosurveillance systems is at what level of certainty of a diagnosis in a single individual or a set of individuals different response actions are warranted. The answer to this question depends on treatability and other cost-benefit considerations that we discuss in Part V of this book.
table 13.3 Differential Diagnosis for Cow Drooling Saliva in a Herd with At Least One Other Cow Drooling Saliva
5.3. Computing Posterior Probabilities Using Bayesian Networks
Readers should be aware that there are other forms of Bayes rules that do not rest on an assumption of conditional independence, given a diagnosis. A developer of a probabilistic expert system may choose to use these alternative forms to improve the diagnostic accuracy of a system. Although mathematically too complex to cover in a brief introductory tutorial to Bayesian inference, the differences among these more complex forms of Bayes rules are simple to explain using a graphical representation called a Bayesian network (Figure 13.10). A Bayesian network comprises a set of nodes and arcs where each node represents a conditional probability distribution for the variable that the node represents, conditioned on its parents (the nodes from which directed arcs connect to the node). For the benefit of statisticians, a Bayesian network is a (compact) factorization of the complete joint probability distribution over all variables represented by the nodes (Neapolitan, 2003). The arcs in a Bayesian network represent the statistical dependence and independence relationships among the variables in the model. Figure 13.10 is a Bayesian network diagram for our mini-BOSSS diagnostic expert system.
The fact that there is no arc between drooling saliva and more than one animal affected indicates that the probability of observing drooling of saliva in a cow, once we know whether the cow has FMD or MCD, is not affected by knowing that other animals have these symptoms and vice versa (knowing that other animals have drooling once we know that this cow has FMD does not change the probability that this animal is drooling). If drooling saliva and more than one animal affected were not independent, given that we know that the cow has FMD, then we could add an arc between them.
Figure 13.11 shows a portion of the Bayesian network that underlies the Pathfinder system, which is a diagnostic expert system for pathologists who are interpreting biopsies of lymph table 13.3 Differential Diagnosis for Cow Drooling Saliva in a Herd with At Least One Other Cow Drooling Saliva
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