## Simple Bayesian Analyses Generalized Affected Versus Unaffected

Another common application of Bayesian analysis in molecular pathology is to calculate the risk that a patient is affected with a particular disease after a negative test result. Again, the need to calculate risk in this scenario stems from the fact that the sensitivities of many genetic tests are less than 100%. Hypothesis 1 (in Table 5-1 and Figure 5-3) in this scenario is that the patient is affected, and Hypothesis 2 is that the patient is unaffected. The prior probability ("A" in Table 5-1 and Figure 5-3) usually derives mostly from signs and symptoms but also may depend on aspects of the patient's history, including family history in diseases with a genetic component. As in the CF example, above, the conditional probabilities ("C" and "D" in Table 5-1 and Figure 5-3) are 1 minus the test sensitivity, and the test specificity, respectively. The remainder of the analysis is accomplished by calculation, with the posterior probabilities ("G" and "H" in Table 5-1) representing 1 minus the negative predictive value, and the negative predictive value, respectively.

For example, suppose that a child with clinically typical spinal muscular atrophy type III (type III SMA; Kugelberg-Welander disease; OMIM #253400) tests negative for the homozygous deletion of the SMN1 gene found in most affected individuals. What is the probability that the child is affected with SMN1 -linked SMA? The Bayesian analysis for this scenario is shown in Table 5-4a. Wirth et al. found that 17 of 131 individuals with clinically typical type III

SMA lacked mutations in both SMN1 alleles (and therefore were considered to have diseases unrelated to SMN1);13 hence, the prior probability that the child is affected with SMN1-linked type III SMA is 114/131, or 0.87. Approximately 6% of individuals with SMN1-linked type III SMA have a deletion of one SMN1 allele and a subtle mutation, undetectable by simple polymerase chain reaction (PCR) testing for a homozygous deletion, in the other SMN1 allele;14 hence, the conditional probability of a negative test result under the hypothesis that the child is affected is 6/100 or 0.06. Homozygous deletions of SMN1, when present, are highly specific for SMN1 -linked SMA; hence, the conditional probability of a negative test result under the hypothesis that the child is unaffected with SMN1-linked SMA approximates 1. Following the simple calculation rules in Table 5-1, the posterior probability that the child is affected with SMN1 -linked type III SMA is approximately 0.29 (Table 5-4a).

Suppose that SMN1 dosage analysis is performed on the child's DNA (i.e.,the SMA carrier test), and the result is that the child has one copy of the SMN1 gene. What is the probability that he or she is affected with SMN1--linked SMA? The Bayesian analysis for this scenario is shown in Table 5-4b. Again, the prior probability that the child is affected with SMN1-linked type III SMA is 0.87. Because approxi-

Table 5-4a. Bayesian Analysis for a Child with Clinically Typical Type III SMA Who Tests Negative for Homozygous Deletions of the SMN1 Gene

Hypothesis

Affected Unaffected

Prior Probability 0.87 0.13

Conditional Probability 0.06 ~1

(of negative test result)

Joint Probability 0.052 0.13

Posterior Probability 0.29 0.71

Table 5-4b. Bayesian Analysis for a Child with Clinically Typical Type III SMA Who Has One Copy of the SMN1 Gene by Dosage Analysis

Hypothesis

Affected Unaffected

Prior probability 0.87 0.13

Conditional probability 0.06 0.026

(of 1-copy test result)

Joint probability 0.052 0.0034

Posterior probability 0.94 0.06

Table 5-4c. Bayesian Analysis for a Child with Clinically Typical Type III SMA Who Has 2 Copies of the SMN1 Gene by Dosage Analysis

Hypothesis Affected Unaffected

0.0009 0.9

0.00078 0.12

### 0.006 0.994

Prior probability Conditional probability (of 2-copy test result) Joint probability Posterior probability mately 6% of individuals with SMN1-linked type III SMA have a deletion of one SMN1 allele and a subtle mutation in the other SMN1 allele that is detectable as a single copy by dosage analysis,14 the conditional probability of a single-copy test result under the hypothesis that the child is affected is again 0.06. However, the carrier frequency for SMA in the general population is approximately 1/38;14 hence, in this scenario the conditional probability of a single-copy test result under the hypothesis that the child is unaffected with SMNl-linked SMA is 1/38 or 0.026. Following the simple calculation rules in Table 5-1, the posterior probability that the child is affected with SMNl-linked type III SMA is approximately 0.94 (Table 5-4b).

Suppose instead that the result of the SMN1 dosage analysis is that the child has two copies of the SMN1 gene. What is the probability that the child is affected with SMNl-linked SMA? The Bayesian analysis for this scenario is shown in Table 5-4c. Again, the prior probability that the child is affected with SMNl-linked type III SMA is 0.87. Only approximately 9 in 10,000 individuals with SMN1-linked type III SMA would be expected to have two subtle, nondeletion mutations, detectable as two gene copies by dosage analysis;14 hence, the conditional probability of a two-copy test result under the hypothesis that the child is affected is approximately 0.0009. Because more than 7% of unaffected individuals have three copies of the SMN1 gene, and approximately 2.5% of unaffected individuals have one copy of the SMN1 gene, for a total of 9.5% of unaffected individuals without two copies of SMN1,14 the conditional probability of a 2-copy test result under the hypothesis that the child is unaffected with SMN1--linked SMA is 90.5/100, or approximately 0.9. Following the simple calculation rules in Table 5-1, the posterior probability that the child is affected with SMN1-linked type III SMA is only approximately 0.006 (Table 5-4c).

Profiling by proteomics, RNA microarrays, or analysis of single-nucleotide polymorphisms (SNPs), or some combination of these, is likely to play an important role in molecular pathology in the future, and clinical test results will be reported, in many cases, as probabilities or relative risks. For example, suppose that a consultand has a 20% lifetime risk of developing a particular disease (based on family history, physical examination, or clinical laboratory test results, or a combination of these) and that his or her pro-teomic profile is 16 times more common in those who go on to develop the disease than in those who do not. What is his or her lifetime risk of developing the disease? The Bayesian analysis for this scenario is shown in Table 5-5. Hypothesis 1 (from Table 5-1) is that the consultand will develop the disease, and Hypothesis 2 is that the consul-tand will not develop the disease. The prior probabilities are 0.2 and 0.8 for Hypotheses 1 and 2,respectively. Because the conditional probability of the proteomic profiling result is 16 times more likely in those who develop the disease than in those who do not, the conditional probabilities ("C" and "D"in Table 5-1) are 16 and 1, respectively. Following the simple calculation rules in Table 5-1, the pos-

Table 5-5. Bayesian Analysis for a Consultand with a 20% Lifetime Risk of Developing a Disease and a Proteomic Profile 16 Times More Common in Those Who Develop the Disease Than in Those Who Do Not

Hypothesis Affected Eventually Never Affected Prior probability 0.2 0.8

Conditional probability 16 1

(of profiling result) Joint probability 3.2 0.8

Posterior probability 0.8 0.2

terior probability that the consultand will develop the disease is 0.8 (Table 5-5).

Note that because posterior probabilities are normalized joint probabilities, the absolute values of the conditional probabilities are unimportant, as long as the ratio (i.e., the odds ratio) between them is correct. This is also true of prior probabilities. For example, in the scenario above, prior probabilities of 1 and 4 can be substituted for 0.2 and 0.8 and the same answer is obtained. Likewise, in the first example of this chapter (Figure 5-1a), prior probabilities of 1 and 1 can be substituted for 1/2 and 1/2, and conditional probabilities of 1 and 8 can be substituted for 1/8 and 1, and the same answer is obtained. Hence, relative risks are easily incorporated into Bayesian analyses.

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