Bayesian Analyses with More Than One Conditional Probability

Often there is more than one test result, or more than one set of pedigree information, or both, that can be incorporated as conditional probabilities in a single Bayesian analysis. For example, consider the pedigree in Figure 5-4a, in which the two maternal great uncles of the consultand were affected with Duchenne muscular dystrophy (DMD; OMIM #310200),a severe X-linked recessive disease caused by mutations in the DMD gene (OMIM #300377). The con-sultand's maternal grandmother's carrier risk was 1/2, her mother's carrier risk was 1/4, and therefore the consul-tand's prior carrier risk is 1/8. Suppose that her carrier testing is negative using a highly specific test (an analysis for heterozygous deletions in the DMD gene) that detects 2/3 of carriers. Suppose also that her serum creatine phos-phokinase (CPK), which is elevated in two-thirds of carriers, is within normal limits. Taking into account her prior probability of 1/8, her normal molecular and CPK test results, and, in addition, her three normal sons, what is the probability that she is a carrier?

The Bayesian analysis for this scenario is shown in Figure 5-4b. Each conditional probability is given its own line. Because the genetic test detects 2/3 of carriers and is highly specific, the conditional probabilities of a negative genetic test result under the hypotheses that she is a carrier and noncarrier are 1/3 and 1, respectively. Because serum CPK is elevated in 2/3 of carriers, the conditional

 -H 1/
 1 a Hypothesis Carrier Non-carrier Prior Probability 1/8 7/8 Conditional probability (of negative genetic test result) 1/3 1 Conditional Probability (of normal CPK result) 1/3 19/20 Conditional Probability (of three normal sons) 1/8 1 Joint Probability 1/576 133/160 Posterior Probability 0.002 0.998

Figure 5-4. (a) Pedigree of a family with individuals affected with DMD (see text). Consultand is indicated by an arrow. (b) Bayesian analysis for the consultand in Figure 5-4a, taking into account her normal carrier test result, her normal CPK test result, and her three normal sons.

Figure 5-4. (a) Pedigree of a family with individuals affected with DMD (see text). Consultand is indicated by an arrow. (b) Bayesian analysis for the consultand in Figure 5-4a, taking into account her normal carrier test result, her normal CPK test result, and her three normal sons.

probability of a normal serum CPK for the hypothesis that she is a carrier is 1/3. Because 5% of noncarrier women have an abnormal serum CPK (i.e., the normal range is defined as comprising 95% of normal individuals),the conditional probability of a normal serum CPK under the hypothesis that she is a noncarrier is 95% or 19/20. Finally, as in Figure 5-1b, the conditional probabilities of three normal sons under the hypotheses that she is a carrier and noncarrier are 1/8 and 1, respectively. The joint probabilities for each hypothesis are the products of the prior probability, and all conditional probabilities, for each hypothesis (Figure 5-4b). Calculation of posterior probabilities then proceeds exactly as in Table 5-1. In this scenario, taking into account her normal test results and her three normal sons, the consultand's carrier risk is lowered from 1/8 to 0.002, or approximately 1/500.