## Simple Bayesian Analyses Generalized Carrier Versus Noncarrier

The preceding Bayesian analyses can be generalized as in Table 5-1. Note that if the correct prior and conditional probabilities can be determined, the rest is simple calculation. Setting up a spreadsheet, as in Table 5-1, facilitates clinical Bayesian analyses.

A very common application of Bayesian analysis in molecular pathology is to calculate carrier risk after a negative test result, as in the second example, above. The need to calculate carrier risk in this scenario stems from the fact that the sensitivity of most carrier tests is, at present, less than 100%; therefore, a negative test result decreases, but does not eliminate, carrier risk. Hypothesis 1 in this scenario is that the consultand is a carrier, and Hypothesis 2 is that the consultand is a noncarrier (Table 5-1). The prior carrier probability ("A" in Table 5-1) depends on whether there is a family history, and if there is, on the relationship of the consultand to the affected family member as shown by the family pedigree. In the absence of a family history, the prior carrier probability is the population carrier risk for that disease. In the case of CF and some other diseases, the appropriate population risk depends on the ethnicity of the consultand. The conditional probabilities ("C" and "D" in Table 5-1) are 1 minus the test sensitivity, and the test specificity, respectively. The remainder of the table is completed through 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. This is shown schematically in Figure 5-3.

For illustration, suppose in the second example above (Figure 5-2) that the consultand's husband is Ashkenazi Jewish, that he has no family history of CF, and that he tests negative for all 23 mutations in the ACMG screening guidelines panel. What is his carrier risk? The carrier risk in Ashkenazi Jewish populations, and therefore the husband's prior carrier risk in the absence of a family history, is approximately 1/25 ("A" in Table 5-1). Thus, his prior probability of being a noncarrier is 24/25 ("B"in Table 5-1). The

Table 5-1. Simple Bayesian Analysis Generalized | ||

Hypothesis | ||

1 |
2 | |

Prior probability |
A |
B = 1 - A |

Conditional probability |
C |
D |

Joint probability |
E = AC |
F = BD |

Posterior probability |
G = E ■ |
(E + F) H = F ■ (E + F) |

Hypothesis 1 (carrier) Prior probability = A

Hypothesis 2 (Noncarrier) Prior probability = B

Hypothesis 1 (carrier) Prior probability = A

Hypothesis 2 (Noncarrier) Prior probability = B

1 - F |
1 - D | ||

(False positive) |
(1 - Specificity) | ||

1 - C |
1 - E | ||

(Sensitivity) |
(True positive) | ||

F = BD |
D | ||

(True negative) |
( Specificity) | ||

C |
E = AC | ||

(1 - Sensitivity) |
(False negative) |

Figure 5-3. Schematic representation of the generalized Bayesian analysis shown in Table 5-1, for the case of a negative carrier test. The small rectangles represent true-positive, false-positive, true-negative, and false-negative rates for a particular consul-tand; that is, the prior probabilities are influenced by factors such as family history or signs and symptoms, and the sensitivity and specificity of the test are influenced by factors such as ethnicity. For a negative carrier test, the posterior carrier probability (1 minus the negative predictive value) is the false-negative rate divided by the sum of the false- and true-negative rates, or E ■ (E + F).

Figure 5-3. Schematic representation of the generalized Bayesian analysis shown in Table 5-1, for the case of a negative carrier test. The small rectangles represent true-positive, false-positive, true-negative, and false-negative rates for a particular consul-tand; that is, the prior probabilities are influenced by factors such as family history or signs and symptoms, and the sensitivity and specificity of the test are influenced by factors such as ethnicity. For a negative carrier test, the posterior carrier probability (1 minus the negative predictive value) is the false-negative rate divided by the sum of the false- and true-negative rates, or E ■ (E + F).

ACMG screening guidelines panel of 23 mutations detects 94% of CF mutations in Ashkenazi Jewish populations,1012 so the conditional probability of a negative test, under the hypothesis that he is a carrier, is 6% = 3/50 ("C" in Table 51). Under the hypothesis that he is a noncarrier, the conditional probability of a negative test approximates 1 ("D" in Table 5-1). (This is generally the case in genetic testing, since noncarriers by definition lack mutations in the relevant disease gene and, hence, unless there are technical problems, essentially always should test negative.) The Bayesian analysis table for this example is shown in Table 5-2. The joint probabilities are the products of the prior and conditional probabilities ("E" and "F" in Table 5-1), and the posterior probabilities ("G" and "H" in Table 5-1) derive from each joint probability divided by the sum of the joint probabilities. The husband's posterior carrier risk after the negative test result is 1/401 (Table 5-2).

What is the risk that the fetus of the mother (consultand) in Figure 5-2 and the father from Table 5-2 is affected with CF? Prior to testing, the risk was the prior probability that the mother was a carrier (2/3), multiplied by the prior probability that the father was a carrier (1/25), multiplied by the probability that the fetus would inherit two disease alleles (1/4), or 2/3 x 1/25 x 1/4 = 1/150. After testing, the risk is the posterior probability that the mother is a carrier (1/6), multiplied by the posterior probability that the father is a carrier (1/401), multiplied by the probability that the fetus would inherit two disease alleles (1/4), or 1/6 x 1/401 x 1/4 = 1/9600.

Often, testing is performed on additional family members, and genetic risks need to be modified accord-

Table 5-2. Bayesian Analysis for an Ashkenazi Jewish Individual Without a Family History of CF Who Tests Negative for the ACMG Screening Guidelines Panel of 23 CFTR Mutations | ||

Hypothesis | ||

Carrier |
Noncarrier | |

Prior probability |
1/25 |
24/25 |

Conditional probability |
3/50 |
1 |

(of negative test | ||

result) | ||

Joint probability |
3/1250 |
24/25 |

Posterior probability |
(3/1250) ■ |
(24/25) ■ (3/1250 |

(3/1250 + 24/25) |
+ 24/25) = 400/401 | |

= 1/401 |

ingly. In the example above, testing of both parents of the mother (consultand) would affect her carrier risk calculations. Detection of mutations in both parents using the same mutation test panel would essentially rule out carrier status for the mother, since we would then know that the sensitivity of the test for the mutations she is at risk of carrying is essentially 100%. Alternatively, if the test results for the mother's parents are positive for only one of her parents (for example, her father) and negative for the other parent (her mother), then the sensitivity of the test for the mutations she is at risk of carrying is essentially 50%. The Bayesian analysis for the mother, modified from Figure 5-2c, is shown in Table 5-3a. The conditional probability of a negative test under the hypothesis that she is a carrier has changed from 1/10 to 1/2, which increases the posterior probability that she is a carrier to 1/2. Taken together with her husband's carrier risk of 1/401 (Table 5-2), the risk that

Table 5-3a. Bayesian Analysis for the Consultand in |
Figure 5-2a | |

After Testing of the Parents (see text) | ||

Hypothesis | ||

Carrier |
Noncarrier | |

Prior probability 2/3 |
1/3 | |

Conditional probability 1/2 |
1 | |

(of negative test | ||

result) | ||

Joint probability 1/3 |
1/3 | |

Posterior probability (1/3) ■ (1/3 + 1/3) (1/3) |
■ (1/3 + 1/3) | |

= 1/2 |
= 1/2 | |

Table 5-3b. Alternative Bayesian Analysis for the Consultand in | ||

Figure 5-2a After Testing of the Parents (see text) | ||

Hypothesis | ||

Carrier | ||

Carrier with |
Carrier with | |

Paternal |
Maternal | |

(detectable) |
(undetectable) | |

Mutation |
Mutation |
Noncarrier |

Prior probability 1/3 |
1/3 |
1/3 |

Conditional 0 |
1 |
1 |

probability | ||

(of negative | ||

test result) | ||

Joint probability 0 |
1/3 |
1/3 |

Posterior probability 0 |
1/2 |
1/2 |

the fetus is affected with CF can be modified to 1/2 x 1/401 x 1/4 = 1/3200.

Another way of conceptualizing the Bayesian analysis described above is to separate the carrier hypothesis into two subhypotheses, as shown in Table 5-3b. (See also "Bayesian Analysis with More Than Two Hypotheses," below.) The two subhypotheses are (1) that the consultand is a carrier with a paternal (detectable) mutation and (2) that she is a carrier with a maternal (undetectable) mutation. The prior probability of each hypothesis is 1/3, that is, half of 2/3. The conditional probability of a negative test result, under the subhypothesis that she is a carrier of a detectable paternal mutation, is 0. The conditional probability of a negative test result, under the subhypothesis that she is a carrier of an undetectable maternal mutation, is 1. As in the generalized Bayesian analysis shown in Table 51, the joint probability for each hypothesis is the product of the prior and conditional probabilities for that hypothesis, and the posterior probability for each hypothesis is the joint probability for that hypothesis divided by the sum of all the joint probabilities. The posterior probability that the consultand has a detectable paternal mutation is 0, and the posterior probability that she has an undetectable maternal mutation is 1/2 (Table 5-3b).

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