2023.02.02 | Questions 25-26
Biostats (1/3) - Bayesian calculations
Hello,
Do you struggle with biostats? If so, then this series of posts is for you!
This is the first in a series of 3 posts that will focus on biostatistics concepts that are tested on the ABMGG and ABGC board exams. Next week we will discuss positive and negative predictive values.
Do you have resources that helped you learn biostats concepts applicable to genetic counseling? Comment below or email me, and I can update this post with those resources.
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-Daniel
Questions
Question 25
A 25 year old female presents for preconception counseling. She had a maternal uncle who died from Tay-Sachs disease. Her genetic carrier screening is negative. If the carrier screening has 80% sensitivity & 100% specificity for Tay-Sachs, what is the probability that she is a carrier for Tay-Sachs?
Question 26
Which of the following statements is correct regarding the carrier screening test for Tay-Sachs mentioned in question 25?
Explanation
I use the following 4-step approach when tackling probability questions involving Bayes theorem. I start by creating a 4x2 table. Each column represents a distinct state — in this case, either carrier or non-carrier status. The first row is for the prior probabilities and the last row for the posterior probabilities. The value within each cell represents a probability (0-1), and the sum of the prior and posterior probabilities must each sum to 1. The middle rows are the “conditional probability” and the “joint probability.” These two rows do not need to sum to 1.
Calculate the prior probabilities. Though these were given to us in the initial problem, you may be expected to calculate this yourself on the exam. The caption explains how we arrived at 1/3 and 2/3.
The 1/3 probability in the blue box can be derived from the pedigree. Both the patient (III-1, indicated by the arrow) and her mother (II-2) are unaffected. Also, the patient’s maternal grandparents (I-1 & I-2) are both carriers for Tay-Sachs, as they have an affected son (the patient’s uncle, II-3). The likelihood that II-2 is a carrier (yellow boxes) is 2/3 (see 2x2 table in lower right corner), because we know that II-2 is unaffected. If II-2 is a carrier, she would have a 50% (1/2) chance of passing on the variant to the patient (III-1). Therefore, the prior probability that the patient is a carrier is 2/3 x 1/2 = 1/3 (blue box). Since the prior probabilities must sum to 1, the prior probability that that patient is not a carrier is 2/3. Calculate the conditional probabilities. This is the trickiest step. Oftentimes, the conditional probabilities will be based on the sensitivity and specificity of the test. In this case, we know the sensitivity of the test is 80% and specificity is 100%. We also know that the conditional probabilities are conditional on a negative test result (because the question told us that the patient’s test result was negative). Therefore, the two conditional probabilities to calculate are that of a false negative (yellow box) and true negative (green box).
Think about each conditional probability separately. The first conditional probability (yellow box) is asking: “given the negative test result, what is the probability that this person is a carrier?” This is asking for the false negative rate (FNR), which is the same as 1 - sensitivity and represents the proportion of carriers who incorrectly receive a negative test result. Since we know that the sensitivity (i.e. the true positive rate (TPR)) is 80%, the FNR is 20% (or 1/5). The second probability is asking for the true negative rate (TNR) (i.e. specificity). In other words, “given the negative test result, what is the probability that this person is not a carrier?” This time, the predicted result and the person’s carrier status are concordant (hence the word “true” in “true negative”). Since the specificity given to us is 100%, the probability we should list in the green box is 1.
Calculate the joint probabilities. This part is relatively straightforward. Take the prior probability and multiply it by the conditional probability to get the joint probability (yellow boxes). Do this within each column. For example, the joint probability for this patient being a carrier is 1/3 x 1/5 = 1/15.
Calculate the posterior probabilities. This part is also relatively straightforward. Remember, the posterior probabilities, like the prior probabilities, must sum to 1. The idea is that we are “normalizing” the joint probabilities so that they sum to 1. Divide each joint probability by the sum of both joint probabilities. For example, for the first column, take the joint probability (1/15) and divide this by the sum of both joint probabilities (1/15 + 2/3). This gives us 1/11 — the probability of being a carrier, given a negative test result — which is our answer!
Question 26
The correct answer for Question 26 is that the false negative rate (FNR) is 20%. This was part of the calculation for the conditional probability (step 2), but we will review this once more. Let’s start with the sensitivity, also known as the true positive rate (TPR). The TPR is given to us and is 80%. The total number of patients who actually are carriers equals the number of true positives (TP) plus the number of false negatives (FN). The TP are the people who are carriers and have a positive test result, which is the intended outcome of this test. The FN, as the name suggests, consist of patients who falsely had a negative test result when they in fact actually are carriers (oops — not what we wanted!). The TPR plus the FNR should equal 1 (or 100%), so the FNR in this case is 20%.
The equations below can help if you are starting from a 2x2 table (as illustrated below):
TPR = TP / (TP + FN) = 80 / (80 + 20) = 80%
FNR = FN / (TP + FN) = 20 / (80 + 20) = 20%
A similar relationship exists between the true negative rate (TNR), aka specificity, and the false positive rate (FPR). The TNR is given to us in this case and is 100%. The total number of patients who actually are not carriers equals the number of true negatives (TN) plus the number of false positives (FP). The TN are the people who are not carriers and have a negative test result, which is the intended outcome of this test. The FP consist of patients who falsely had a positive test result when they in fact are not carriers (oops — not what we wanted!). The TNR plus the FPR should equal 1 (or 100%), therefore the FPR is 0%.
A similar set of equations below can help if you are starting from a 2x2 table (as illustrated below):
TNR = TN / (TN + FP) = 100 / (100 + 0) = 100%
FPR = FP / (TN + FP) = 0 / (100+ 0) = 0%
The 2x2 table below illustrates the concept of 80% sensitivity and 100% specificity within a population of patients who underwent carrier screening:
Learning objective
The (posterior) probability of being a carrier for a disease depends on multiple factors (e.g. prior probability, test result, test sensitivity/specificity) and can be calculated in a 4-step process using Bayes theorem. Specificity and sensitivity are measures of a test’s performance. The false negative and false positive rates can be derived from the test’s sensitivity and specificity, respectively.
2023 ABMGG General Exam Blueprint | VII. Reproductive Genetics → a. Carrier screening (p. 4)
Additional resources
Bayesian Analysis and Risk Assessment in Genetic Counseling and Testing (2004)
Bayesian analysis of pedigrees (YouTube)
Confusion matrix (Wikipedia)