Bayes’ Theorem: Determining Disease Risk Using Probability
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Bayes’ theorem is a probability formula. Here it is:
In simple terms: A and B are events. The theorem states that the probability of event A given that event B has occurred is equal to the probability of B given A, times the probability of A, divided by the probability of B. That was a mouthful. This is why we love notation lol.
Okay. So this simple formula can be applied to genetics problems. Specifically, if we want to determine the likelihood of someone having a specific genotype based on diagnostic test results and family history, we can use Bayesian analysis, which is based on this theorem.
First, you must identify the two events (A and B). In the context of a genetics problem, these are likely two genotypes that we are trying to determine the probabilities of. You then find the prior probability (Pprior) and conditional probability (Pconditional). Pprior is the probability that the patient has the genotype based on Hardy-Weinberg calculations or family history/pedigree calculations. Pconditional is the probability that the patient has the genotype based on information about the accuracy/sensitivity/specificity/error rate/whatever info is given about the diagnostic test. The joint probability (Pjoint) is the product of Pprior and Pconditional. The ratios of the Pjoints tell you about the likelihood of the patient having either genotype based on the test result.
That probably didn’t make much sense. Let’s do a problem. (sauce: USABO Semifinal Exam 2008)
Here’s an old article I wrote about this, before actually knowing what Bayes’ theorem was! Goes to show that there are multiple correct approaches to each problem. https://www.sixfootscience.com/brain-snips/sensitivity-vs-specificity-in-disease-diagnosis/
Here’s what the Bayes’ approach would look like:
The Pprior were determined using Hardy-Weinberg. The Pconditional were determined using the definitions of sensitivity and specificity, and the values for each given in the problem. If you have questions about how I got these, please don’t hesitate to reach out; I’d be happy to explain :)
Anyways, we need to take the Pjoint ratio, specifically for the case of getting a negative result of AA (because the question asks for probability of being “not a carrier”) as follows:
So that’s the answer! Here’s some resources to learn more about this:
Very helpful wikipedia: https://en.m.wikipedia.org/wiki/Bayes%27_theorem
Sample problem from Nikolai’s genetics: https://youtu.be/ewa1doHvR5w
Stonks: https://www.investopedia.com/terms/b/bayes-theorem.asp
