A logical question we’d like to answer here is, “What is the probability of disease given a positive test.” <\/em><\/strong>Written in probability notation we are asking, p(Disease | Positive)<\/strong><\/em>.<\/p>\n Using the data in the table, we can quickly compute this as 25 people who were positive and had the disease divided by 28 total positive tests. 25\/28 = 89.3%<\/strong><\/em><\/p>\n Of course, we could also compute this using Bayes Theorem:<\/p>\n p(Disease | Positive) = p(Disease) * p(Positive | Disease) \/ p(Positive)<\/strong><\/em><\/p>\n We store the necessary values in R objects and then compute the result<\/p>\n This checks out. We get the exact same result as when we did 25\/28.<\/p>\n Okay, how did we get here? Why does it work out like that?<\/strong><\/span><\/p>\n The math works out because we start with two different joint probabilities, p(A n B) and p(B n A). Or, in our case, p(Disease n Positive) and p(Positive n Disease). Formally, we can write these as follows:<\/p>\n We’ve already stored the necessary probabilities in specific elements, above. But here it what they both look like using our 2×2. First I’ll calculate it with the counts directly from the table and then calculate it with the R elements that we stored. You’ll see the answers are the same.<\/p>\n Well, would you look at<\/strong> that! <\/strong><\/em>The two joint probabilities are equal to each other. We can formally test that they are equal to each other by setting up a logical equation in R.<\/p>\n![\"\"](\"https:\/\/optimumsportsperformance.com\/blog\/wp-content\/uploads\/2024\/02\/Screenshot-2024-02-25-at-11.26.57\u202fAM.png\")
<\/a><\/p>\n\r\n### What we want to know: p(Disease | Positive)\r\n# p(Disease | Positive) = p(Disease) * p(Positive | Disease) \/ p(Positive)\r\n# p(A | B) = p(A) * p(B|A) \/ p(B)\r\n\r\np_disease <- 30\/73\r\np_positive_given_disease <- 25\/30\r\np_positive <- 28\/73\r\n\r\np_no_disease <- 43\/73\r\np_positive_given_no_disease <- 3\/43\r\n\r\np_disease_given_positive <- (p_disease * p_positive_given_disease) \/ (p_disease * p_positive_given_disease + p_no_disease * p_positive_given_no_disease) \r\np_disease_given_positive\r\n<\/pre>\n
<\/a><\/p>\n<\/a><\/p>\n\r\n## Joint Probability 1: p(Positive n Disease) = p(Positive | Disease) * p(Disease)\r\n\r\n25\/30 * 30\/73\r\n\r\np_positive_given_disease * p_disease\r\n\r\n## Joint Probability 2: p(Disease n Positive) = p(Disease | Positive) * p(Positive)\r\n\r\n25\/28 * 28\/73\r\n\r\np_disease_given_positive * p_positive\r\n<\/pre>\n
<\/a><\/p>\n\r\n## These two joint probabilities are equal!\r\n# p(Positive n Disease) = p(Disease n Positive)\r\n\r\n(p_positive_given_disease * p_disease) == (p_disease_given_positive * p_positive)\r\n<\/pre>\n