2a:null 2b:["$","title",null,{"children":"Bayes' theorem for disease | Chris Wallace | Observable"}] 30:Tbc5,md`The accuracy of the test is ${Number(100*accuracyOfTest).toFixed(0)}%. The question of interest to us is "What is the probability that someone actually has the disease given that they have tested positive." That sounds like a job for Bayes' theorem. Writing it out: > ${tex`P(\rm{disease} | \rm{test\ positive}) = \frac{P(\rm{test\ positive} | \rm{disease}) P(\rm{disease})}{P(\rm{test \ positive})}`} We already know ${tex `P(\rm{test\ positive}|\rm{disease})`}, the probability that someone will test positive given that they have the disease: it's just the accuracy of the test, ${Number(100*accuracyOfTest).toFixed(0)}% (or ${Number(accuracyOfTest).toFixed(2)}). We also know the probability that someone has the disease, ${tex `P(\rm{disease})`} = ${probInfected}, since we set that above. We can calculate the probability of testing positive using the sum rule of probabilities (that the probability of an event is the sum of all the ways it can happen). There are only two ways to test positive. Either one has the disease and tests positive, or one does not have the disease and tests positive. The total probability of testing positive is just the sum of those two possibilities: > ${tex`P(\rm{test \ positive}) = P(\rm{test \ positive} | \rm{disease})P( \rm{disease}) + P(\rm{test \ positive}| \rm{no \ disease})P(\rm{no \ disease})`} The probability of not having the disease is just one minus the probability of having the disease, we can substitute in those numbers. > ${tex `P(\rm{test \ positive}) = ${Number(accuracyOfTest).toFixed(3)}*${Number(probInfected).toFixed(3)}+${Number(1-accuracyOfTest).toFixed(3)}*${Number(1-probInfected).toFixed(3)}`} So before we even know whether the person has the disease or not, we know the probability of them testing positive is about ${Number(probInfected*accuracyOfTest + (1-probInfected)*(1-accuracyOfTest)).toFixed(3)}. Substituting everything into Bayes' rule, we find that the probability of someone having the disease given that they've tested positive is ${Number( accuracyOfTest * probInfected/(probInfected*accuracyOfTest + (1-probInfected)*(1-accuracyOfTest)) ).toFixed(2)}. In the case of a rare disease (try a probability of infection of, say, 0.001) and an effective test (an accuracy of, say, 99%), do you find the result surprising? Below we show a population of 400 people who are all potentially infected with a disease. We're assuming the disease is independently and identically distributed among everyone. That means that one person having the disease does not make another more likely to have it. It's probably not a good model of infectious disease - maybe you could try dreaming up something it might be a good model for. Adjust the sliders here or above to see new randomly generated sample outcomes.`31:[] 2e:{"id":"d506ec61cdf9b62d","slug":"bayes-theorem-for-disease","title":"Bayes' theorem for disease","thumbnail":"610cbce74c6d1fad9d0680341517be6304c795faebb0042e11985485428e16d3","access_level":"public","create_time":"$D2018-02-01T22:35:27.879Z","update_time":"$D2022-02-18T14:38:18.615Z","view_time":null,"first_public_time":"$D2022-02-18T14:34:36.265Z","fork":false,"stars":3,"starred":false,"owner":{"type":"team","id":"0ac9abcf38e64f7b","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":"starter_2024"},"version":496,"publish_version":496,"role":null,"authored":false,"authors":[{"type":"individual","id":"b6d36fa8c41d8f92","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":null}],"members":[{"type":"individual","id":"b6d36fa8c41d8f92","login":"cjwallace","name":"Chris Wallace","avatar_url":"https://avatars-next.observableusercontent.com/ddff0ab0c8f4670e38fe8871e77257cde4558310a5516931b79aa56e27b59005","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Bayes' theorem for disease","cells":[{"id":0,"value":"md`# Bayes' theorem for disease`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":491,"value":"_This is a modification of an earlier notebook: [Bayes' Theorem](https://observablehq.com/@cjwallace/bayes-theorem), and was written around the same time (February 2018), despite it's late (February 2022) publishing date._","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":4,"value":"md`The ability to reason under uncertainty is essential to many areas of enquiry: science, business, life. Fortunately, formal mathematical tools for such have long since been developed.\n\nRecall that for two events _A_ and _B_, Bayes' theorem may be written,\n> ${tex`P(A|B) = \\frac{P(B|A)P(A)}{P(B)}`}\n\nOr, in words:\n> the probability of _A_ _given_ _B_ is the product of the probability of _B_ given _A_ and the total (marginal) probability of _A_, divided by the total probability of _B_.\n\nLet's make this more concrete with an example in medical diagnostics.\n\nSuppose there is a disease outbreak in an enclosed population. Maybe it's an airborn virus that turns people into zombies. Maybe it's hiccups. You can choose the probability of a person being infected by dragging this slider.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":95,"value":"viewof infectedSlider1 = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":165,"value":"md`The probability of a person being infected is ${probInfected}.\n\nNow let's say this is a subtle disease, but you've put your best scientists on it and they've come up with a test. While we're playing Elder God, you can also choose the accuracy of a test for the disease. An accuracy of 100% means that the test is always correct and 50% means it is no better than random. (Thought experiment: according to this formulation, why can the test not be less than 50% accurate?)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":100,"value":"viewof accuracySlider1 = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":184,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":480,"value":"viewof infectedSlider2 = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":441,"value":"md`The probability of a person being infected is ${probInfected}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":398,"value":"viewof accuracySlider2 = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":437,"value":"md`The accuracy of the test is ${Number(100*accuracyOfTest).toFixed(0)}%`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":43,"value":"chart = {\n const viz = d3.select(DOM.svg(600, 180));\n viz.append('g')\n\t .selectAll('circle')\n .data(circles)\n .enter()\n .append('circle')\n \t.attr('cx', d => grid(d.x))\n \t.attr('cy', d => grid(d.y))\n \t.attr('r' , '4')\n \t.attr('fill', d => d.colour)\n \n return viz.node()\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":410,"value":"md` are people who don't have the disease and tested negative - the general populace.\n are people who don't have the disease and tested positive - \"false alarms\".\n are people who have the disease and tested positive - correct diagnoses.\n are people who have the disease and tested negative - missed diagnoses.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":416,"value":"md`#### Supporting code\nBelow here in the notebook is the code that makes the interactive parts tick.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":104,"value":"import {slider} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":31,"value":"d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":45,"value":"grid = d3.scaleLinear()\n .domain([0, 40])\n .range([20, 580])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":53,"value":"circles = {\n var arr = []\n for (let i = 1; i <= 40; ++i) {\n for (let j = 1; j <= 10; ++j) {\n\t var col = '#eaeded' // light grey\n \n let infected = (Math.random() <= probInfected) ? true : false\n \n if (infected) {\n\t let test_positive = Math.random() <= accuracyOfTest\n\t col = test_positive ?\n \t \t'#f5b7b1' : // light red\n \t \t'#a93226' // dark red\n } else {\n let test_positive = Math.random() > accuracyOfTest\n\t col = test_positive ?\n \t\t '#f7dc6f' : // yellow\n \t\t\t col\t // still light grey\n }\n \n \t arr.push({x: i,\n y: j,\n colour: col\n })\n \t}\n }\n return arr\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":385,"value":"mutable accuracyOfTest = 0.99","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":401,"value":"{\n if (accuracySlider1 !== mutable accuracyOfTest) {\n mutable accuracyOfTest = accuracySlider1;\n viewof accuracySlider2.value = accuracySlider1;\n viewof accuracySlider2.dispatchEvent(new CustomEvent(\"input\"));\n }\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":403,"value":"{\n if (accuracySlider2 !== mutable accuracyOfTest) {\n mutable accuracyOfTest = accuracySlider2;\n viewof accuracySlider1.value = accuracySlider2;\n viewof accuracySlider1.dispatchEvent(new CustomEvent(\"input\"));\n }\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":359,"value":"mutable probInfected = 0.01","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":371,"value":"{\n if (infectedSlider1 !== mutable probInfected) {\n mutable probInfected = infectedSlider1;\n viewof infectedSlider2.value = infectedSlider1;\n viewof infectedSlider2.dispatchEvent(new CustomEvent(\"input\"));\n }\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":395,"value":"{\n if (infectedSlider2 !== mutable probInfected) {\n mutable probInfected = infectedSlider2;\n viewof infectedSlider1.value = infectedSlider2;\n viewof infectedSlider1.dispatchEvent(new CustomEvent(\"input\"));\n }\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"