2d:null 2e:["$","title",null,{"children":"Combining Multiple Tests | Blake Elias | Observable"}] 33:T434,{ var table = html`

Outcomes to report positive	Threshold ${tex`x`}	False negative rate ${tex`P(\bar{T_{\text{c}}} \| D)`}	False positive rate ${tex`P(T_{\text{c}} \| \bar{D})`}

`; // Threshold ${tex`x`}
Such that:
${tex`P(D|t_1, t_2) \geq x`} critical_threshold_results.forEach(function(result) { var row = html`` var conditions_cell = html``; const disjunction = disjunction_string(result["diagnosis_conditions"], false); conditions_cell.appendChild(tex`${disjunction}`); row.appendChild(conditions_cell); row.appendChild(html`${result["threshold"]}`); row.appendChild(html`${result["false_negative_rate"]}`); row.appendChild(html`${result["false_positive_rate"]}`); table.appendChild(row); }); return table; }34:T43a,md`## Conclusion We see that combining multiple tests affords the ability to more carefully modulate the false-negative and false-positive rates of the test for a disease. In particular, it is possible to achieve a lower the false-negative rate - that is, to identify more of the patients who have a disease - than either of the two individual tests. This comes at the expense of having a higher false-positive rate than both of the individual tests. In this setup, combining tests does not yield a more accurate diagnosis on both axes simultaneously. It simply allows more flexibility on the trade-off between the two types of error, and the abililty to design a test which more heavily prioritizes one over the other. In our experiments, the false-negative rate of the *combined* test can be anywhere from 4- to 10-times lower than that of either individual test. For highly infectious and deadly rare diseases, we recommend such a minimization of false-negatives (while not allowing false-positives to become 100%), so as to identify as many cases of the disease as possible. `35:[] 31:{"id":"080dd99d20ec65bf","slug":"combining-multiple-tests","title":"Combining Multiple Tests","thumbnail":"18329e43739ad781474de014e269b7989843f1093c43271a6d6a69033f9875e8","access_level":"public","create_time":"$D2020-04-26T01:10:23.274Z","update_time":"$D2020-10-10T03:23:11.917Z","view_time":null,"first_public_time":"$D2020-04-26T04:33:23.117Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"614157e65006a2d4","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":"starter_2024"},"version":1762,"publish_version":1762,"role":null,"authored":false,"authors":[{"type":"individual","id":"01ac8a6a43bd3e70","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":null}],"members":[{"type":"individual","id":"01ac8a6a43bd3e70","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Combining Multiple Tests","cells":[{"id":0,"value":"md`# Combining Multiple Tests`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":660,"value":"md`Suppose there are two methods by which one can test a person for a disease. (E.g. in the case of COVID, both PCR tests and chest CT scans can be used as diagnosis.) If each test is somewhat faulty (i.e. has some false-negative and false-positive rate), how much can we decrease the error by combining the two tests?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":214,"value":"md`## Infection Rates`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"md`Consider the following events:\n * ${tex`D`}: Patient has disease, ${tex`\\bar{D}`}: Patient does not have disease \\n\n * ${tex`T_1`}: Patient tests positive via test method 1, ${tex`\\bar{T_1}`}: Patient tests positive via test method 1 \\n\n * ${tex`T_2`}: Patient tests positive via test method 2\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":239,"value":"md`Drag the sliders below to change the (conditional) probabilities of each event.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":564,"value":"md`\n### Base infection rate among those tested:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1187,"value":"tex`P(D) = ${p_d}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"viewof p_d = slider({\n min: 0,\n max: 1,\n step: .001,\n value: p_d_default,\n format: \",\",\n description: \"\" //Prior belief that patient has disease. The base infection rate, among those tested.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"md`### Test method 1 (PCR test):`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"viewof t1_false_pos_false_neg = xySlider({\n xmin: 0, xmax: .5, xvalue: 0.1, xstep: .001, \n ymin: 0, ymax: .5, yvalue: 0.3, ystep: .001\n})\n// t1_false_pos_false_neg","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":95,"value":"tex`\n\\text{False negative rate: } P(\\bar{T_1} | D) = ${p_not_t1_given_d} \\\\\n\\text{False positive rate: } P(T_1 | \\bar{D}) = ${p_t1_given_not_d} \\\\`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1752,"value":"md`[[Source 1](https://www.medrxiv.org/content/10.1101/2020.04.07.20051474v1), \n[Source 2](https://www.the-scientist.com/news-opinion/false-negatives-in-quick-covid-19-test-near-15-percent-study-67451),\n[Source 3](https://www.ncbi.nlm.nih.gov/pubmed/32270882)]. Numbers vary depending on which exact test kit and methodology is used.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":141,"value":"md`### Test Method 2 (CT Scans):`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":140,"value":"viewof t2_false_pos_false_neg = xySlider({\n xmin: 0, xmax: .5, xvalue: 1 - 0.958, xstep: .001, \n ymin: 0, ymax: .5, yvalue: 1 - 0.898, ystep: .001\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"md`\n${tex`\n\\text{False negative rate: } P(\\bar{T_2} | D) = ${p_not_t2_given_d}\\\\\n\\text{False positive rate: } P(T_2 | \\bar{D}) = ${p_t2_given_not_d} \\\\\n`}\n[[Source](https://pubs.rsna.org/doi/full/10.1148/radiol.2020200905)]:\n> The per-exam sensitivity and specificity for detecting COVID-19 in the independent test set was 114 of 127 (90% [95% CI: 83%, 94%]) and 294 of 307 (96% [95% CI: 93%, 98%])\n\n${tex`\nP(\\bar{T_2} | D) = 1 - \\frac{114}{127} = ${(1 - 114 / 127)}\\\\\nP(T_2 | \\bar{D}) = 1 - \\frac{294}{307} = ${(1 - 294/307)}\n`}\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1584,"value":"md`## Error Rate of Combination\n\nIf the above two tests are combined, the following (false-negative, false-positive) combinations are achievable:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":828,"value":"vl.markCircle() // Make a scatter chart\n .data(all_threshold_results) // Using the cars data (below)\n .encode(\n vl.x().fieldQ(\"false_negative_rate\"), // For x, use the Horsepower field\n vl.y().fieldQ(\"false_positive_rate\"), // For y, use the Miles_per_Gallon field\n vl.tooltip().field(\"threshold_tooltip\") // For tooltips, show the Name field\n )\n .render() // Draw the chart\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1579,"value":"md`The combination scheme having the smallest non-zero false-negative rate is:\n\n${/*tex`T_c = ${disjunction_string(critical_threshold_results[1][\"diagnosis_conditions\"], true)}`*/\n\ntex`T_1 \\lor T_2`\n}\n\nThat is, if either of the two test results is positive, declare the patient positive.\n\n\nThis has:\n\n* False-negative rate: ${tex`P(\\bar{T_c} | D) = ${critical_threshold_results[1][\"false_negative_rate\"]}`}, \n* False-positive rate: ${tex`P(T_c | \\bar{D}) = ${critical_threshold_results[1][\"false_positive_rate\"]}`}\n\n\n**By combining two tests that had false-negative rates of ${p_not_t1_given_d * 100}% and ${p_not_t2_given_d * 100}%, we are able to lower the false-negative rate to ${critical_threshold_results[1][\"false_negative_rate\"] * 100}%.**\n`\n\n// Compared with the rates for each individual test, this combined test has a lower false-negative rate and a higher false-positive rate. The goal was to have the lowest possible false-negative rate -- i.e., to ensure we detect .\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":852,"value":"md`## How to Combine Tests`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1531,"value":"md`We need a decision rule which takes a pair of test results and outputs a binary decision. That is, we need a function ${tex`f : \\{0, 1\\} \\times \\{0, 1\\} \\rightarrow \\{0, 1\\}`}. \n\nThere are ${tex`2^4 = 16`} such functions. However, we will see that only 4 are worth considering.`\n//md`There are 2 tests, each with 2 possible outcomes, giving 4 total combinations of possible test results.","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1553,"value":"md`\nGiven the results of the two tests, and that ${tex`T_1`} and ${tex`T_2`} are conditionally independent given ${tex`D`} [written as ${tex`(T_1 \\perp\\!\\!\\!\\perp T_2) | D`}], the posterior probability of ${tex`D`} is:\n\n${\ntex`\nP(D|\\bar{T_1} \\text{ and } \\bar{T_2}) = ${p_d_given_not_t1_and_not_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D}|\\bar{T_1} \\text{ and } \\bar{T_2}) = ${p_not_d_given_not_t1_and_not_t2.toFixed(4)}\\\\\n\nP(D|T_1 \\text{ and } \\bar{T_2}) = ${p_d_given_t1_and_not_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D}|T_1 \\text{ and } \\bar{T_2}) = ${p_not_d_given_t1_and_not_t2.toFixed(4)} \\\\\n\nP(D|\\bar{T_1} \\text{ and } T_2) = ${p_d_given_not_t1_and_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D}|\\bar{T_1} \\text{ and } T_2) = ${p_not_d_given_not_t1_and_t2.toFixed(4)} \\\\\n\nP(D|T_1 \\text{ and } T_2) = ${p_d_given_t1_and_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D}|T_1 \\text{ and } T_2) = ${p_not_d_given_t1_and_t2.toFixed(4)}\n`\n}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":679,"value":"md`We can impose a threshold on this probability to produce a binary answer:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":704,"value":"tex`T_{\\text{c}} = \\begin{cases}\n1 & \\text{ if } P(D \\space|\\space t_1, t_2) \\geq x \\\\\n 0 & \\text{ otherwise }\n\\end{cases}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1372,"value":"md`where ${tex`0 < x < 1`}.\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1435,"value":"md`Large values for ${tex`x`} require us to be very certain a patient has the disease before we declare them positive. Hence, raising ${tex`x`} lowers the false-positive rate...`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1304,"value":"vl.markCircle() // Make a scatter chart\n .data(all_threshold_results) // Using the cars data (below)\n .encode(\n vl.x().fieldQ(\"threshold\"), // For x, use the Horsepower field\n vl.y().fieldQ(\"false_positive_rate\"), // For y, use the Miles_per_Gallon field\n vl.tooltip().field(\"threshold_tooltip\") // For tooltips, show the Name field\n )\n .render() // Draw the chart\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1439,"value":"md`...and raises the false-negative rate:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1307,"value":"vl.markCircle() // Make a scatter chart\n .data(all_threshold_results) // Using the cars data (below)\n .encode(\n vl.x().field(\"threshold\"), // For x, use the Horsepower field\n vl.y().field(\"false_negative_rate\"), // For y, use the Miles_per_Gallon field\n vl.tooltip().field(\"threshold_tooltip\") // For tooltips, show the Name field\n )\n .render() // Draw the chart\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1462,"value":"md`The jump discontinuities in this graph are due to the discrete logic for combining tests. The 16 possible test rules have collapsed down to 4:`\n\n\n\n// For 2 tests, each with 2 possible outcomes (positive or negative), there are 4 possible test result combinations. Each such combination must be labeled \"positive\" or \"negative\". Hence, the choice of ${tex`x \\in [0, 1]`} is really a choice between 4 levels of stringency:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":962,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1448,"value":"md`Combining the two plots above, we can directly plot the trade-off between the false-positive and false-negative rates of the combined test:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1637,"value":"vl.markCircle() // Make a scatter chart\n .data(all_threshold_results) // Using the cars data (below)\n .encode(\n vl.x().fieldQ(\"false_negative_rate\"), // For x, use the Horsepower field\n vl.y().fieldQ(\"false_positive_rate\"), // For y, use the Miles_per_Gallon field\n vl.tooltip().field(\"threshold_tooltip\") // For tooltips, show the Name field\n )\n .render() // Draw the chart\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1691,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1164,"value":"md`\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n\n-----\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1229,"value":"md`## Appendix A: Calculations\nThe text and code below shows how the above results were calculated, using Bayes Rule and the laws of conditional probability.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":944,"value":"all_threshold_results = all_thresholds.map(threshold_evaluation).slice().sort(function(a, b) {return a[\"threshold\"] - b[\"threshold\"];})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1309,"value":"critical_threshold_results = critical_thresholds.map(threshold_evaluation).slice().sort(function(a, b) {return a[\"threshold\"] - b[\"threshold\"];})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1601,"value":"function disjunction_string(diagnosis_conditions, oneline) {\n var conditions_strings = diagnosis_conditions.map(function(condition, index) {\n //const prefix_spaces = (index > 0) ? \"\\\\space\" : \"\";\n //const prefix_newline = \"\"; // (index > 0 && index % 2 == 1)\n //const prefix_newline = (index > 0) ? (\"\\\\\\\\ \\\\hspace{\" + index*.2 + \"cm}\") : \"\";\n\n const prefix_newline = (index > 0 && index % 2 == 0 && !oneline) ? \"\\\\\\\\\" : \"\";\n\n return prefix_newline + \"(\" + \n (condition[\"t1\"] ? \"\" : \"\\\\bar\") + \"{T_1} \\\\land \" +\n (condition[\"t2\"] ? \"\" : \"\\\\bar\") + \"{T_2} )\";\n });\n var disjunction_string = conditions_strings.join(\"\\\\lor\"); // \\\\text{ OR }\n if (disjunction_string.trim() == \"\") {\n disjunction_string = \"\\\\varnothing\";\n }\n return disjunction_string;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1288,"value":"all_thresholds = {\n var thresholds = [];\n for (var x = 0; x < 1; x += 0.01) {\n thresholds.push(x);\n }\n return thresholds;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1292,"value":"critical_neighborhood = {\n var critical_neighborhood = [];\n critical_thresholds.forEach(function(threshold) {\n critical_neighborhood = critical_neighborhood.concat([threshold - 0.0001]);\n critical_neighborhood = critical_neighborhood.concat([threshold]);\n critical_neighborhood = critical_neighborhood.concat([threshold + 0.0001]);\n });\n return critical_neighborhood;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":933,"value":"// sensible_thresholds = ; // p_d\n\ncritical_thresholds = [p_d_given_not_t1_and_not_t2, p_d_given_t1_and_not_t2, p_d_given_not_t1_and_t2, p_d_given_t1_and_t2].map(t => t + 0.0001).concat([p_d_given_not_t1_and_not_t2 - 0.0001]); // p_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":936,"value":"function threshold_evaluation(threshold) {\n const result = {};\n const n = combined_false_negative(threshold);\n const p = combined_false_positive(threshold);\n result[\"false_negative_rate\"] = n[\"error_rate\"].toFixed(4);\n result[\"false_positive_rate\"] = p[\"error_rate\"].toFixed(4);\n result[\"diagnosis_conditions\"] = p[\"diagnosis_conditions\"];\n result[\"threshold\"] = threshold.toFixed(4);\n result[\"threshold_tooltip\"] = \"threshold = \" + result[\"threshold\"] + \"; false_negative_rate = \" + result[\"false_negative_rate\"] + \"; false_positive_rate = \" + result[\"false_positive_rate\"];\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":916,"value":"function combined_false_negative(threshold) {\n var p_neg_given_d = 0;\n var diagnosis_conditions = [];\n \n if (p_d_given_not_t1_and_not_t2 < threshold) {\n p_neg_given_d += p_not_t1_and_not_t2_given_d;\n diagnosis_conditions.push({\"t1\": false, \"t2\": false});\n }\n \n if (p_d_given_t1_and_not_t2 < threshold) {\n p_neg_given_d += p_t1_and_not_t2_given_d;\n diagnosis_conditions.push({\"t1\": true, \"t2\": false});\n }\n \n if (p_d_given_not_t1_and_t2 < threshold) {\n p_neg_given_d += p_not_t1_and_t2_given_d;\n diagnosis_conditions.push({\"t1\": false, \"t2\": true});\n }\n \n if (p_d_given_t1_and_t2 < threshold) {\n p_neg_given_d += p_t1_and_t2_given_d;\n diagnosis_conditions.push({\"t1\": true, \"t2\": true});\n }\n \n return {\"diagnosis_conditions\": diagnosis_conditions,\n \"error_rate\": p_neg_given_d};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":918,"value":"function combined_false_positive(threshold) {\n var p_pos_given_not_d = 0;\n var diagnosis_conditions = [];\n \n if (p_d_given_not_t1_and_not_t2 >= threshold) {\n p_pos_given_not_d += p_not_t1_and_not_t2_given_not_d;\n diagnosis_conditions.push({\"t1\": false, \"t2\": false});\n }\n \n if (p_d_given_t1_and_not_t2 >= threshold) {\n p_pos_given_not_d += p_t1_and_not_t2_given_not_d;\n diagnosis_conditions.push({\"t1\": true, \"t2\": false});\n }\n \n if (p_d_given_not_t1_and_t2 >= threshold) {\n p_pos_given_not_d += p_not_t1_and_t2_given_not_d;\n diagnosis_conditions.push({\"t1\": false, \"t2\": true});\n }\n \n if (p_d_given_t1_and_t2 >= threshold) {\n p_pos_given_not_d += p_t1_and_t2_given_not_d;\n diagnosis_conditions.push({\"t1\": true, \"t2\": true});\n }\n \n return {\"diagnosis_conditions\": diagnosis_conditions,\n \"error_rate\": p_pos_given_not_d};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":228,"value":"md`### Posterior belief conditioned on test results`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":444,"value":"md`#### Single Test:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1197,"value":"tex`\nP(D | \\bar{T_1}) = ${p_d_given_not_t1.toFixed(4)} \\hspace{1cm} P(\\bar{D} | \\bar{T_1}) = ${p_not_d_given_not_t1.toFixed(4)} \\\\\nP(D | T_1) = ${p_d_given_t1.toFixed(4)} \\hspace{1cm} P(\\bar{D} | T_1) = ${p_not_d_given_t1.toFixed(4)} \\\\\n\nP(D | \\bar{T_2}) = ${p_d_given_not_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D} | \\bar{T_2}) = ${p_not_d_given_not_t2.toFixed(4)} \\\\\nP(D | T_2) = ${p_d_given_t2.toFixed(4)} \\hspace{1cm} P(\\bar{D} | T_2) = ${p_not_d_given_t2.toFixed(4)} \\\\\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":419,"value":"md`Consider a patient who is tested using test method 1, and who gets a positive result. We can compute our posterior belief of their having the disease, as follows:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"tex`P(D | T_1) = \\frac{P(T_1 | D) P(D)}{P(T1)} = \\frac{P(T_1 | D) P(D)}{ P(T_1 | D) P(D) + P(T_1 | \\bar{D}) P(\\bar{D}) } = ${p_d_given_t1}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":665,"value":"tex`P(\\bar{D} | \\bar{T_1}) = ${p_not_d_given_not_t1}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"md`If we had instead tested the patient using test method 2, a positive result would give us the following posterior belief:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":423,"value":"tex`P(D|T_2) = \\frac{P(T_2 | D) P(D)}{P(T2)} = \\frac{P(T_2 | D) P(D)}{ P(T_2 | D) P(D) + P(T_2 | \\bar{D}) P(\\bar{D}) } = ${p_d_given_t2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":592,"value":"md`Note that:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":582,"value":"tex`P(D) = ${p_d}, \\text{ so } P(D|T_1) \\geq P(D) \\text{ and } P(D|T_2) \\geq P(D)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":585,"value":"md`This means a positive test result (via either method) increases our belief that the patient has the disease. This is true even if the posterior P(D|T_1) is less than 0.5. It just means we have increased our belief that they have the disease, even if they still most likely don't have it.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":448,"value":"md`#### Two Tests, Both Positive:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":244,"value":"tex`\\text{Assuming } T_1 \\text{ and } T_2 \\text{ are conditionally independent given } D \\text{ [that is, } (T_1 \\perp\\!\\!\\!\\perp T_2) | D \\text{], } \\newline\n\\text{a positive result on \\textbf{both tests} yields higher belief about the patient's diseased state, } \\newline\n\\text{compared with a positive result on either test alone:}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"tex`P(D|T_1 \\text{ and } T_2) = ${p_d_given_t1_and_t2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":463,"value":"md`This value is calculated as follows, using the laws of conditional probability:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":458,"value":"tex`P(D | T_1 \\text{ and } T_2) = \\frac{P(T_1 \\text{ and } T_2 | D) P(D)}{P(T_1 \\text{ and } T_2)} \\\\\nP(D | T_1 \\text{ and } T_2) = \\frac{P(T_1 \\text{ and } T_2 | D) P(D)}{P(T_1 \\text{ and } T_2 | D) P(D) + P(T_1 \\text{ and } T_2 | \\bar{D}) P(\\bar{D})} \\\\\n\n\nP(D | T_1 \\text{ and } T_2) = \\frac{P(T_1 | D) P(T_2 | D) P(D)}{P(T_1 | D) P(T_2 | D) P(D) + P(T_1 | \\bar{D}) P(T_2 | \\bar{D}) P(\\bar{D})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"md`#### Both Tests Negative:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":250,"value":"md`Likewise, a negative result on both tests yields higher belief about the patient's *non-diseased* state than either test alone:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"tex`P(\\bar{D}|\\bar{T_1}) = ${p_not_d_given_not_t1} \\\\\nP(\\bar{D}|\\bar{T_2}) = ${p_not_d_given_not_t2} \\\\\nP(\\bar{D}|\\bar{T_1} \\text{ and } \\bar{T_2}) = ${p_not_d_given_not_t1_and_not_t2}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":336,"value":"md`#### Disagreeing Test Results:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":375,"value":"md`Even when the two tests disagree, the accumulated evidence can still tend strongly toward one side, depending on the false-positive and false-negative rates of each test (this can be seen by trying different values in the sliders above). Note that:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":379,"value":"tex`P(\\bar{D}|\\bar{T_1} \\text{ and } T_2) \\leq P(\\bar{D}|\\bar{T_1})`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":392,"value":"md`and:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":395,"value":"tex`P(\\bar{D}|T_1 \\text{ and } \\bar{T_2}) \\leq P(\\bar{D}|\\bar{T_2})`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":398,"value":"md`That is, with a single negative test, we would have been more sure that the patient was non-diseased. If we then get a second test that comes back positive, we become less confident in the non-diseased status.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":671,"value":"md`### Error Rates of Combined Test`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":823,"value":"function error_rate(threshold) {\n var d = {\n \"threshold\": threshold,\n \"False negative rate\": combined_false_negative(threshold),\n \"False positive rate\": combined_false_positive(threshold),\n //\"tooltip\": \"Combined test: \\n false negative rate: ${false_negative_rate},\\n false positive rate: ${false_positive_rate}\"\n };\n return d;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":758,"value":"data = [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0].map(error_rate);//.concat([{'false_negative_rate': p_not_t1_given_d, 'false_positive_rate': p_t1_given_not_d}, {'false_negative_rate': p_not_t2_given_d, 'false_positive_rate': p_t2_given_not_d}]);\n// [[combined_false_negative(0.2), combined_false_positive(0.2)]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1717,"value":"md`\n-----\n\n-----\n\n-----\n\n-----\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":545,"value":"md`## Appendix B: On Computing *P(D)*`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":483,"value":"md`The default value for *P(D)* -- the prior belief -- was taken to be ${p_d_default} in the slider at the beginning. This might seem awfully high: could ${100 * p_d_default}% of the population really have this disease?\n\nNote, however, that *P(D)*, our prior belief that a patient has the disease, is different than the prevalence of the disease in the population. This is because we are considering our belief about a patient *who is getting tested*, and simply comparing our belief before seeing the test result vs. after. Everywhere we have used the expression *P(D)*, it would be more precise to write:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":634,"value":"tex`P(D \\space | \\text{ patient has been sent for testing})`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":627,"value":"md`The prevalence of the disease among this group of patients (those who get tested), may be different than in the general population. The very fact that a patient is getting tested *at all* changes our belief that they have the disease.\n\nPrevalence in the general population could be hard to estimate due to un-reported cases, especially if the disease is novel and/or poorly understood. But prevalence in the pool of patients who get tested can be approximated as the fraction of tests which come back positive:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":493,"value":"tex`P(D) = \\frac{\\text{number of positive tests}}{\\text{number of total tests}}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":497,"value":"md`So, while it is unlikely that ${p_d_default*100}% of the population will have some disease, it is quite possible that ${p_d_default*100}% *of those tested* will come back positive. This, of course, must be corrected for false-positives and false negatives. What we really want is:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":499,"value":"tex`P(D) = \\frac{\\text{ tested patients who have disease }}{\\text{number of total tests}} = \\frac{\\text{(true positives) } + \\text{ (false negatives)}}{\\text{number of total tests}},`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":507,"value":"md`where the quantities in the numerator can be computed from published false-negative and false-positive rates. This all assumes, of course, that the criteria for selecting patients to test remains constant over time (e.g. testing all patients who report certain symptoms). If these criteria change (e.g. to include asymptomatic people who have com in contact with a symptomatic person), then we can no longer use the prevalence computed for the previous group, and will have to re-estimate *P(D)*.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":467,"value":"md`_______\n_______\n_______\n_______\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"md`## Appendix C: Intermediate probability calculations`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":471,"value":"md`(Below: intermediate \"scratch\" results for the above computations. Only of interest if editing the code.)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":561,"value":"p_d_default = 0.2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":63,"value":"p_not_d = 1 - p_d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"md`### Basics for test method 1`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":125,"value":"p_t1_given_not_d = t1_false_pos_false_neg[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":127,"value":"p_not_t1_given_not_d = 1 - p_t1_given_not_d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":131,"value":"p_not_t1_given_d = t1_false_pos_false_neg[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":134,"value":"p_t1_given_d = 1 - p_not_t1_given_d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":182,"value":"p_t1 = p_t1_given_d * p_d + p_t1_given_not_d * p_not_d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":261,"value":"p_not_t1 = 1 - p_t1","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":171,"value":"md`### Basics for test method 2`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1137,"value":"p_not_d_given_t2 = 1 - p_d_given_t2;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":159,"value":"p_t2_given_not_d = t2_false_pos_false_neg[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":166,"value":"p_not_t2_given_not_d = 1 - p_t2_given_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":176,"value":"p_not_t2_given_d = t2_false_pos_false_neg[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":179,"value":"p_t2_given_d = 1 - p_not_t2_given_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":187,"value":"p_t2 = p_t2_given_d * p_d + p_t2_given_not_d * p_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":278,"value":"p_not_t2 = 1 - p_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":163,"value":"md`### Conditional probability`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"md`#### Single test`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"p_d_given_t1 = (p_t1_given_d * p_d) / (p_t1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1121,"value":"p_not_d_given_t1 = 1 - p_d_given_t1;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":259,"value":"p_d_given_not_t1 = (p_not_t1_given_d * p_d) / (p_not_t1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":257,"value":"p_not_d_given_not_t1 = 1 - p_d_given_not_t1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":190,"value":"p_d_given_t2 = (p_t2_given_d * p_d) / (p_t2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":276,"value":"p_d_given_not_t2 = (p_not_t2_given_d * p_d) / (p_not_t2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"p_not_d_given_not_t2 = 1 - p_d_given_not_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":296,"value":"md`#### Two tests, both positive`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"p_t1_and_t2_given_d = p_t1_given_d * p_t2_given_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":222,"value":"p_t1_and_t2_given_not_d = p_t1_given_not_d * p_t2_given_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":201,"value":"p_t1_and_t2 = p_t1_and_t2_given_d * p_d + p_t1_and_t2_given_not_d * p_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":199,"value":"p_d_given_t1_and_t2 = p_t1_and_t2_given_d * p_d / p_t1_and_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":754,"value":"p_not_d_given_t1_and_t2 = 1 - p_d_given_t1_and_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":306,"value":"md`#### Two tests, both negative`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"p_not_t1_and_not_t2_given_d = p_not_t1_given_d * p_not_t2_given_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":317,"value":"p_not_t1_and_not_t2_given_not_d = p_not_t1_given_not_d * p_not_t2_given_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":289,"value":"p_not_t1_and_not_t2 = p_not_t1_and_not_t2_given_d * p_d + p_not_t1_and_not_t2_given_not_d * p_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":321,"value":"p_d_given_not_t1_and_not_t2 = p_not_t1_and_not_t2_given_d * p_d / p_not_t1_and_not_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":326,"value":"p_not_d_given_not_t1_and_not_t2 = 1 - p_d_given_not_t1_and_not_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":343,"value":"md`#### Test 1 positive, test 2 negative`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":345,"value":"p_t1_and_not_t2_given_d = p_t1_given_d * p_not_t2_given_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":347,"value":"p_t1_and_not_t2_given_not_d = p_t1_given_not_d * p_not_t2_given_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":349,"value":"p_t1_and_not_t2 = p_t1_and_not_t2_given_d * p_d + p_t1_and_not_t2_given_not_d * p_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":351,"value":"p_d_given_t1_and_not_t2 = p_t1_and_not_t2_given_d * p_d / p_t1_and_not_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":353,"value":"p_not_d_given_t1_and_not_t2 = 1 - p_d_given_t1_and_not_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"md`#### Test 1 negative, test 2 positive`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":358,"value":"p_not_t1_and_t2_given_d = p_not_t1_given_d * p_t2_given_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":362,"value":"p_not_t1_and_t2_given_not_d = p_not_t1_given_not_d * p_t2_given_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":364,"value":"p_not_t1_and_t2 = p_not_t1_and_t2_given_d * p_d + p_not_t1_and_t2_given_not_d * p_not_d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":366,"value":"p_d_given_not_t1_and_t2 = p_not_t1_and_t2_given_d * p_d / p_not_t1_and_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":368,"value":"p_not_d_given_not_t1_and_t2 = 1 - p_d_given_not_t1_and_t2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"md`## Appendix D: Imports, etc.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"import {slider} from '@jashkenas/inputs'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"import {xySlider} from '98bbb19bf9e859ee'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":815,"value":"import { vl } from \"@vega/vega-lite-api\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"