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from flask import Flask, request, url_for |
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from scipy.optimize import bisect |
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import math |
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import numpy as np |
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app = Flask(__name__) |
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app.secret_key = 'Whatever' |
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@app.route('/') |
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def index(): |
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return """A Bayesian <a href="%s">calculator</a> |
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for Down Syndrome odds in twin pregnancies.""" % (url_for('twin_calculator_input'),) |
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def mathjax(): |
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return """<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"> |
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</script>""" |
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def introduction(): |
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return """ |
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This calculator might be of interest to you if: |
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<ol> |
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<li> you are expecting twins and |
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<li> received a "positive" test result for Down Syndrome. |
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</ol> |
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But I am not a doctor (not the right kind, anyway) so you should seek medical advice. |
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<p> |
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<b>Computing the probability of both your twins having Down Syndrome</b> (idealized model) |
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<p> |
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Because the blood tests cannot distinguish between the two foetuses, the results are |
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difficult to interpret. Yet they tend to be communicated in the same manner as results for single |
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foetus pregnancies - which I personally find confusing. What is intended here is a calculation to estimate the probability of giving birth to |
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zero, one or two Down Syndrome children in a twin pregnancy after a positive blood test for Down Syndrome has |
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been received. There are many caveats, but let's proceed... |
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""" |
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def input_form(): |
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return """ |
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<p> |
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<b> User inputs </b> |
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<form method = "POST" action="%s"> |
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The calculation requires your age: <input type = "number" value = "35" min = "20" max = "49" step="1" name="age" /> (please adjust). |
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<p> |
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We will also need a probability of identical twins equal to |
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<input type = "number" value = "0.1" min = "0.01" step="0.01" name="prob_identical" /> (please adjust). |
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This should be a reasonable estimate of the probability that your twins are identical, not fraternal. |
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In other words, based on previous examinations (and not taking into account the blood test, which would be circular), roughly what probability |
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did your ostetrician assign to your twins being identical? Please enter a number between 0 and 1. |
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(The default setting of 0.1 is very roughly the population incidence of identical versus fraternal |
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twins. However it is usually possible for a medical professional to form an opinion in your individual case.) |
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<p> |
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We will assume Down Syndrome odds of 1 in <input type = "number" value = "150" min = "1" step="1" name="odds_down" /> (please adjust). |
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This is the number that was provided in the integrated test. (If your test was "positive" it generally |
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implies the odds are 200/1 or less, but the terminology "positive" seems largely meaningless without the number itself). |
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<p> |
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Once you have adjusted these inputs, please <input type="submit" value="submit" />. |
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</form> |
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""" % (url_for('twin_calculator_output'),) |
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@app.route('/twins_calculator_input') |
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def twin_calculator_input(): |
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return mathjax() + introduction() + input_form() |
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def priorOdds( age ): |
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# Taken from http://downsyndrome.about.com/od/diagnosingdownsyndrome/a/Matagechart.htm |
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table = { 20:1667, 21:1429,22:1429, 23:1429, 24:1250, |
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25:1250, 26:1176,27:1111, 28:1053, 29:1000, |
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30: 952, 31:909, 32:769, 33:625, 34: 500, |
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35: 385, 36:294, 37:227, 38:175, 39:137, |
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40: 106, 41:82, 43:50, 44:38, 45:30, |
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46: 23, 47:18, 48:18, 49:11 } |
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return table[age] |
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muNormal = 0.0 |
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muDown = 1.0 |
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def implyMarker( odds_ratio ): |
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""" Given odds of n:1 , interpret as a marker measurement """ |
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def markerRatio(x ): |
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return math.exp( -(x-muDown)*(x-muDown)/2. ) / math.exp( -(x-muNormal)*(x-muNormal)/2. ) |
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def markerRatioError( x ): |
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return markerRatio(x) - odds_ratio |
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a = muNormal - 10.0 |
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b = muDown + 10.0 |
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marker = bisect( markerRatioError, a, b) |
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return marker |
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@app.route('/twins_calculator_output', methods = ['POST']) |
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def twin_calculator_output(): |
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# Interpret the odds ratio supplied as a lab result |
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age = int(request.form["age"]) |
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odds_down = float(request.form["odds_down"]) |
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odds_prior = priorOdds( age ) |
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odds_ratio = odds_prior / (2*odds_down) |
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prob_identical = float(request.form["prob_identical"]) |
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marker = implyMarker( odds_ratio ) |
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# Posterior probabilities for the four possibilities |
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prob_down = 1/(2*odds_down) |
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pDD = prob_identical*prob_down + (1-prob_identical)*prob_down*prob_down |
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pDU = (1-prob_identical)*prob_down*(1-prob_down) |
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pUD = (1-prob_identical)*prob_down*(1-prob_down) |
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pUU = (1-prob_identical)*(1-prob_down)*(1-prob_down) + prob_identical*(1-prob_down) |
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probs = [ pDD, pDU, pUD, pUU ] |
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markers = [ (muDown,muDown), (muDown,muNormal), (muNormal,muDown), (muNormal,muNormal) ] |
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posteriors = list() |
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for (marker1, marker2),prior in zip( markers, probs ): |
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meanMarker = 0.5*marker1 + 0.5*marker2 |
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prob = math.exp( -( marker-(meanMarker )*( marker-meanMarker )/2. ) ) |
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posterior = prior*prob |
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posteriors.append( posterior ) |
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p0 = posteriors[3] / np.sum( posteriors ) |
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p1 = (posteriors[1]+posteriors[2]) / np.sum( posteriors ) |
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p2 = posteriors[0] / np.sum( posteriors ) |
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return """ <b> Results </b> |
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<p> |
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Based on an age of %s the prior odds of Down Syndrome are %s:1. |
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<p> |
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So the updated odds of %s:1 might (I presume) be interpreted as odds of |
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roughly %s:1 for each foetus and thus |
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as an odds ratio of %s and, furthermore, as |
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a marker measurement of %s in standardized terms. |
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<p> |
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However, assuming a fifty/fifty mix in the blood test (i.e. half coming from each foetus, which |
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is admittedly quite an assumption) the |
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twin probabilities may actually be as follows. |
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<p> |
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<table border="1" cellpadding="3" cellspacing="0" width="300"> |
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<thead> |
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<tr> |
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<th>Outcome</th> |
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<th>Probability</th> |
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</tr> |
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</thead> |
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<tbody> |
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<tr> |
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<th> Neither has Down Syndrome</th> |
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<td> %s percent</td> |
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</tr> |
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<tr> |
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<th> One has Down Syndrome</th> |
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<td> %s percent</td> |
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</tr> |
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<tr> |
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<th> Both have Down Syndrome</th> |
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<td> 1 in %s </td> |
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</tr> |
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</tbody> |
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</table> |
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<p> |
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These probabilites should be taken with a grain of salt. There are many assumptions, including |
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a prior probability of identical twins equal to %s percent. Perhaps the biggest simplification, however, |
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is that the test result is from a single marker. In reality it is from multiple markers. |
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<p> |
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As a final caveat, I may well have misinterpreted the odds ratio supplied to patients |
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who have twins. Here I have assumed that the same scale is used for single pregnancies in converting |
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from marker levels to an odds ratio. I'd be happy to be corrected on that one. |
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<p> |
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Please provide feedback at github where the |
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<a href="https://gist.github.com/notbanker/9eb7908a9cee4e81df72">code</a> for this little app is posted. |
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""" % ( age, odds_prior, odds_down, 2*odds_down, int(10*odds_ratio)/10., int(100*marker)/100., |
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int(p0*1000)/10., int(p1*1000)/10., int(1/p2), int(prob_identical*100) ) |
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notbanker revised this gist