document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191803>Question 1191803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A certain disease has an incidence rate of 0.9%. If the false negative rate is 4% and the false positive rate is 5%, compute the probability that a person who tests positive actually has the disease. </SPAN>\n" );
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document.write( "Edit: ikleyn is using the wrong numbers in her solution. The 1-0.05 = 0.95 is NOT the true positive rate. The true positive rate is 1 - 0.04 = 0.96 = 96%<BR>\n" );
document.write( "The false negative rate is indeed relevant. \r<BR>\n" );
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document.write( "Edit2: Her solution has been fixed. Thank you ikleyn.\r<BR>\n" );
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document.write( "D = person has the disease<BR>\n" );
document.write( "~D = person does not have the disease<BR>\n" );
document.write( "P(D) = 0.009<BR>\n" );
document.write( "P(~D) = 1-P(D) = 1 - 0.009 = 0.991\r<BR>\n" );
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document.write( "Terms to know:<ul><li>Positive test result = the test claims the person has the disease (the claim may be true or false)</li><li>Negative test result = the test claims the person does not have the disease (the claim may be true or false)</li><li>False negative = when the test says \"negative\", but the reality is that the person actually has the disease</li><li>False positive = when the test says \"positive\", but the reality is that the person does not actually have the disease</li></ul>T = person tests positive<BR>\n" );
document.write( "~T = person tests negative<BR>\n" );
document.write( "False negative rate = 4% = 0.04<BR>\n" );
document.write( "P(~T given D) = 0.04<BR>\n" );
document.write( "Its complementary event is<BR>\n" );
document.write( "P(T given D) = 1 - P(~T given D) = 1 - 0.04 = 0.96\r<BR>\n" );
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document.write( "False positive rate = 5% = 0.05<BR>\n" );
document.write( "P(T given ~D) = 0.05<BR>\n" );
document.write( "Its complementary event is<BR>\n" );
document.write( "P(~T given ~D) = 1 - P(T given ~D) = 1 - 0.05 = 0.95\r<BR>\n" );
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document.write( "Summary so far:<BR>\n" );
document.write( "P(D) = 0.009 = incidence rate<BR>\n" );
document.write( "P(~D) = 0.991<BR>\n" );
document.write( "P(~T given D) = 0.04 = false negative rate<BR>\n" );
document.write( "P(T given D) = 0.96 = true positive rate<BR>\n" );
document.write( "P(T given ~D) = 0.05 = false positive rate<BR>\n" );
document.write( "P(~T given ~D) = 0.95 = true negative rate\r<BR>\n" );
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document.write( "Let's compute the probability of testing positive<BR>\n" );
document.write( "Use the law of total probability<BR>\n" );
document.write( "P(T) = P(T and D) + P(T and ~D)<BR>\n" );
document.write( "P(T) = P(T given D)*P(D) + P(T given ~D)*P(~D)<BR>\n" );
document.write( "P(T) = 0.96*0.009 + 0.05*0.991<BR>\n" );
document.write( "P(T) = 0.05819\r<BR>\n" );
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document.write( "Now we can use Bayes Theorem to get the following.<BR>\n" );
document.write( "P(D given T) = probability person has disease given they tested positive<BR>\n" );
document.write( "P(D given T) = P(T given D)*P(D)/P(T)<BR>\n" );
document.write( "P(D given T) = 0.96*0.009/0.05819<BR>\n" );
document.write( "P(D given T) = 0.14847912012373<BR>\n" );
document.write( "P(D given T) = <font color=red>0.1485</font><BR>\n" );
document.write( "You may be asking yourself \"why is this value so low?\"<BR>\n" );
document.write( "It's not a typo or a mistake that we got something this small. <BR>\n" );
document.write( "The reason why is because the disease incidence rate (0.9% = 0.009) is very small. <BR>\n" );
document.write( "The more rare the disease, the lower the P(D given T) value will get.<BR>\n" );
document.write( "This makes sense because the person is more likely to be clear of the disease even if the test says \"positive\" because the incidence rate is so low. <BR>\n" );
document.write( "In cases like this, it might be recommended to do a second medical test or perhaps an alternative test all together.\r<BR>\n" );
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document.write( "We can create a table like this to help get a better understanding of everything<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Tests Positive</TD><TD >Tests Negative</TD><TD >Total</TD></tr><tr><TD >Has Disease</TD><TD >864</TD><TD >36</TD><TD >900</TD></tr><tr><TD >Does Not Have Disease</TD><TD >4,955</TD><TD >94,145</TD><TD >99,100</TD></tr><tr><TD >Total</TD><TD >5,819</TD><TD >94,181</TD><TD >100,000</TD></tr>\n" );
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document.write( "I'm considering a hypothetical city of 100,000 people<BR>\n" );
document.write( "0.9% has the disease so 0.009*100,000 = 900 people have the disease and 100,000-900 = 99,100 do not.<BR>\n" );
document.write( "Of the 900 people who have the disease, 4% of the tests say \"negative\" and claims the person doesn't have the disease when it should say \"positive\". <BR>\n" );
document.write( "4% of 900 = 0.04*900 = 36 people get false negative results. 900-36 = 864 people get true positive results, i.e correct positive results.\r<BR>\n" );
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document.write( "That fills up the first row of the table.<BR>\n" );
document.write( "The second row has 99,100 people who don't have the disease. <BR>\n" );
document.write( "5% of this sub-population will get a false positive unfortunately<BR>\n" );
document.write( "5% of 99,100 = 0.05*99100 = 4955<BR>\n" );
document.write( "Thankfully, the remaining 99,100 - 4,955 = 94,145 people get true negative tests.\r<BR>\n" );
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document.write( "To fill out the rest of the table, add up the items in each separate column<BR>\n" );
document.write( "column1: 864 + 4,955 = 5,819	<BR>\n" );
document.write( "column2: 36 + 94,145 = 94,181<BR>\n" );
document.write( "These sums are the totals for those who tested positive and negative in that order.\r<BR>\n" );
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document.write( "The table shows that 5,819 people tested positive. Of those who tested positive, 864 actually have the disease.<BR>\n" );
document.write( "864/5819 = <font color=red>0.1485</font> approximately<BR>\n" );
document.write( "So this is one way to use an example to create a hypothetical population and help back up the results of the previous section.<BR>\n" );
document.write( "You can change the \"100,000\" to any large number of your choice.\r<BR>\n" );
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document.write( "Answer: Approximately <font color=red>0.1485</font><BR>\n" );
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