# Bayes Theorem & Diagnostic Tests Screening Tests

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1 Bayes heorem & Screening ests Bayes heorem & Diagnostic ests Screening ests Some Questions If you test positive for HIV, what is the probability that you have HIV? If you have a positive mammogram, what is the probability that you have breast cancer? Given that you have breast cancer, what is the probability that you will live? 1 2 Example: Coronary artery disease Remember: Present D Absent D Conditional Probability of A given B Positive A A Negative 3 [From the book, Medical Statistics page, and the 2x2 table from data of Weiner et al (1979)] 4 Present D Absent D Present D Absent D Positive Positive Negative Negative he disease Prevalence in these patients D ) /.70 Predictive Value of a Positive est: he probability of having the disease given that a person has a positive test is given by: ) / D ).88 ) / Predictive Value of a Negative est: D ) ) ) 6 Screening ests - 1

2 Bayes heorem & Screening ests Present D Absent D Present D Absent D Positive Positive Negative Negative hings to notice: (Prevalence) D ) ) ) / / /.7 Likewise, (Overall percentage that tested positive) ) D ) D ) / / /.63 7 Calculate the following: D ) ) ) D ) ) 8 Present D Absent D Positive Sensitivity and Specificity Negative Sensitivity: he probability of a person testing positive, given that (s)he has the disease. D ) / D ) /.80 D ) / 9 10 Present D Absent D Positive Negative Specificity: he probability of a person testing negative given that (s)he does not have the disease. D ) / D ) /.74 D ) / Note that sensitivity is not affected by disease prevalence. For example, if number of people with true coronary artery disease has tripled from to 69 so that the new prevalence is now 69/(69).68, then we should expect that three times as many patients would test positive. hus 3x 244 should have a positive result. 11 he sensitivity would be 244/69.80 Before: / Screening ests - 2

3 Bayes heorem & Screening ests Positive Present D Absent D Sensitivity & False Negatives Negative False Negative rate 1 sensitivity D ) / D ) /.2 D ) / False Positive rate 1 specificity D ) / D ) /.26 D ) / 13 For our problem, since the sensitivity is.8, the false negative rate is Interpretation: % of the time a person will actually have the disease when the test says that he/she does not. Likewise, since specificity is.74, the false positive rate is Sensitivity vs Specificity In a perfect world, we want both to be high. he two components have a see-saw relationship. Comparisons of tests accuracy are done with Receiver Operator Characteristic (ROC) curves. ROC Sensitivity est Positive Criteria Sensitivity Specificity Specificity 16 About ROC Curve Area under the ROC represents the probability of correctly distinguishing a normal from an abnormal subject on the relative ordering of the test ratings. wo screening tests for the same disease, the test with the higher area under its ROC curve is considered the better test, unless there is particular level of sensitivity or specificity is especially important in comparing the two tests. How can we get the predictive value of a positive test from sensitivity? After all, at least as a patient, we want to know that probability that we have a disease given that we have just tested positive! Bayes heorem Screening ests - 3

4 Bayes heorem & Screening ests Bayes heorem A A A and B Solving the first equation as follows, P ( A A Substituting this in for the second equation, we have A Bayes' heorem P ( B Bayes' heorem A P ( B Applying this to our formula, we have sensitivity prevalence D ) D ) (.8)(.7) D ). 89 ).63 ested positive In words, the predictive value of a positive test is equal to the sensitivity (.8) times prevalence (.7) divided by percentage who test positive (.63). 19 PV Example Suppose that 84% of hypertensives and 23% of normotensives are classified as hypertensive by an automated blood-pressure machine. If it is estimated that the prevalence of hypertensives is %, what is the predictive value of a positive test? What is the predictive value of a negative test? Sensitivity.84 Specificity Pretty Good est!?? ) D ).48 D ) (.2)(.84) (.8)(.23) Finding Disease Prevalence in a Specified Population Want to know: what percentage of a certain group has HIV? Use a screening test on a sample of n individuals from the group. Know the sensitivity and specificity from prior experience. PV.9 What if only 2% prevalence? Finding Prevalence Suppose n people test positive. Using Bayes heorem we can derive the formula: D ) ( n / n) D ) D ) D ) Relative Risk and the Odds Ratio Important measures useful for comparing the probabilities of disease in two different groups Screening ests - 4

5 Bayes heorem & Screening ests Conditional Probability S S C (.2) (.0) 2 C) (.2) C S).2/..4 C S ' ).0/..1 C (.3) 4 (.4) 7 C)(.7) A A S) (.) S ) (.) 100 () C S) What is C S )? 4 (Relative Risk ) 2 (Exposed) (Not Exposed) Relative Risk (Disease) a c a c Relative Risk (RR) (No Disease) b d b d a b c d disease exposed) disease unexposed) a/(ab) c/(cd) n 26 Odds he odds of an event to occur is the probability of an event to occur divided by the probability of the event not to occur. Example: In casting a balanced die experiment, the odds of a 6 to occur is (1/6) / (/6) 1/. 27 (Exposed) (Not Exposed) Odds Ratio (Disease) a c a c (No Disease) b d b d a b c d disease exposed)/[1- disease exposed)] Odds Ratio (OR) disease unexposed)/[1- disease unexp.)] a/b ad c/d cb n 28 Odds Ratio S S C (.2) (.0) 2 C) (.2) C (.3) 4 (.4) 7 C)(.7) S) (.) S ) (.) 100 () he odds ratio of getting cancer for smoking versus non-smoking is: OR (/) / (/4) 6 For rare diseases, the odds ratio and the relative risk are almost the same. Relative Risk (RR) disease exposed) disease unexposed) Small 0 disease exposed)/[1- disease exposed)] Odds Ratio (OR) disease unexposed)/[1- disease unexp.)] Relative Risk: RR (/) / (/) 4 29 Screening ests -

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