The 95% Accurate Test: Reading a Positive Result
A class in Barpeta is working through an application of conditional probability using a health-department poster about a rapid screening test. The poster claims the test is '95% accurate'. The data sheet gives the details: sensitivity 95%, specificity 90%, and the condition affects 2% of the women in the screening age group. The question asks: 'A woman tests positive. What is the probability that she actually has the condition?' Anima writes: 'Out of 1,000 women, 20 have the condition. The test finds 95% of them, so 19 are correctly detected. Since the test is 95% accurate, the remaining 980 are cleared correctly. Therefore a woman who tests positive has a 95% chance of having the condition.' The teacher also wants to use the same data for a second part set in a referral clinic, where 20% of those tested have the condition.
1. Build the 1,000-woman table and evaluate Anima's answer. What is the correct probability, and precisely where does her reasoning break?
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Starting with 1,000 women and a 2% prevalence: 20 have the condition, 980 do not.
Of the 20 with the condition, 95% test positive: 19 true positives, 1 false negative. Of the 980 without it, specificity 90% means 90% test negative: 882 true negatives, and 98 false positives.
Total positives = 19 + 98 = 117. P(condition | positive) = 19/117 = 0.162, about 16%.
Anima's error is in the sentence 'since the test is 95% accurate, the remaining 980 are cleared correctly'. That silently sets the false-positive rate to zero, contradicting the stated specificity of 90%, which guarantees 98 false positives among the healthy. The deeper error is confusing P(positive | condition) with P(condition | positive). Sensitivity of 95% is a property of the test given the disease; the question asks about the disease given the test. Because the condition is rare, the small percentage of errors applied to a very large healthy group (98 women) swamps the true cases (19). An answer that quotes 95% has answered a different question.
2. Complete the second part for the referral clinic and explain, in terms of the formula, why the same test performs so differently in the two settings.
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At 20% prevalence in the clinic, take 1,000 attendees: 200 have the condition, 800 do not. True positives = 0.95 x 200 = 190. False positives = 0.10 x 800 = 80. Total positives = 270. P(condition | positive) = 190/270 = 0.704, about 70%.
The formula shows directly why: P(D|+) = (sensitivity x prevalence) / [(sensitivity x prevalence) + ((1 - specificity) x (1 - prevalence))]. The numerator grows in proportion to prevalence, while the false-positive term is driven by the size of the healthy group. Raising prevalence from 2% to 20% shrinks that healthy group tenfold, so the same 10% false-positive rate generates far fewer misleading results. The test itself has not improved; the prior has. Sensitivity and specificity are stable properties of the instrument, but the predictive value of a positive result is a property of the instrument and the population together.
3. The head of department says the class should simply remember 'low prevalence means low predictive value'. Give one design change that would raise the predictive value without changing the test, and one limitation of the 95% accuracy claim in the poster.
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Design change: do not apply the test to the general population of 2%. Screen first with a cheap questionnaire or risk score so that only a high-risk subgroup (say 20% prevalence) is tested, or use a two-stage protocol in which everyone who tests positive is confirmed by a second, more specific test. Either route raises the proportion of positives that are genuine, without touching sensitivity or specificity. The trade-off must be stated: pre-screening misses some cases among people not selected for testing, so the gain in predictive value is paid for in reach.
Limitation of the poster: '95% accurate' is ambiguous, because accuracy is not one number. It conflates sensitivity and specificity, and neither equals the probability that a positive result is correct. A test with 95% sensitivity and 90% specificity can be accurate in the technical sense and still mislead most of the women it flags in a low-prevalence population. A well-argued answer says which quantity the poster ought to report, and for whom: sensitivity and specificity for the clinician, predictive values for the person being screened.