contingency table 2x2
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2x2 Contigency table for medical screening test given sensitivity, specificity, and base rate
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January 2011
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by Edwardson Tan
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base ratefrequency of the disease/condition in the population (expressed as fraction of population)
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sensitivityP(test is positive | person has the condition)
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specificityP(test is negative | person doesn't have the condition)
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ProbabilitiesPerson has the disease/conditionTotalsConditional Probabilites
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YesNoTest result correctTest result erroneous
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test positiveABA + BA / (A + B)P(yes | positive)B / (A + B)False positive = P(no | positive)
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test negativeCDC + DD / (C + D)P(no | negative)C / (C + D)False negative = P(yes | negative)
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TotalsA+CB+D1
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A = P(true & positive)
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B = P(false & positive)
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C = P(true & negative)
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D = P(false & negative)
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Sensitivity = A / (A+C)
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Specificity = D / (B+D)
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base rate = A+C
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A+B+C+D = 1
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A = sensitivity * base rate
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C = (1 - sensitivity) * base rate
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B + D = 1 - base rate
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B = (1- specificity) * (1 - base rate)
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D = specificity * (1 - base rate)
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P(has condition | test positive) = true positive = P(A | A+B) = A / (A+B)
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P(no condition | test negative) = true negative = P(D | C+D) = D / (C+D)
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P(no condition | test positive) = false positive = P(B | A+B) = B / (A + B)
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P(has condition | test negative) = false negative = P(C | C+D) = C / (C + D)
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true positive = (sensitivity * base rate) / [(sensitivity * base rate) + (1 - specificity)*(1 - base rate)]
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true negative = [specificity * (1 - base rate)] / [specificity * (1 - base rate) + (1 - sensitivity) * base rate]
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false positive = 1 - true positive rate
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false negative = 1 - true negative rate
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EXAMPLE
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base rate0.05frequency of the disease/condition in the population (expressed as fraction of population)
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sensitivity0.9P(test is positive | person has the condition)
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specificity0.9P(test is negative | person doesn't have the condition)
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sample size1000number of people (used for second table below)
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ProbabilitiesPerson has the disease/conditiontotalsConditional Probabilites
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YesNoTest result correctTest result erroneous
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test positive0.0450.0950.1432.14%P(has condition | positive)67.86%False positive = P(no condition | positive)
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test negative0.0050.8550.8699.42%P(no condition | negative)0.58%False negative = P(has condition | negative)
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totals0.050.951
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Number of personsPerson has the disease/conditiontotals
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YesNo
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test positive4595140
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test negative5855860
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totals509501000
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cells A to D are computed by multiplying probabilities of corresponding cell by sample size
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