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First, to show why it is important to understand how the different aspects of a test’s accuracy are related and not related, here is an example of what is called the “base rate neglect” fallacy.
If the test in question has a false-positive rate of 1 in 20, that does not that mean that 19 in 20 results are true positive, as the base rate neglect fallacy would assume. The false-positive rate means that if 1,000 people were tested, 50 of them (1 in 20) would have false-positive results, and if the case prevalence is only 1 in 1,000 (and assuming the test has a false-negative rate of 0), the probability of a positive result is 51 in 1,000. So the probability that a positive result is true (ie, the true-positive rate) is only 1 (the case prevalence) in 51 (the case prevalence plus the false-positive rate).
A test’s true-positive (positive result for positive case), true-negative (negative result for negative case), false-positive (positive result for negative case), and false-negative (negative result for positive case) rates are all independent from each other and determined by the population-level case prevalence.
A test’s sensitivity is its true-positive rate, i.e., the proportion of true-positive results in the population: the number of true-positive results divided by the population-level case prevalence (which would be represented by the sum of the true-positive and false-negative results).
Sensitivity of 100% would mean a false-negative rate of 0. The higher the false-negative rate, the lower the sensitivity. High-sensitivity tests are useful for “ruling in” the disease or condition being tested for.
A test’s specificity is its true-negative rate, i.e., the proportion of true-negative results in the population: the number of true-negative results divided by the inverse of the case prevalence (which would be represented by the sum of the true-negative and false-positive results).
Specificity of 100% would mean a false-positive rate of 0. The higher the false-positive rate, the lower the specificity. High-specificity tests are useful for “ruling out” the disease or condition being tested for.
A cut-off point for a test’s results is determined by graphing them with true-positive results (sensitivity) on the y-axis and false-positive results (1 − specificity) on the x-axis. The value on the curve that is closest to both 100% true positive and 0 false positive (the upper left corner of the graph) is the best cut-off. The area under the curve is a measure of the test’s accuracy.
Positive predictive value is the probability of that a positive result is true: the number of true-positive results divided by the number of positive results (which would be represented by the sum of the true-positive and false-positive results) (as opposed to the number of cases in the population, as for sensitivity).
Negative predictive value is the probability of that a negative result is true: the number of true-negative results divided by the number of negative results (which would be represented by the sum of the true-negative and false-negative results) (as opposed to the number of noncases in the population, as for specificity).