Sensitivity, Specificity, and Predictive Values

Two Different Questions About a Diagnostic Test

A diagnostic test can be evaluated by asking how often it gives a particular result for people whose disease status is already known. It can also be used to ask how likely a patient is to have a disease after receiving a particular result. These sound like similar questions, but they condition on different information, so they have different answers.

For example, “Among patients who have the disease, what fraction test positive?” describes a test’s ability to detect disease. “Among patients who test positive, what fraction have the disease?” describes what a positive result means for a patient. Keeping the denominator in view is the most reliable way to distinguish these measures.

The reference standard is the method used to classify disease status for evaluating the test. In a diagnostic table, a true positive (TP) is a positive test result for someone classified as having the disease; a false negative (FN) is a negative result for someone with the disease. A true negative (TN) is a negative result for someone without the disease; a false positive (FP) is a positive result for someone without the disease. “True” and “false” describe whether the test result agrees with the reference classification.

Which Group Is the Denominator?

Sensitivity and specificity begin with disease status and ask what the test reports. For sensitivity, restrict attention to people with the disease, then count the fraction who test positive. For specificity, restrict attention to people without the disease, then count the fraction who test negative. These are probabilities of test outcomes conditional on disease status.

Predictive values begin with the test result and ask about disease status. For PPV, restrict attention to people who tested positive, then count the fraction who have the disease. For NPV, restrict attention to people who tested negative, then count the fraction who do not have the disease. These are probabilities of disease status conditional on the test result.

The formulas follow directly from conditional probability. In each fraction, the denominator is the group named after the vertical bar, and the numerator counts members of that group who also satisfy the event before the bar.

For instance, the sensitivity denominator is \(\mathrm{TP}+\mathrm{FN}\), the total number classified as diseased, because sensitivity asks about the test results within that group. The PPV denominator is \(\mathrm{TP}+\mathrm{FP}\), the total number who tested positive, because PPV asks how many of those positive results correspond to disease. The same true positives appear in both numerators, but the denominators differ.

Worked Example: Choose the Measure That Matches the Question

Calculating Sensitivity and Specificity From a Table

Here is a short derivation of the count formulas. By the definition of conditional probability, \(P(A\mid B)=P(A\cap B)/P(B)\), provided \(P(B)>0\). In a finite group of tested people, the probabilities can be represented by corresponding proportions. Among those classified as diseased, the people who test positive are the true positives, and everyone classified as diseased is either a true positive or a false negative. Therefore sensitivity is \(\mathrm{TP}/(\mathrm{TP}+\mathrm{FN})\). Among those without disease, the true negatives are the negative results and the full group consists of true negatives and false positives. Therefore specificity is \(\mathrm{TN}/(\mathrm{TN}+\mathrm{FP})\).

Predictive Values: What a Result Says About Disease Status

PPV and NPV answer questions that are often central to an individual patient. PPV is the fraction of positive results that occur in people who have the disease; NPV is the fraction of negative results that occur in people who do not. Their denominators are all positive results and all negative results, respectively. They are therefore not interchangeable with sensitivity and specificity.

Predictive values depend on how common the disease is in the population being tested, as well as on test performance. To see why, consider positive results: they can arise from people with disease who test positive, and from people without disease who test positive. If disease is uncommon in the tested population, the group without disease may be large enough that false positives make up a substantial share of all positive results, even when specificity is high. If disease is more common, true positives may make up a larger share. The corresponding pattern for negative results affects NPV.

Sensitivity and specificity are often called test characteristics because they condition on disease status; PPV and NPV condition on the result and describe the composition of result groups in a particular tested population. The distinction does not mean sensitivity and specificity are guaranteed to remain identical in every setting: patient mix, disease severity, test procedures, and the reference standard can affect observed performance. Predictive values especially should not be transferred from one population to another without considering the prevalence in each.

Complements and Common Mistakes

Sensitivity and specificity also have useful complementary interpretations. Among people with the disease, the fraction who test negative is the false-negative rate, \(1-\text{sensitivity}\). Among people without the disease, the fraction who test positive is the false-positive rate, \(1-\text{specificity}\). These complements still condition on disease status; they are not the probability that a patient has or does not have disease after receiving a result.

• Reversing a conditional probability. “Sensitivity is the chance of having disease after a positive test” uses \(P(D+\mid T+)\), which is PPV. Sensitivity is \(P(T+\mid D+)\).
• Using all tested people as the denominator. Sensitivity uses only the disease-positive group; specificity uses only the disease-negative group.
• Treating a high sensitivity as a high PPV. A high sensitivity means few diseased people test negative. It does not by itself tell what fraction of positive results come from diseased people.
• Reporting predictive values without the population context. PPV and NPV describe result groups in the population where they were assessed and can change when prevalence changes.