Sensitivity

Sensitivity

Sensitivity, in diagnostic test evaluation, is the proportion of people who truly have the specified target condition and receive a positive result from the index test. It describes detection among cases under a stated population, reference standard and test threshold. It is also called the true-positive rate. This entry uses the diagnostic meaning of “sensitivity”; sensitivity analysis of a model is a different method.

Definition and denominator

Classify each eligible person by an index test result and the presence or absence of the target condition according to an appropriate reference standard. True positives (TP) have the condition and test positive; false negatives (FN) have the condition but test negative. Estimated sensitivity is

$$\widehat{\mathrm{Se}}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}},$$

provided $\mathrm{TP}+\mathrm{FN}>0$. The denominator is all reference-positive people, not all tested people or all positive tests. In probability notation, $\mathrm{Se}=\Pr(T^+\mid D^+)$ for a specified test, threshold, condition and setting. A sensitivity of 90% means that about 90 of 100 people with the condition test positive under the conditions represented by the estimate; it does not mean that a positive test has a 90% chance of being correct.

The target condition must be defined precisely: active disease, a particular stage, an infection during a testing window, or a future outcome are different targets. Specify whether indeterminate test results are excluded, treated as positive or negative, or reported separately. Such rules can materially change the denominator and the apparent performance.

Worked 2-by-2 example

Suppose 1,000 people are tested, of whom 100 have the target condition according to a credible reference standard. The index test identifies 90 of those 100, while 10 test negative. Of the 900 without the condition, 180 test positive and 720 test negative. These are illustrative complete data, not a claim about a real diagnostic product.

Sensitivity is $90/(90+10)=0.90$, or 90%. Specificity, a different measure, is $720/(720+180)=0.80$, or 80%. Among those who tested positive, the positive predictive value in this sample is $90/(90+180)=1/3\approx33.3\%$; the negative predictive value is $720/(720+10)\approx98.6\%$. These different denominators answer different questions. A test can have high sensitivity while many positive results are false positives in a low-prevalence population.

The example's condition prevalence is $100/1{,}000=10\%$. If sensitivity and specificity remained 0.90 and 0.80 in another population with 1% prevalence, the positive predictive value implied by Bayes' rule would be

$$\Pr(D^+\mid T^+)=\frac{0.90(0.01)}{0.90(0.01)+(1-0.80)(0.99)}\approx0.0435,$$

or about 4.35%. The calculation assumes those accuracy parameters really transport to that population; spectrum and setting differences may invalidate that assumption. Sensitivity is conditional on disease status and does not algebraically contain prevalence, but its estimated value can still change across populations when disease severity, stage, coexisting conditions or test implementation change.

Thresholds and test pathways

For a quantitative test where a result above cutoff $c$ is positive, sensitivity at $c$ is $\Pr(X\geq c\mid D^+)$. Lowering a positive threshold typically detects more true cases and also creates more false positives, reducing specificity. This trade-off can be displayed across thresholds with a receiver operating characteristic curve; a single sensitivity value is meaningful only with its threshold and paired specificity. Direction reverses if lower measurements indicate the condition.

A sequence of tests changes pathway performance. If everyone first receives a screening test and only initial positives get a confirmatory test, the sensitivity of the final positive pathway depends on the joint results among people with disease; it cannot generally be obtained by multiplying two separately reported sensitivities unless the needed conditional independence or conditional probabilities are justified. Report whether the index test is used alone, as a triage test, in parallel, or after another test.

“Clinical sensitivity” for detecting the defined condition should be distinguished from analytical sensitivity such as a laboratory limit of detection. A device may detect very low analyte concentrations yet have lower clinical sensitivity at an early disease stage or when collection is poor. Likewise, a high area under an ROC curve does not specify performance at the threshold actually used for action.

Study design and uncertainty

Estimate sensitivity in people representing the intended use population and spectrum of disease, using a valid reference standard applied without differential verification where possible. If only index-positive people receive the definitive reference test, missed cases among index negatives may go unidentified and sensitivity can be biased. Blinding between index and reference readings reduces review bias, and a reference standard that incorporates the index test risks incorporation bias. Report timing between tests because the condition can change.

Sampling variability matters: the effective denominator for sensitivity precision is the number with the condition, not the overall 1,000 alone. With 90 positives among 100 reference-positive people, a binomial confidence interval should be reported with the 90% point estimate and method specified. Correlated observations, clustered sites, repeated tests, missing results and imperfect reference standards require appropriate methods rather than an automatic independent-binomial interval. A perfect-looking estimate based on a few cases is not evidence of perfect detection in future patients.

Subgroups may differ by age, disease stage, symptom status, specimen handling or operator. Assess performance in clinically relevant groups with uncertainty, and avoid treating a subgroup percentage based on a handful of cases as stable. When the reference standard is imperfect, observed positive agreement with it need not equal true clinical sensitivity. Describe the resulting uncertainty and do not relabel an agreement statistic as sensitivity without adequate justification.

Decisions and health economics

False negatives can delay treatment and lead to downstream health losses, while efforts to increase sensitivity may produce more false positives, confirmatory procedures, costs and harms. Choosing a threshold therefore requires the expected consequences of both kinds of error, disease prevalence, treatment effects and resource constraints. Sensitivity alone cannot tell whether a screening programme is effective or cost-effective.

In a decision model, apply sensitivity to the condition-positive population and specificity to the condition-negative population. For prevalence $\pi$ among $N$ eligible people, expected true positives are $N\pi\mathrm{Se}$ and false negatives $N\pi(1-\mathrm{Se})$ under the model assumptions; expected false positives are $N(1-\pi)(1-\mathrm{Sp})$. Preserve the same target definition, time horizon and pathway as the source study. Propagate uncertainty in prevalence and test performance, and consider correlation or between-setting heterogeneity rather than using one fixed number uncritically.

Reporting and source guidance

Report the target condition, intended-use population, index test and cutoff, reference standard, eligibility, flow and missing/indeterminate results, complete 2-by-2 counts, sensitivity and specificity with uncertainty, and the consequences of test errors. The FDA guidance on reporting diagnostic test studies discusses reference standards and agreement when truth is uncertain. STARD 2015 sets out transparent reporting items for diagnostic accuracy studies. The Cochrane Handbook for Diagnostic Test Accuracy Reviews guides synthesis and appraisal across studies. Apply those methods to the actual clinical question rather than transferring the invented 90% from this example.