Sensitivity — Sample Size Calculator

Determines the sample size required to estimate the sensitivity of a diagnostic test with a specified precision and confidence level, based on the expected sensitivity.

Key takeaways

What is calculated: Determine the diseased subjects required, and hence the total enrolment.
Quantities you must supply: Objective; Expected sensitivity; Absolute precision; Disease prevalence in the sampled population; Significance level.
Type of calculation: Power-based: it answers how much data is needed to detect a stated effect. Power 0.80 at α = 0.05 is the usual target; confirmatory studies use 0.90.

Method used for Sensitivity

Sensitivity — Buderer prevalence-adjusted method Sizes the diseased group first — the number of confirmed positives needed to pin down how often the test finds true cases — and then converts that into the total to screen using the prevalence. It applies whenever the deliverable is the detection ability of a test rather than a comparison between groups. The prevalence is what governs feasibility: only diseased subjects carry information, so at a prevalence of 5% the screening burden is twenty times the figure that actually counts, and recruiting known positives separately is usually the difference between a study that can be run and one that cannot.

When to use

Use when the primary objective is to determine how accurately the study can estimate the probability that a diagnostic test correctly identifies individuals who truly have the disease.

Go to the calculator →

Sample size

Objective

Expected sensitivity

Absolute precision (± d)

Disease prevalence in the sampled population

Significance level (α)

Type I error rate.

Anticipated dropout / non-response (proportion)

Final n is inflated by 1/(1 − dropout).

Reset to defaults

Awaiting inputs

—

Set the parameters and calculate.

Methodology — Sensitivity — Buderer prevalence-adjusted method

Formula used Sensitivity — Buderer prevalence-adjusted method

n_{d i s} = \frac{z_{1 - α / 2}^{2} \cdot S e (1 - S e)}{d^{2}}

Precision objective

n_{d i s} = \frac{{[z_{1 - α} \sqrt{S e_{0} q_{0}} + z_{1 - β} \sqrt{S e \cdot q}]}^{2}}{{(S e - S e_{0})}^{2}}

Hypothesis test objective

N = \frac{n_{d i s}}{φ}

Total enrolment

where —

Se: Expected sensitivity of the test. 0 – 1 exclusive
d: Absolute precision, in precision mode. 0.0001 – 0.5; commonly 0.05
φ: Disease prevalence in the sampled population. 0 – 1
α: Probability of rejecting a true null hypothesis (Type I error). 0.0001 – 0.5; conventionally 0.05, or 0.025 one-sided for regulatory non-inferiority
ℓ: Expected proportion of enrolled units lost before analysis. 0 – 0.95

How this method works

Sensitivity is a proportion estimated only among truly diseased subjects, so the calculation applies to that subgroup. Total enrolment follows by dividing by the disease prevalence.

Sensitivity is estimated only among truly diseased subjects, so the formula applies to that subgroup and total enrolment follows by dividing by prevalence — the point Buderer made and which is still routinely overlooked.

In low-prevalence screening this binds hard: 139 diseased animals at 5% prevalence means enrolling nearly 2,800. Enriched sampling of confirmed cases is the usual answer, but precludes estimating predictive values from the same sample.

Null hypothesis. In test mode, H0: Se = Se0, the minimum acceptable sensitivity. In precision mode there is no hypothesis.

Alternative hypothesis. In test mode, H1: Se > Se0 (one-sided).

Calculation procedure

State the sensitivity expected, $S e$ , and either the precision wanted or the minimum acceptable value to rule out.
Only diseased subjects inform sensitivity, so the calculation is for the number of true positives $n_{d i s}$ .
For precision, evaluate $n_{d i s} = \frac{z_{1 - α / 2}^{2} S e (1 - S e)}{d^{2}}$ ; for a hypothesis test use the null and alternative variances.
Convert to the total to screen using the prevalence: $N = n_{d i s} / φ$ .
Round up, then inflate for indeterminate results. Enriching the sample with known positives is the usual way to make $N$ affordable.

Exact or approximate

Approximate — a normal approximation to the binomial within the diseased subgroup.

The formula shown always matches what was computed: when a one-sided test is selected the rendered quantile changes from z_1−α/2 to z_1−α, and where several methods exist, the formula follows the method selected in the calculator.

Advantages & limitations

Advantages

Makes the prevalence constraint explicit.
Supports both precision and hypothesis-testing objectives.
Reports the diseased subgroup separately from total enrolment.

Limitations

In low-prevalence screening the total enrolment becomes very large.
Enriched sampling of confirmed cases solves that but precludes estimating predictive values and can bias sensitivity through spectrum effects.
Partial verification of only test-positives inflates apparent sensitivity.

Assumptions

The reference standard is applied to all subjects regardless of index test result.
Disease status is classified without error by the reference standard.
Subjects are independent and representative of the target population.
The stated prevalence applies to the sampled population.

Applicable adjustments

Supports dropout inflation.

Conclusion

Report both the diseased subgroup and the total enrolment with the assumed prevalence. If enriched sampling is planned, state it and acknowledge the spectrum effect.

References

Buderer NMF. Statistical methodology: I. Incorporating the prevalence of disease into the sample size calculation for sensitivity and specificity. Acad Emerg Med. 1996;3(9):895–900. Link
Flahault A, Cadilhac M, Thomas G. Sample size calculation should be performed for design accuracy in diagnostic test studies. J Clin Epidemiol. 2005;58(8):859–862. Link
Hajian-Tilaki K. Sample size estimation in diagnostic test studies of biomedical informatics. J Biomed Inform. 2014;48:193–204. Link