Proportions: Correlated and Paired
Calculate the number of case patients required for a matched case-control study when the anticipated association is expressed as an odds ratio. The calculation accounts for the control exposure probability, number of controls matched to each case, and correlation of exposure between matched case-control subjects, using the asymptotic method described by Dupont.
This calculator implements the asymptotic matched case-control sample-size method described by William D. Dupont for testing an odds ratio of 1 against a specified alternative odds ratio. The method allows one or more matched controls per case and incorporates both the exposure probability among sampled controls and the correlation of exposure between matched case and control subjects.
PASS documents this procedure as a test for two correlated proportions in a matched case-control design and identifies Dupont (1988) as the source of the formulas. PASS also provides a validation example in which P0 = 0.60, φ = 0.20, OR = 3.0, α = 0.05, and power = 0.80 produce 80 cases for one control per case and 50 cases for three controls per case.
First, the anticipated exposure probability among cases is obtained from the control exposure probability and the specified odds ratio:
Let q1 = 1 − p1 and q0 = 1 − p0. The joint exposure probabilities are determined from the marginal probabilities and the case-control correlation:
For M matched controls per case, the calculation then forms the probability tk that exactly k exposed subjects occur in a matched set. The resulting expected score and variance under the specified odds ratio are combined with the corresponding null-score variance to obtain the required number of case patients.
With the conditional score quantities defined from these probabilities, the sample-size calculation has the general form
where β = 1 − power. The result is expressed as the number of cases. The corresponding number of controls is M × N, and the total number of subjects is N(1 + M).
PASS's published validation example specifies power = 0.80, α = 0.05, P0 = 0.60, OR = 3.0, and φ = 0.20. For M = 1, PASS reports a required sample size of 80 cases; for M = 3 it reports 50 cases. The first scenario is used as the fixed implementation check for this calculator.
The JavaScript implementation reproduces the published 80-case result after conversion of the continuous Dupont estimate to a whole-number case sample. PASS's documentation states that its M = 1 and M = 3 results agree with Dupont's published example.
Dupont, W. D. (1988). Power Calculations for Matched Case-Control Studies. Biometrics, 44(4), 1157–1168. The paper derives power and sample-size calculations in terms of control exposure probability, case-control exposure correlation, odds ratio, alpha, beta, and the number of matched controls.
NCSS/PASS. Tests for Two Correlated Proportions in a Matched Case-Control Design, Chapter 155. The PASS documentation gives the computational formulas, parameter definitions, worked examples, and validation against Dupont (1988).
Mander, A. (2005; revised 2006). SAMPSI_MCC: Stata module to calculate Sample Size or Power for Matched Case-Control Studies. The module explicitly identifies Dupont (1988) as the theoretical basis and uses P0, alternative OR, matching ratio, alpha, power, and φ as inputs.