Non-Inferiority and Equivalence
Two one-sided tests (TOST) for equivalence of two independent proportions using the pooled Z-test. Enter the observed proportions, sample sizes, and lower and upper equivalence margins to determine whether the observed difference is statistically contained within the prespecified equivalence region.
This calculator implements the equivalence test for the difference between two independent proportions using the pooled Z-test. This is one of the equivalence-test procedures documented in PASS for two independent proportions. PASS lists the pooled Z-test alongside unpooled, continuity-corrected, Farrington-Manning, Miettinen-Nurminen, and Gart-Nam alternatives.
Equivalence is assessed with the two one-sided tests (TOST) framework. The null hypothesis states that the difference is at or beyond at least one equivalence boundary, while the alternative states that the difference lies strictly between the lower and upper margins.
Let p̂1 and p̂2 be the observed proportions, with sample sizes n1 and n2. For either equivalence boundary δ0, the pooled Z statistic is
For the lower margin D0.L, the test is H01: D ≤ D0.L versus HA1: D > D0.L. Its p-value is the upper-tail probability of the corresponding Z statistic.
For the upper margin D0.U, the test is H02: D ≥ D0.U versus HA2: D < D0.U. Its p-value is the lower-tail probability of the corresponding Z statistic.
Equivalently, for this symmetric two-sided equivalence test, the corresponding 100(1 − 2α)% confidence interval for P1 − P2 is contained entirely within the prespecified equivalence interval. With α = 0.05, this corresponds to a 90% confidence interval.
The validation inputs use the D0.U = 0.03, D0.L = −0.03, P2 = 0.10, D1 = 0.00, α = 0.05 scenario reported in the PASS validation documentation for the pooled Z-test. PASS reports N1 = N2 = 2,165 for its 90% power calculation. Applying the actual two-sided TOST to those fixed observed proportions gives:
pooled proportion = 0.100000; pooled SE = 0.00911816; zlower = 3.29014; plower = 0.000501; zupper = −3.29014; pupper = 0.000501; TOST p = 0.000501.
Therefore both one-sided tests reject their respective null hypotheses, and the observed difference is equivalent at α = 0.05.
A statistically significant result in an ordinary test of equality is not sufficient to establish equivalence. Equivalence requires both one-sided tests to reject the null hypotheses that the difference lies at or beyond an equivalence boundary. The margins should therefore be specified before examining the observed results and should have a clinical or scientific justification.
PASS / NCSS. Equivalence Tests for the Difference Between Two Proportions. PASS Sample Size Software documentation. The documentation specifies the hypotheses and pooled Z-test statistic used for equivalence testing of two independent proportions.
Tubert-Bitter, P., Manfredi, R., Lellouch, J., & Bégaud, B. (2000). “Sample size calculations for risk equivalence testing in pharmacoepidemiology.” Journal of Clinical Epidemiology, 53(12), 1268–1274. doi:10.1016/S0895-4356(00)00252-3.
The PASS documentation uses the Tubert-Bitter et al. example as a validation reference for equivalence testing with the pooled Z-test.