Multiplicity and Multiple Testing
Adjust an existing sample-size requirement for multiple hypothesis tests using the Bonferroni correction. Enter the sample size required for one unadjusted test, the overall significance level, number of tests, and desired power. The calculator applies the corresponding Bonferroni alpha level and preserves the specified power under a normal-approximation sample-size calculation.
The Bonferroni method controls the family-wise Type I error rate by dividing the desired overall alpha level by the number of hypothesis tests. Thus, for K tests, each individual test is performed at alpha/K. PASS documentation describes this same adjustment: the Bonferroni-adjusted alpha is substituted for the original alpha in the relevant power and sample-size formula.
This calculator is intentionally procedure-neutral. It starts with an already-established sample size for one test and adjusts that requirement for multiplicity. Under the normal approximation, sample size is proportional to the square of the sum of the critical normal quantile and the power quantile. Replacing the original alpha with alpha/K therefore gives:
Here, zpower = z1-beta, where beta = 1 - power. The adjusted sample size is always rounded upward because a fractional participant is impossible and rounding downward could leave the design below the requested sample-size requirement.
Inputs: unadjusted N = 64, overall alpha = 0.05, K = 3 tests, power = 0.80, two-sided.
Bonferroni-adjusted alpha = 0.05 / 3 = 0.0166667.
Original critical value = z0.975 = 1.95996.
Adjusted critical value = z0.9916667 = 2.39398.
Power quantile = z0.80 = 0.84162.
Adjustment factor = [(2.39398 + 0.84162) / (1.95996 + 0.84162)]2 = 1.33384.
Adjusted N = 64 × 1.33384 = 85.3655, therefore N = 86.
The validation target is therefore 86 participants when the unadjusted requirement is 64 participants, with three two-sided tests at an overall alpha of 0.05 and 80% power.
PASS contains many procedure-specific power and sample-size calculations. Those procedures may use t, chi-square, noncentral F, exact, or other distributions rather than the normal approximation used here. In those cases, the most precise implementation is to substitute alpha/K directly into the underlying procedure-specific PASS formula. This calculator is therefore a general Bonferroni sample-size adjustment tool rather than a reproduction of every individual PASS sample-size procedure.
Leon, A. C. (2004). Multiplicity-adjusted sample size requirements: A strategy to maintain statistical power with Bonferroni adjustments. Journal of Clinical Psychiatry, 65(11), 1511–1514. DOI: 10.4088/JCP.v65n1111.
NCSS, LLC. PASS Sample Size Software Documentation. PASS 2026 documentation. The PASS documentation describes Bonferroni adjustment as dividing the original alpha by the number of tests and substituting that adjusted alpha into the relevant power/sample-size calculation.