Means: Many Groups (ANOVA)
Calculate the balanced sample size needed when several treatment means will each be compared with a common control using Dunnett's procedure. This implementation follows the balanced-case sample-size algorithm documented in PASS and based on Hsu's simultaneous-confidence-interval approach.
Dunnett's procedure is designed for the special multiple-comparison situation in which several treatment means are each compared with one common control mean. Because every treatment-control comparison shares the control mean, the comparisons are correlated; simply dividing alpha among independent tests does not reproduce Dunnett's procedure.
For the balanced one-way design, with k total groups and n subjects per group, the pooled error degrees of freedom are df = k(n - 1). The simultaneous two-sided Dunnett critical constant q is obtained numerically from the multivariate Studentized distribution associated with the treatment-versus-control comparisons.
Here, ω is the specified minimum detectable difference and σ is the assumed common within-group standard deviation. The PASS/Hsu sample-size calculation uses a modified definition of power: the probability that the simultaneous Dunnett confidence intervals both cover the true treatment-control differences and are sufficiently narrow to have half-width no greater than ω/2.
The calculator evaluates this integral numerically. It then searches over integer values of n and returns the smallest per-group sample size for which the calculated power is at least the requested target. Total sample size is kn.
This calculator follows the definition used by the PASS Multiple Comparisons module for this sample-size procedure. It is not the ordinary probability of rejecting at least one null hypothesis. PASS defines power here as the probability that the simultaneous confidence intervals are correct and sufficiently narrow. This distinction matters when reproducing PASS's sample-size results.
Inputs: 4 groups; alpha = 0.05; target power = 0.90;
minimum detectable difference = 0.50; SD = 0.50.
Expected PASS result: 53 subjects per group,
total N = 212, achieved power = approximately 0.9146.
Independent implementation check: the numerical integration
used here gives n = 53 and achieved power approximately 0.9147;
n = 52 gives approximately 0.8938. Thus 53 is the first integer
group size to reach the requested 0.90 power, reproducing the PASS
sample-size result.
The calculation assumes a one-way normal model with a common unknown variance and equal sample sizes in all groups. The treatment-control comparisons are simultaneous and two-sided. The minimum detectable difference is an absolute difference between a treatment population mean and the control population mean.
Dunnett, C. W. (1955). A Multiple Comparison Procedure for Comparing Several Treatments with a Control. Journal of the American Statistical Association, 50(272), 1096–1121.
Hsu, J. C. (1996). Multiple Comparisons: Theory and Methods. Chapman & Hall/CRC. The PASS Multiple Comparisons documentation identifies Hsu's work as the principal basis for its multiple-comparison sample-size calculations.
NCSS / PASS. Multiple Comparisons, Chapter 575, PASS Sample Size Software documentation. The documented balanced Dunnett example uses k = 4, power = 0.90, alpha = 0.05, minimum detectable difference = 0.50, SD = 0.50, and reports n = 53 per group with achieved power 0.9146.