Means: Correlated, Paired, and Cross-Over Designs
Calculate the minimum number of paired observations needed to achieve a specified power for a one- or two-sided paired t-test. The calculation uses the noncentral t distribution and the standard deviation of the paired differences.
The paired t-test reduces each pair of observations to a single difference, D = X1 − X2. The planning calculation therefore uses the hypothesized mean paired difference δ1 and the standard deviation of those differences, σ. The test has n − 1 degrees of freedom and, under the specified alternative, the test statistic follows a noncentral t distribution with noncentrality parameter δnc = √n · δ1 / σ.
For a two-sided test, the null hypothesis is H0: δ = 0 versus H1: δ ≠ 0, and the rejection region is |t| > t1−α/2, df. For a one-sided test, the corresponding critical value uses α in the selected tail. Power is calculated from the exact noncentral t distribution at the specified alternative, and the calculator searches upward from n = 2 until the first integer sample size reaches the requested power.
Mean paired difference (δ1) is the difference that the study is designed to detect. SD of paired differences (σ) is the expected standard deviation of X1 − X2, not the SD of either measurement separately. If separate SDs and the within-pair correlation are available, PASS documents the relationship σ2 = σ12 + σ22 − 2ρσ1σ2.
The paired t-test assumes continuous paired differences that are normally distributed and arise from a simple random sample of pairs. The planning calculation treats the specified SD of paired differences as the population value used to determine power.
NCSS, LLC. PASS Sample Size Software: Paired T-Tests, Chapter 485. PASS documentation. The procedure describes sample-size and power calculations for paired t-tests using the mean paired difference, standard deviation of paired differences, alpha, and power, with the t distribution and noncentral t distribution used for the test and power calculations.
O'Brien, R. G., & Muller, K. E. (1993). Unified power analysis for t-tests through multivariate hypotheses. In L. K. Edwards (Ed.), Applied Analysis of Variance in Behavioral Science (pp. 297–344). Marcel Dekker. This is cited in the nQuery documentation as a reference for its paired t-test power and sample-size computations.