Means: Correlated, Paired, and Cross-Over Designs
Calculates the required number of subjects when the same subjects are measured at two time points. The two-time-point repeated-measures problem reduces to a paired t-test on the within-subject change, with the variance of the change determined from the two time-point standard deviations and their correlation. Runs entirely in your browser.
With exactly two measurements per subject, the repeated-measures analysis can be expressed in terms of each subject's change score, D = Y2 − Y1. The resulting hypothesis test is a paired t-test of the mean change. PASS describes paired-data analysis in terms of the differences between the paired observations, and nQuery describes a two-measurement repeated-measures design as the simplest repeated-measures setting, for which paired methods are appropriate.
If the standard deviations at the two time points are σ1 and σ2, and the within-subject correlation is ρ, the variance of the change is
The calculator therefore converts the more directly interpretable time-point standard deviations and correlation into the standard deviation of the paired differences. This is the same variance relationship documented by PASS for pre-post repeated observations.
The paired t statistic has n − 1 degrees of freedom and noncentrality parameter
For each candidate integer n, the calculator evaluates the exact noncentral-t power for the selected one- or two-sided alternative. It then searches for the smallest n for which the calculated power reaches the requested target.
This is preferable to simply rounding a normal-approximation formula because the paired t-test uses an estimated standard deviation and therefore has n − 1 degrees of freedom.
Inputs: two-sided α = 0.05; target power = 0.80; expected mean change = 0.60; null change = 0; SD at Time 1 = 2.00; SD at Time 2 = 2.00; correlation = 0.20.
Implied SD of change: 2.5298.
Validated result: 142 subjects.
At n = 142, exact two-sided noncentral-t power is approximately 0.8015; n = 141 does not reach 0.80.
The assumptions in this validation example are taken from a PASS paired-means worked example: PASS uses a mean shift of 0.6, standard deviation 2, and correlation 0.20 in its paired-data example. The 142-subject result above is independently obtained by solving those assumptions for 80% power with the exact noncentral-t calculation implemented here.
This calculator is for a single repeated-measures cohort in which every subject contributes measurements at two specified time points and the primary effect is the mean within-subject change. It assumes approximately normally distributed change scores and a common planning value for the two time-point standard deviations and their correlation.
A two-group pre-post study, in which the primary question is whether the change differs between treatment groups, is a different design. PASS provides a separate procedure for that situation, based on comparing the paired changes between the two groups.
NCSS, LLC / PASS. Tests for Paired Means and PASS statistical software documentation. PASS documentation identifies the paired t-test as the standard method for testing the mean difference between two correlated means.
NCSS, LLC / PASS. Tests for Two Groups of Pre-Post Scores, Chapter 432. The documentation gives σD2 = σ12 + σ22 − 2ρσ1σ2 for the variance of paired differences and explains the reduction of the two-time-point repeated-measures problem to analysis of the change scores.
Statistical Solutions, Ltd. nQuery Advisor Version 5.0 User's Guide, Chapter 13, “Repeated Measures and Crossover Designs.” The guide describes two measurements per subject as the simplest repeated- measures setting and identifies paired methods for this design.