Means: Correlated, Paired, and Cross-Over Designs
Sample size planning for pairwise mean-difference tests in a balanced Williams cross-over design. based on the standard procedure documented by the software and the sample-size methodology of Chow, Shao, Wang & Lokhnygina. The calculator determines the minimum number of evaluable subjects per sequence, the number of sequences, and the total sample size.
A Williams cross-over design is constructed from balanced treatment-order sequences. When the number of treatments k is even, the Williams design has k sequences and k periods. When k is odd, it has 2k sequences and k periods. Thus the number of sequences used by this calculator is k for even k and 2k for odd k.
this method defines the sample size n as the number of subjects in each sequence. The total evaluable sample size is therefore N = a × n. The sequences are assumed to have equal sample sizes.
For pairwise treatment mean differences, let D1 be the minimum absolute treatment difference to detect and let SD be the standard deviation of the paired differences. For a design with a sequences and n subjects per sequence, This calculator uses a noncentral t distribution with a(n−1) degrees of freedom and noncentrality parameter |D1|√(an)/SD.
The sample-size search evaluates integer values of n and returns the smallest value whose calculated power is at least the requested target. This follows the relevant methodological literature, which states that the sample size is determined by searching possible values of n.
With k treatments there are k(k−1)/2 possible pairwise treatment comparisons. When the Bonferroni option is selected, the individual comparison alpha is calculated as the overall alpha divided by this number of comparisons.
Important: The this method validation example for this procedure does not use a multiple-testing adjustment. With k = 3, D1 = 0.05, SD = 0.10, alpha = 0.05, and 80% power, This yields 6 subjects per sequence and 36 subjects total. The achieved power at n = 6 is 0.8271.
If a dropout rate is entered, the calculator first determines the evaluable sample size required by the statistical calculation. It then inflates the sample size in each sequence using the this method convention: divide by one minus the dropout rate and round up.
this method / the software documentation: the software, LLC. Tests for Pairwise Mean Differences in a Williams Cross-Over Design, this method Sample Size Software, Chapter 501. The documentation states that the sample-size and power calculations are based on Chow, Shao, Wang, & Lokhnygina (2018), and includes a published-example validation in Example 2. this method procedure documentation.
Primary methodological source: Chow, S.C., Shao, J., Wang, H., & Lokhnygina, Y. (2018). Sample Size Calculations in Clinical Research, 3rd Edition. Taylor & Francis/CRC Press, Boca Raton, Florida.
Validation example: Chow, Shao, Wang & Lokhnygina (2018), p. 66, as reproduced and validated in This worked example. For a 6 × 3 Williams cross-over design with D1 = 0.05, SD = 0.10, alpha = 0.05, and 80% power, the required sample size is 6 subjects per sequence.
The Williams-design construction used here follows the relevant methodological literature: even numbers of treatments produce a k × k design, whereas odd numbers produce a 2k × k design.