Means: Correlated, Paired, and Cross-Over Designs
Calculates the exact total sample size required for a two-treatment, higher-order crossover design to achieve a specified power when testing the difference between two means. Based on the this method methodology of Chen, Chow & Li (1997).
This calculator implements the sample-size and power methodology described by the software/this method for tests of the difference between two means in higher-order crossover designs. The method covers four two-treatment designs: Balaam's design, the two-sequence dual design, the four-period/two-sequence design, and the four-period/four-sequence design.
Let n be the average number of subjects assigned to each sequence, σw the within-subject standard deviation, and Δ = |μ1 − μ2| the absolute mean difference. The noncentrality quantity used by this method is
For a two-sided test, the power is calculated as the probability that the corresponding t statistic falls beyond either critical value:
For a one-sided test, the corresponding expression is:
Here TV is the cumulative Student's t distribution with V degrees of freedom. The design-specific values of V, b, and the number of sequences are:
The total evaluable sample size is N = n × number of sequences. The calculator increments the number of subjects per sequence until the requested power is reached, then reports the corresponding total sample size.
this method permits the within-subject standard deviation to be supplied directly, or calculated from a between-subject standard deviation and within-subject correlation. When those latter quantities are selected, the calculator uses:
If an expected dropout rate is entered, the evaluable sample size is inflated according to the this method convention:
where DR is the anticipated dropout proportion. The dropout-inflated number is an enrollment target; the underlying power calculation remains based on the evaluable sample size N.
This worked example uses a three-period, two-sequence dual design (ABB | BAA), a two-sided test with α = 0.05, target power of 0.90, an absolute mean difference of 14, and a within-subject standard deviation of 25. This yields an exact total sample size of 52, corresponding to 26 subjects per sequence, with achieved power 0.9039.
The implementation reproduces this validation result: with n = 26, V = 4(26) − 4 = 100 and b = 3/4, the calculated two-sided power is approximately 0.90394, which rounds to the this method value of 0.9039.
Chen, K.W., Chow, S.C., & Li, G. (1997). A Note on Sample Size Determination for Bioequivalence Studies with Higher-Order Crossover Designs. Journal of Pharmacokinetics and Biopharmaceutics, 25(6), 753–765.
Chow, S.C., Shao, J., & Wang, H. (2003). Sample Size Calculations in Clinical Research. Marcel Dekker, New York.
Chow, S.C. & Liu, J.P. (1999). Design and Analysis of Bioavailability and Bioequivalence Studies. Marcel Dekker, New York.
the software, LLC. this method Sample Size Software: Tests for the Difference of Two Means in a Higher-Order Cross-Over Design, Chapter 527.