The replicated crossover design (2 by 2\(m\) crossover) is a commonly used design to test variabilities. When \(m \gt 1\), it has the benefit of allowing estimates of inter-subject variability.

The power and sample size calculators for tests of variabilities on this site use the methods explained in Chinchilli *et al.* and Chow *et al.*. Using the notation of Chow *et al.*, we consider an experiment where we are comparing two treatments: a test treatment denoted \(T\) and a reference treatment denoted \(R\). The study is to be conducted using a 2 by 2\(m\) crossover design with \(n_i\) subjects in each sequence \(i\).

Consider the following mixed effects model:

\[ x_{ijkl} = \mu_k + \gamma_{ikl} + S_{ijk} + \epsilon_{ijkl} \]where \(\mu_k\) is the treatment effect for treatment \(k\), \(\gamma_{ijk}\) is the fixed effect of the \(l^{th}\) replicate of treatment \(k\) in the \(i^{th}\) sequence subject to the constraint that the sum of the fixed effects is zero, and \(S_{ijk}\) is the random effect of the \(j^{th}\) subject in the \(i^{th}\) sequence and \( \epsilon_{ijkl} \), the error term, is normally distributed with mean 0 and variance \( \sigma^2_{Wk} \). Each vector \((S_{ijT}, S_{ijR})'\) is bivariately normally distributed with mean \((0, 0)'\), covariance

\[ \Sigma_B = \begin{bmatrix} \sigma^2_{BT} & \rho \sigma_{BT} \sigma_{BR} \\ \rho \sigma_{BT} \sigma_{BR} & \sigma^2_{BR} \\ \end{bmatrix} \]and independant from \epsilon_{ijkl}. The values \(\sigma^2_{WT}\) and \(\sigma^2_{WR}\) are the intra-subject variabilities for treatments \(T\) and \(R\), respecively while \(\sigma^2_{BT}\) and \(\sigma^2_{BR}\) are the inter-subject variabilities for treatments \(T\) and \(R\).

Under this model, we are able to test total variability, intra-subject variability and inter-subject variability. In the case of inter-subject variability, we must ensure that \(m > 1\).

- Chinchilli, V.M. and Esinhart, J.D. (1996), "Design and analysis of intra-subject variability in cross-over experiments,"
*Statistics in Medicine,*15, 1619-1634. - Chow, S., Shao, J., & Wang, H. (2003),
*Sample size calculations in clinical research,*New York: Marcel Dekker.