# The Replicated Crossover Design for Testing Variabilities

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 $$m > 1$$.

## References

• 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.