# The Parallel Design with Replicates for Testing Variabilities

Adding replicates to a parallel design allows you to asses more than just the total variability. You can also asses the inter- and intra-subject variability. To discuss the design, we will use the notation from Chow et al. which concerns itself with comparing two treatments: a test treatment, denoted $$T$$, and a reference treatment, denoted $$R$$.

Responses in this design will be modeled using a mixed model described as follows: let $$x_{ijk}$$ be the $$k^{th}$$ replicate ($$k=1, \dots, m$$) from the $$j^{th}$$ subject ($$j=1, \dots, n_i)$$ under the $$i^{th}$$ treatment ($$i=T, R)$$. We assume that

$x_{ijk} = \mu_i + S_{ij} + \epsilon_{ijk}$

where $$\mu_i$$ is the treatment effect, $$S_{ij}$$ is the random effect of the $$j^{th}$$ subject under the $$i^{th}$$ treatment and $$\epsilon_{ijk}$$ is the random error with mean 0 and variance $$\sigma^2_{Bi}$$. Further, we assume that for a given $$i$$, $$S_{ij}$$ is an i.i.d normal random variabile with mean 0 and variance $$\sigma^2_{Bi}$$.

Under this model, we are able to test total variability, intra-subject variability and inter-subject variability

## References

• Chow, S., Shao, J., & Wang, H. (2003), Sample size calculations in clinical research, New York: Marcel Dekker.