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

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