The staircase/diagonal cluster randomised trial design has J sequences of clusters
observed over J time periods, each sequence has `nper` clusters.
The first cluster is observed only once, in the first period in the treatment state.
All the remaining clusters are observed twice, once in the control and once in the treatment state
staggered over the course of the trial. In each time period there is therefore one cluster/sequence
in the control state and one in the treatment state, so that the design produces a "staircase".
The assumed generalised linear mixed model for the staircase/diagonal cluster trial is, for
individual i, in cluster j, at time t:
$$y_{ijt} \sim F(\mu_{ijt},\sigma)$$
$$\mu_{ijt} = h^-1(x_{ijt}\beta + \alpha_{1j} + \alpha_{2jt} + \alpha_{3i})$$
$$\alpha_{p.} \sim N(0,\sigma^2_p), p = 1,2,3$$
Defining \(\tau\) as the total model variance, then the intraclass correlation
coefficient (ICC) is
$$ICC = \frac{\sigma_1 + \sigma_2}{\tau}$$
the cluster autocorrelation coefficient (CAC) is :
$$CAC = \frac{\sigma_1}{\sigma_1 + \sigma_2}$$
and the individual autocorrelation coefficient as:
$$IAC = \frac{\sigma_3}{\tau(1-ICC)}$$
When CAC and/or IAC are not specified in the call, then the respective random effects
terms are assumed to be zero. For example, if IAC is not specified then \(\alpha_{3i}\)
does not appear in the model, and we have a cross-sectional sampling design; if IAC
were specified then we would have a cohort.
For non-linear models, such as Poisson or Binomial models, there is no single obvious choice
for `var_par` (\(\tau\) in the above formulae), as the models are heteroskedastic. Choices
might include the variance at the mean values of the parameters or a reasonable choice based
on the variance of the respective distribution.