The complete stepped-wedge cluster randomised trial design has J sequences of clusters
observed over J+1 time periods, each sequence has `nper` clusters.
The first time period has all clusters in the control state, and the final time period has
all clusters in the treatment state, with one sequence switching between the two each
period to create the "step".
The assumed generalised linear mixed model for the stepped-wedge 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.
If the user specifies more than one value for icc, cac, or iac, then a ModelSpace is returned
with Models with every combination of parameters. This can be used in particular to generate
a design space for optimal design analyses.