parallel_crt: Generate a parallel cluster design

Description

Generate a parallel cluster randomised trial design model

Usage

parallel_crt(
  J,
  M,
  t,
  ratio = 0.5,
  beta = c(rep(0, t), 0),
  icc,
  cac = NULL,
  iac = NULL,
  var = 1,
  family = stats::gaussian()
)

Value

A Model object

Arguments

J: Integer indicating the number of sequences such that there are J+1 time periods
M: Integer. The number of individual observations per cluster-period, assumed equal across all clusters
t: Integer. The number of time periods.
ratio: Numeric value indicating the proportion of clusters assigned to treatment. Default is 0.5.
beta: Vector of beta parameters to initialise the design, defaults to all zeros.
icc: Intraclass correlation coefficient.
cac: Cluster autocorrelation coefficient, optional.
iac: Individual autocorrelation coefficient, optional.
var: Assumed overall variance of the model, used to calculate the other covariance, see details
family: a family object

Details

The complete parallel cluster randomised trial design has J clusters observed over T time periods. A proportion (`ratio`) of the clusters are assigned to treatment condition for the duration of the trial and the rest are control.

The assumed generalised linear mixed model for the parallel 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.

Examples

Run this code

#generate a simple design with only cluster random effects and 6 clusters in 3 time periods
# with 10 individuals in each cluster-period
des <- parallel_crt(J=6,M=10,t=3,icc=0.05)
# same design but with a cohort of individuals
des <- parallel_crt(J=6,M=10,t=3,icc=0.05, iac = 0.1)
# same design, but with two clusters per sequence and specifying the initial parameters
des <- parallel_crt(J=6,M=10,t=3,beta = c(rnorm(3,0,0.1),-0.1),icc=0.05, iac = 0.1)
# specifying multiple values of the variance parameters will return a design space 
# with all designs with all the combinations of the variance parameter
des <- parallel_crt(J=6,M=10,t=3,icc=c(0.01,0.05), cac = c(0.5,0.7,0.9), iac = 0.1)