AngusMcLure
commented
1 year ago Here's my first attempt which we can refine as required:
form: To calculate design effects for cluster surveys with individual (not pooled) testing, we need only consider the pairwise-correlation between any two units from the same cluster which is summarised by the parameter $\rho$ (correlation). However, in pool or group-tested cluster designs, because there can be more than two units in a pool/group, we must have a model of the higher order correlation structure between multiple units from the same cluster. This could in principle be done in many ways. However, the most practical and analytically simplest way to achieve this is by modelling the prevalence of the marker at each cluster, $\Theta_j$, as being drawn from some distribution, with units from a given cluster being statistically independent once conditioned on the cluster-level prevalence. The mean of this distribution is the population prevalence, $E[\Theta_j] = \theta$ and the variance of this distribution, $V[\Theta_j]$ is related to correlation as $\rho = \frac{V[\Theta_j]}{\theta ( 1-\theta)}$. Any distribution with support on $[0,1]$ can used to model the distributional form for the cluster-level prevalences, $\Theta_j$. We consider four examples, each of which is a two-parameter of family of distributions: beta , beta distribution; logitnorm, the logit-normal distribution; cloglognorm the complementary-log-log-normal distribution; and discrete, a discrete distribution with mass at $0$, $\theta$, and $1$.
Could you please supply the text for this Angus?