Microbiome Abundance Modeling Protocol

Assumption 0.1. $E\left(O_{ijk}\mid A_{ijk}\right)=c_{jk}{A}_{ijk}Var\left(O_{ijk}\mid A_{ijk}\right)={\sigma }_{w,ijk}^2,$ where ${\sigma }_{w,ijk}^2$  = variability between specimens within the kth sample from the jth group. Therefore, ${\sigma }_{w,ijk}^2$ characterizes the within-sample variability. Typically, researchers do not obtain more than one specimen at a given time in most microbiome studies. Consequently, variability between specimens within sample is usually not estimated.
According to Assumption 0.1, in expectation the absolute abundance of a taxon in a random sample is in constant proportion to the absolute abundance in the ecosystem of the sample. In other words, the expected relative abundance of each taxon in a random sample is equal to the relative abundance of the taxon in the ecosystem of the sample.
Assumption 0.2. For each taxon i, A_ijk, j = 1, …, g, k = 1, …, n_j, are independently distributed with $E\left(A_{ijk}\mid {\theta }_{ij}\right)={\theta }_{ij}Var\left(A_{ijk}\mid {\theta }_{ij}\right)={\sigma }_{b,ij}^2,$ where ${\sigma }_{b,ij}^2$  = between-sample variation within group j for the ith taxon.
The Assumption 0.2 states that for a given taxon, all subjects within and between groups are independent, where θ_ij is a fixed parameter rather than a random variable.