TABLE 2.

Variations in the MMGLM for the observation model defined by Equation (8) and ecological process model defined by Equation (5) used as our study scenarios, with one covariate used for each model.

Classification probability type	Study scenario	Ecological process model	Observation process model
Heterogeneous	Variable/covariate	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}$	$\ln \left(\frac{{\mathrm{\Omega }}_{\mathit{jks}}}{{\mathrm{\Omega }}_{\mathit{jKs}}}\right)=\left({\omega }_{0\mathit{jk}}-{\omega }_{0\mathit{jK}}\right)+\psi z_{1s}\left({\omega }_{1\mathit{jk}}-{\omega }_{1\mathit{jK}}\right)$
	Fixed covariate	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}$	$\ln \left(\frac{{\mathrm{\Omega }}_{\mathit{jks}}}{{\mathrm{\Omega }}_{\mathit{jKs}}}\right)=\left({\omega }_{0\mathit{jk}}-{\omega }_{0\mathit{jK}}\right)+\psi z_{1s}{\omega }_{1\mathit{jj}},$
			where ${\omega }_{1\mathit{jk}}=0$ for $j\ne k$
	Fixed intercov	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}$	$\ln \left(\frac{{\mathrm{\Omega }}_{\mathit{jks}}}{{\mathrm{\Omega }}_{\mathit{jKs}}}\right)=\left({\omega }_{0\mathit{jk}}-{\omega }_{0\mathit{jK}}\right)+\psi z_{1s}{\omega }_{1\mathit{jj}},$
			where ${\omega }_{1\mathit{jk}}=0$ for $j\ne k$
			and ${\omega }_{0\mathit{jk}}$ is the same for $j=k$
Homogeneous	Intercept	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}$	$\ln \left(\frac{{\mathrm{\Omega }}_{\mathit{jks}}}{{\mathrm{\Omega }}_{\mathit{jKs}}}\right)={\omega }_{0\mathit{jk}}$
	Constant	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}$	${\mathrm{\Omega }}_{\mathit{jk}}\sim \text{Dirichlet}\left({\alpha }_{\mathit{jk}}\right),$
			where ${\alpha }_{\mathit{jk}}\sim \mathrm{Exp}\left(1\right)$
	Main	$\ln \left({\lambda }_{\mathit{js}}\right)={\beta }_{0j}+x_{1s}{\beta }_{1j}+x_{2s}{\beta }_{2j}+\psi z_{1s}{\beta }_{3j}$	${\mathrm{\Omega }}_{\mathit{jk}}\sim \text{Dirichlet}\left({\alpha }_{\mathit{jk}}\right),$
			where ${\alpha }_{\mathit{jk}}\sim \mathrm{Exp}\left(1\right)$

Note: There are three heterogeneous models: covariate, fixed covariate and fixed intercov and three homogeneous models: intercept, constant and main. The definitions of the parameters in this table are described in Sections 2.1.1, 2.1.2, 2.1.3.