the between studies model takes the form

$$\begin{aligned}
\label{copula-between-norm}
&&\Bigl(F\bigl(X_{11};l(\pi_{11}),\de_{11}\bigr),\ldots,
F\bigl(X_{1t};l(\pi_{1t}),\de_{1t}\bigr),\ldots,F\bigl(X_{1T};l(\pi_{1T}),\de_{1T}\bigr)\nonumber\\
&&F\bigl(X_{01};l(\pi_{01}),\de_{01}\bigr),\ldots,
F\bigl(X_{0t};l(\pi_{0t}),\de_{0t}\bigr),\ldots,F\bigl(X_{0T};l(\pi_{0T}),\de_{0T}\bigr)\Bigr)\sim C(\cdot;\thbf),\end{aligned}$$

where $F\bigl(\cdot;l(\pi),\de\bigr)$ is the cdf of the univariate distribution of the random effect and $C(\cdot;\thbf)$ is an one-factor copula with dependence parameter vector $\thbf=(\th_{11},\ldots,\th_{1t},\ldots,\th_{1T},\th_{11},\ldots,\th_{1t},\ldots,\th_{1T})$. That is, we assume that $U_{11},\ldots,U_{1T},U_{01},\ldots,U_{0T}$, where $U_{kt}=F\bigl(X_{kt};l(\pi_{kt}),\de_{kt}\bigr),\,k=0,1,\,t=1,\ldots,T$, are conditionally independent given another standard uniform latent variable/factor $U$. Then we can easily get the joint density of the one-factor copula using 1-dimensional numerical integration, viz., $$c(u_{11},\ldots,u_{1t},\ldots,u_{1T},u_{01},\ldots,u_{0y},\ldots,u_{0T};\thbf)=\int_0^1\biggr\{\prod_{t=1}^T\biggr[c(u_{1t},u;\th_{1t})c(u_{0t},u;\th_{0t})\biggl]\biggl\}du.$$ It is seen that the $2T$-variate density decomposes in a product of $2T$ bivariate copula densities that link the random effects $U_{kt},\,k=0,1,\,t=1,\ldots,T$ with $U$. The parameters $(\pi_{11},\ldots,\pi_{1t},\ldots,\pi_{1T})$ $:=\bfpi_1$ and $(\pi_{01},\ldots,\pi_{0t},\ldots,\pi_{0T}):=\bfpi_0$ are those of main interest denoting the meta-analytic parameters for the sensitivities and specificities, respectively, while the univariate parameters $(\de_{11},$ $\ldots,\de_{1t},\ldots,\de_{1T}):=\debf_1$ and $(\de_{01},\ldots,\de_{0t},$ $\ldots,\de_{0T}):=\debf_0$ are of secondary interest denoting the between-study variabilities for the sensitivities and specificities, respectively. The copula parameter vector $\thbf$ has parameters of the random effects model and they are separated from the univariate parameters $\bfpi_1,\bfpi_0,\debf_1,\debf_0$.

The models in (\[withinBinom\]) and (\[copula-between-norm\]) together specify an one-factor copula mixed model with joint likelihood

$$\begin{aligned}
\label{mixed-cop-likelihood}
&L(\bfpi_1,\bfpi_0,\debf_1,\debf_0,\thbf)\nonumber\\
&
=\prod_{i=1}^N\int_{[0,1]^{2T}}\biggl\{\prod_{t=1}^T\biggl[ g\biggl(y_{i11t};y_{i+1t},l^{-1}\Bigl(F^{-1}\bigl(u_{1t};l(\pi_{1t}),\de_{1t}\bigr)\Bigr)\biggr) g\biggl(y_{i00t};y_{i+0t},
l^{-1}\Bigl(F^{-1}\bigl(u_{0t};\nonumber\\&l(\pi_{0t}),\de_{0t}\bigr)\Bigr)\biggr)\biggr]
\int_0^1\biggl\{\prod_{t=1}^T\biggl[c(u_{1t},u;\th_{1t})c(u_{0t},u;\th_{0t})\biggr]\biggr\}du\biggr\}du_{11},\ldots,du_{1T}\,du_{01},\ldots,du_{0T}\nonumber\\
&=
\prod_{i=1}^N\int_0^1\biggl\{\prod_{t=1}^T\biggl[ \int_0^1 \biggl\{g\biggl(y_{i11t};y_{i+1t},l^{-1}\Bigl(F^{-1}\bigl(u_{1t};l(\pi_{1t}),\de_{1t}\bigl)\Bigr)\biggr)c(u_{1t},u;\th_{1t})\biggr\}du_{1t}\nonumber\\
&\int_0^1\biggl\{g\biggl(y_{i00t};y_{i+0t},l^{-1}\Bigl(F^{-1}\bigl(u_{0t};l(\pi_{0t}),\de_{0t}\bigl)\Bigr)\biggr)c(u_{0t},u;\th_{0t})\biggr\}du_{0t}\biggr]\biggr\}du\end{aligned}$$

where $g\bigl(y;n,\pi\bigr)=\binom{n}{y}\pi^y(1-\pi)^{n-y},\quad y=0,1,\ldots,n,\quad 0<\pi<1,$ is the binomial 