probability mass function (pmf). It is shown that the joint likelihood is represented as an one-dimensional integral of a function which in turn is a product of $2T$ one-dimensional integrals. As a result, $2T$-dimensional numerical integration has been avoided.

Our general statistical model allows for selection of bivariate copulas and univariate margins independently, i.e., there are no constraints in the choices of parametric bivariate copulas and univariate margins. In line with our previous contributions in copula mixed models [@Nikoloulopoulos2015b; @Nikoloulopoulos2015c; @Nikoloulopoulos-2016-SMMR; @Nikoloulopoulos2018-AStA; @nikoloulopoulos-2018-smmr; @Nikoloulopoulos-2018-3dmeta-NE; @Nikoloulopoulos-2018-4dmeta-NE] we use

-   bivariate parametric copulas with different tail dependence behaviour, namely the bivariate normal (BVN) with intermediate tail dependence, Frank with tail independence, and Clayton with positive lower tail dependence. For the latter we also use its rotated versions to provide negative upper-lower tail dependence (Clayton rotated by 90$^\circ$), positive upper tail dependence (Clayton rotated by 180$^\circ$) and negative lower-upper tail dependence (Clayton rotated by 270$^\circ$);

-   the choices of $F\bigl(\cdot;l(\pi),\de\bigr)$ and $l$ that are given in Table \[choices\]. With a beta distribution we work on the original scale of the latent sensitivities and specificities.

   $F\bigl(\cdot;l(\pi),\de\bigr)$     $l$          $\pi$        $\de$
  --------------------------------- ---------- --------------- ----------
             $N(\mu,\s)$              logit     $l^{-1}(\mu)$     $\s$
         Beta$(\pi,\gamma)$          identity       $\pi$       $\gamma$

  : \[choices\]The choices 