copula mixed model. In Section \[true\], we gauge the small-sample efficiency of the ML method in Section \[computation\] and investigate the misspecification of either the parametric margin or bivariate copula of the random effects distribution. In Section \[miss\], we investigate the mixed model misspecification by using the D-vine copula mixed model proposed by [@nikoloulopoulos-2018-smmr] as the true model.

\[true\]Misspecification of the parametric margin or bivariate pair-copulas
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We randomly generated $B = 10^4$ samples of size $n = 20, 50, 100$ from an one-factor copula mixed model with both normal and beta margins that jointly meta-analyses $T=\{2,3,4\}$ diagnostic tests.

The simulation process is as below:

1.  Simulate $(u_{11},\ldots,u_{1T},u_{01},\ldots,u_{0T})$ from an factor copula $C(;\taubf)$; $\taubf$ is converted to the copula parameter vector $\thbf$ of BVN, Frank or (rotated) Clayton copulas via the relations $$\label{tauBVN}
    \tau=\frac{2}{\pi}\arcsin(\th),$$ $$\label{tauFrank}
    \tau=\left\{\begin{array}{ccc}
    1-4\theta^{-1}-4\theta^{-2}\int_\theta^0\frac{t}{e^t-1}dt &,& \th<0\\
    1-4\theta^{-1}+4\theta^{-2}\int^\theta_0\frac{t}{e^t-1}dt &,& \th>0\\
    \end{array}\right.,$$ and $$\label{tauCln}
    \tau=\left\{\begin{array}{rcl}
    \th/(\th+2)&,& \mbox{by 0$^\circ$ or 180$^\circ$ }\\
    -\th/(\th+2)&,& \mbox{by 90$^\circ$ or 270$^\circ$}\\
    \end{array}\right.,$$ in [@HultLindskog02], [@genest87], and , respectively.

2.  For each test $t$ in $1,\ldots,T$ convert to proportions via $$\begin{aligned}
    x_{1t}&=&l^{-1}\biggl(F^{-1}\Bigl(u_{1t};l(\pi_{1t}),\de_{1t}\Bigr)\biggr) \\
    x_{0t}&=&l^{-1}\biggl(F^{-1}\Bigl(u_{0t};l(\pi_{0t}),\de_{0t}\Bigr)\biggr)\end{aligned}$$

3.  Simulate the study size $n$ from a shifted gamma distribution, i.e., $n\sim \mbox{sGamma}(\a=1.2,\b=0.01,\mbox{lag}=30)$ and round off to the nearest integer.

