The standard errors (SE) of the ML estimates can be also obtained via the gradients and the Hessian computed numerically during the maximization process.

For one-factor copula mixed models of the form with joint likelihood as in (\[mixed-cop-likelihood\]), numerical evaluation of the joint pmf can be achieved with the following steps:

1.  Calculate Gauss-Legendre quadrature points $\{u_q: q=1,\ldots,n_q\}$ and weights $\{w_q: q=1,\ldots,n_q\}$ in terms of standard uniform.

2.  Numerically evaluate the joint pmf $$\begin{aligned}
    &\int_0^1\biggl\{\prod_{t=1}^T\biggl[ \int_0^1 \biggl\{g\biggl(y_{i11t};y_{i+1t},l^{-1}\Bigl(F^{-1}\bigl(u_{1t};l(\pi_{1t}),\de_{1t}\bigl)\Bigr)\biggr)c(u_{1t},v;\th_{1t})\biggr\}du_{1t}\\
    &\int_0^1\biggl\{g\biggl(y_{i00t};y_{i+0t},l^{-1}\Bigl(F^{-1}\bigl(u_{0t};l(\pi_{0t}),\de_{0t}\bigl)\Bigr)\biggr)c(u_{0t},v;\th_{0t})\biggr\}du_{0t}\biggr]\biggr\}dv \end{aligned}$$ in a double sum:

    $$\begin{aligned}
    &\sum_{q_1=1}^{n_q}\biggl\{w_{q_1}\prod_{t=1}^T\biggl[\sum_{q_2=1}^{n_q} \biggl\{w_{q_2}g\biggl(y_{i11t};y_{i+1t},l^{-1}\Bigl(F^{-1}\bigl(C^{-1}(u_{q_2}|u_{q_1};\th_{1t});l(\pi_{1t}),\de_{1t}\bigl)\Bigr)\biggr)\biggr\}\\
    &\sum_{q_2=1}^{n_q}\biggl\{w_{q_2}g\biggl(y_{i00t};y_{i+0t},l^{-1}\Bigl(F^{-1}\bigl(C^{-1}(u_{q_2}|u_{q_1};\th_{0t});l(\pi_{0t}),\de_{0t}\bigl)\Bigr)\biggr)\biggr\}\biggr]\biggr\},\end{aligned}$$

    where $C^{-1}(v|u;\th)$ is the inverse conditional bivariate copula cdf. Note that the independent quadrature points $\{u_{q_1}: q_1=1,\ldots,n_q\}$ and $\{u_{q_2}: q_2=1,\ldots,n_q\}$ have converted to dependent quadrature points that have an one-factor copula distribution $C(\cdot;\thbf)$.

With Gauss-Legendre quadrature, the same nodes and weights are used for different functions; this helps in yielding smooth numerical derivatives for numerical optimization via quasi-Newton [@nash90]. Our one-factor copula mixed model for meta-analysis of multiple diagnostic tests is straightforward computationally as it requires the calculation of a double summation over the quadrature points.

\[miss-section\]Small-sample efficiency – Misspecification of the random effects distribution
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In this section, we study the small-sample efficiency and robustness of the ML estimation of the one-factor 