FactorCopulaREMADA {CopulaREMADA}

Maximum likelihood estimation of 1-factor copula mixed models for joint meta-analysis of T diagnostic tests

Description

The estimated parameters can be obtained by using a quasi-Newton method applied to the logarithm of the joint likelihood. This numerical method requires only the objective function, i.e., the logarithm of the joint likelihood, while the gradients are computed numerically and the Hessian matrix of the second order derivatives is updated in each iteration. The standard errors (SE) of the ML estimates can be also obtained via the gradients and the Hessian computed numerically during the maximization process.

Usage

FactorCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2)
                               
FactorCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2)

Arguments

TP

an n\times T matrix where n is the number of studies. Column j has the number of true positives for test j for j=1\ldots T

FN

an n\times T matrix where n is the number of studies. Column j has the number of false negatives Column j has the number of true positives for test j for j=1\ldots T

FP

an n\times T matrix where n is the number of studies. Column j has the number of false positives Column j has the number of true positives for test j for j=1\ldots T

TN

an n\times T matrix where n is the number of studies. Column j has the number of true negatives Column j has the number of true positives for test j for j=1\ldots T

gl

a list containing the components of Gauss-Legendre nodes gl$nodes and weights gl$weights

mgrid

a list containing two matrices with the rows of the output matrix x are copies of the vector gl$nodes; columns of the output matrix y are copies of the vector gl$nodes. For more details see also meshgrid

qcond1

function for the inverse conditional copula cdfs that link the factor with the latent sensitivities

tau2par1

function for maping Kendall's tau to copula parameter at the copulas that link the factor with the latent sensitivities

qcond2

function for the inverse conditional copula cdfs that link the factor with the latent specificities

tau2par2

function for maping Kendall's tau to copula parameter at the copulas that link the factor with the latent specificities

Value

A list containing the following components:

minimum

the value of the estimated minimum of the negative log-likelihood

estimate

the MLE

gradient

the gradient at the estimated minimum of of the negative log-likelihood

hessian

the hessian at the estimated minimum of the negative log-likelihood

code

an integer indicating why the optimization process terminated

iterations

the number of iterations performed

For more details see nlm

References

Nikoloulopoulos, A.K. (2022) An one-factor copula mixed model for joint meta-analysis of multiple diagnostic tests. Journal of the Royal Statistical Society: Series A (Statistics in Society), 185 (3), 1398–1423. doi:10.1111/rssa.12838.

Examples

data(arthritis)
attach(arthritis)
TP=cbind(TP1,TP2)
TN=cbind(TN1,TN2)
FP=cbind(FP1,FP2)
FN=cbind(FN1,FN2)


nq=25
gl=gauss.quad.prob(nq,"uniform")
mgrid=meshgrid(gl$n,gl$n)
qcond1=qcondcln
qcond2=qcondcln270
tau2par1=tau2par.cln
tau2par2=tau2par.cln270

out=FactorCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2)
se=sqrt(diag(solve(out$hessian)))

detach(arthritis)