hybridCopulaREMADA {CopulaREMADA}

Maximum likelhood estimation for hybrid copula mixed models for combining case-control and cohort studies in meta-analysis of 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

hybridCopulaREMADA.norm(TP,FN,FP,TN,type,gl,mgrid1,mgrid2,
               qcondcop12,qcondcop13,
               tau2par12,tau2par13,qcond,tau2par)
hybridCopulaREMADA.beta(TP,FN,FP,TN,type,gl,mgrid1,mgrid2,
               qcondcop12,qcondcop13,
               tau2par12,tau2par13,qcond,tau2par) 

Arguments

TP

the number of true positives

FN

the number of false negatives

FP

the number of false positives

TN

the number of true negatives

type

a scalar indicating the study type: 1: Cohort study; 2: Case-control study.

gl

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

mgrid1

a list containing three-dimensional arrays. For more details see meshgrid

mgrid2

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

qcondcop12

function for the inverse of conditional copula cdf at the (1,2) bivariate margin of the vine

qcondcop13

function for the inverse of conditional copula cdf at the (1,3) bivariate margin of the vine

tau2par12

function for maping Kendall's tau at the (1,2) bivariate margin of the vine to copula parameter

tau2par13

function for maping Kendall's tau at the (1,3) bivariate margin of the vine to copula parameter

qcond

function for the inverse of conditional copula cdf

tau2par

function for maping Kendall's tau to the bivariate copula parameter

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. (2018) Hybrid copula mixed models for combining case-control and cohort studies in meta-analysis of diagnostic tests. Statistical Methods in Medical Research, 27, 2540–2553. doi:10.1177/0962280216682376.

See Also

VineCopulaREMADA, CopulaREMADA

Examples

# simulate the data from N=25 cohort studies
N=25
p=c(0.8,0.7,0.4)
g=c(0.1,0.1,0.05)
taus=c(-0.5,-0.3,-0.0001)
qcondcop12=qcondcop23=qcondcop13=qcondcln90
tau2par12=tau2par23=tau2par13=tau2par.cln90
simdat1=rVineCopulaREMADA.beta(N,p,g,taus,0,0,
qcondcop12,qcondcop13,qcondcop23,tau2par12,tau2par13,tau2par23)
# simulate data from the N=25 case-control studies
tau=0.5
p=p[-3]
g=g[-3]
simdat2=rCopulaREMADA.beta(N,p,g,tau,rcln,tau2par.cln)
# combine the data
TP=c(simdat1$TP,simdat2$TP)
TN=c(simdat1$TN,simdat2$TN)
FP=c(simdat1$FP,simdat2$FP)
FN=c(simdat1$FN,simdat2$FN)
type=rep(c(1,2),each=N)

# fit the hybrid copula mixed model
nq=21
gl=gauss.quad.prob(nq,"uniform")
mgrid1<- meshgrid(gl$n,gl$n,gl$n,nargout=3)
mgrid2<- meshgrid(gl$n,gl$n)

qcond=qcondcln
tau2par=tau2par.cln
est=hybridCopulaREMADA.beta(TP,FN,FP,TN,type,gl,mgrid1,mgrid2,
qcondcop12,qcondcop13,tau2par12,tau2par13,qcond,tau2par)