Help for package CopulaREMADA

Version:

1.7.5

Date:

2025-05-27

Title:

Copula Mixed Models for Multivariate Meta-Analysis of Diagnostic Test Accuracy Studies

Author:

Aristidis K. Nikoloulopoulos

[aut, cre]

Maintainer:

Aristidis K. Nikoloulopoulos <A.Nikoloulopoulos@uea.ac.uk>

Depends:

R (≥ 3.5.0), statmod, matlab, tensor, mc2d

Description:

The bivariate copula mixed model for meta-analysis of diagnostic test accuracy studies in Nikoloulopoulos (2015) <doi:10.1002/sim.6595> and Nikoloulopoulos (2018) <doi:10.1007/s10182-017-0299-y>. The vine copula mixed model for meta-analysis of diagnostic test accuracy studies accounting for disease prevalence in Nikoloulopoulos (2017) <doi:10.1177/0962280215596769> and also accounting for non-evaluable subjects in Nikoloulopoulos (2020) <doi:10.1515/ijb-2019-0107>. The hybrid vine copula mixed model for meta-analysis of diagnostic test accuracy case-control and cohort studies in Nikoloulopoulos (2018) <doi:10.1177/0962280216682376>. The D-vine copula mixed model for meta-analysis and comparison of two diagnostic tests in Nikoloulopoulos (2019) <doi:10.1177/0962280218796685>. The multinomial quadrivariate D-vine copula mixed model for meta-analysis of diagnostic tests with non-evaluable subjects in Nikoloulopoulos (2020) <doi:10.1177/0962280220913898>. The one-factor copula mixed model for joint meta-analysis of multiple diagnostic tests in Nikoloulopoulos (2022) <doi:10.1111/rssa.12838>. The multinomial six-variate 1-truncated D-vine copula mixed model for meta-analysis of two diagnostic tests accounting for within and between studies dependence in Nikoloulopoulos (2024) <doi:10.1177/09622802241269645>. The 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard (Nikoloulopoulos, 2025) <doi:10.1093/biomtc/ujaf037>.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Packaged:

2025-05-27 15:52:45 UTC; aris

NeedsCompilation:

Repository:

CRAN

Date/Publication:

2025-05-27 16:20:05 UTC

Copula Mixed Models for Multivariate Meta-Analysis of Diagnostic Test Accuracy Studies

Description

Fits copula mixed models for multivariate meta-analysis of diagnostic test accuracy studies proposed in Nikoloulopoulos (2015, 2017, 2018a, 2018b, 2019, 2020a, 2020b, 2022, 2024, 2025).

Details

This package contains R functions to implement:

The copula mixed model for meta-analysis of diagnostic test accuracy studies and produce SROC curves and summary operating points (a pair of average sensitivity and specificity) with a confidence region and a predictive region (Nikoloulopoulos, 2015, 2018a). All the analyses presented in Section 7 of Nikoloulopoulos (2015) are given as code examples in the package;
The vine copula mixed model for meta-analysis of diagnostic test accuracy studies accounting for disease prevalence and non-evaluable subjects (Nikoloulopoulos, 2017, 2020a);
The hybrid vine copula mixed model for meta-analysis of diagnostic test accuracy case-control and cohort studies (Nikoloulopoulos, 2018b);
The D-vine copula mixed model for meta-analysis and comparison of two diagnostic tests (Nikoloulopoulos, 2019).
The multinomial quadrivariate D-vine copula mixed model for diagnostic studies meta-analysis accounting for non-evaluable subjects (Nikoloulopoulos, 2020b).
The one-factor copula mixed model for joint meta-analysis of multiple diagnostic tests (Nikoloulopoulos, 2022).
The multinomial six-variate D-vine copula mixed model for for meta-analysis of two diagnostic tests accounting for within and between studies dependence (Nikoloulopoulos, 2024).
The 1-truncated D-vine copula mixed model for for meta-analysis of diagnostic test accuracy studies without a gold standard (Nikoloulopoulos, 2025).

Author(s)

Aristidis K. Nikoloulopoulos.

References

Nikoloulopoulos, A.K. (2015) A mixed effect model for bivariate meta-analysis of diagnostic test accuracy studies using a copula representation of the random effects distribution. Statistics in Medicine, 34, 3842–3865. doi:10.1002/sim.6595.

Nikoloulopoulos, A.K. (2017) A vine copula mixed effect model for trivariate meta-analysis of diagnostic test accuracy studies accounting for disease prevalence. Statistical Methods in Medical Research, 26, 2270–2286. doi:10.1177/0962280215596769.

Nikoloulopoulos, A.K. (2018a) On composite likelihood in bivariate meta-analysis of diagnostic test accuracy studies. AStA Advances in Statistical Analysis, 102, 211–227. doi:10.1007/s10182-017-0299-y.

Nikoloulopoulos, A.K. (2018b) 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.

Nikoloulopoulos, A.K. (2019) A D-vine copula mixed model for joint meta-analysis and comparison of diagnostic tests. Statistical Methods in Medical Research, 28(10-11):3286–3300. doi:10.1177/0962280218796685.

Nikoloulopoulos, A.K. (2020a) An extended trivariate vine copula mixed model for meta-analysis of diagnostic studies in the presence of non-evaluable outcomes. The International Journal of Biostatistics, 16(2). doi:10.1515/ijb-2019-0107.

Nikoloulopoulos, A.K. (2020b) A multinomial quadrivariate D-vine copula mixed model for diagnostic studies meta-analysis in the presence of non-evaluable subjects. Statistical Methods in Medical Research, 29(10), 2988–3005. doi:10.1177/0962280220913898.

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.

Nikoloulopoulos, A.K. (2024) Joint meta-analysis of two diagnostic tests accounting for within and between studies dependence. Statistical Methods in Medical Research, 33(10), 1800–1817. doi:10.1177/09622802241269645.

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

The computing tomography data

Description

Data on 17 studies of computed tomography (CT) for the diagnosis of lymph node metastasis in women with cervical cancer, one of three imaging techniques in the meta-analysis in Scheidler et al. (1997). Diagnosis of metastatic disease by CT relies on nodal enlargement.

Usage

data(CT)

Format

A data frame with 17 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

Scheidler, J., H. Hricak, K. K. Yu, L. Subak, and M. R. Segal. (1997) Radiological evaluation of lymph node metastases in patients with cervical cancer: A meta-analysis. Journal of the American Medical Association, 278, 1096–1101.

Maximum likelhood estimation for copula mixed models for diagnostic test accurracy studies

Description

For copula mixed models for diagnostic test accuracy studies numerical evaluation of the MLE is easily done with the following steps:

1. Calculate Gauss-Legendre quadrature points gl$nodes and weights gl$weights.

2. Convert from independent uniform quadrature points to dependent uniform quadrature points that have distribution 'cop'. The inverse of the conditional distribution qcondcop corresponding to the copula 'cop' is used to achieve this.

3. Numerically evaluate the joint probability mass function with the bivariate integral in a double sum.

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. Our comparisons show that n_q=15 is adequate with good precision to at least at four decimal places.

Usage

CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcond,tau2par)
CopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcond,tau2par) 
countermonotonicCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid) 
countermonotonicCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid)

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

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

qcond

function for the inverse of conditional copula cdf

tau2par

function for maping Kendall's tau to 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

Examples

nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n)

data(LAG)
attach(LAG)
c270est.b=CopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcondcln270,tau2par.cln270)
detach(LAG)

data(MRI)
attach(MRI)
c270est.n=CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcondcln270,tau2par.cln270)
detach(MRI)

data(CT)
attach(CT)
est.n=countermonotonicCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid)
est.b=countermonotonicCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid)
detach(CT)

The down syndrome data

Description

Data on N=11 studies from the systematic review that examined the screening accuracy of shortened humerus and shortened femur of the fetus markers (two out of seven ultrasonographic markers or their combination in detecting Down syndrome in Smith-Bindman et al., 2001).

Format

A data frame with 11 observations on the following 8 variables.

down_n_00: the number of the test results in the diseased where the shortened humerus outcome is negative and the shortened femur outcome is negative
down_n_01: the number of the test results in the diseased where the shortened humerus outcome is negative and the shortened femur outcome is positive
down_n_10: the number of the test results in the diseased where the shortened humerus outcome is positive and the shortened femur outcome is negative
down_n_11: the number of the test results in the diseased where the shortened humerus outcome is positive and the shortened femur outcome is positive
nodown_n_00: the number of the test results in the non-diseased where the shortened humerus outcome is negative and the shortened femur outcome is negative
nodown_n_01: the number of the test results in the non-diseased where the shortened humerus outcome is negative and the shortened femur outcome is positive
nodown_n_10: the number of the test results in the non-diseased where the shortened humerus outcome is positive and the shortened femur outcome is negative
nodown_n_11: the number of the test results in the non-diseased where the shortened humerus outcome is positive and the shortened femur outcome is positive

References

Smith-Bindman R, Hosmer W, Feldstein V et al. Second-trimester ultrasound to detect fetuses with down syndrome: A meta-analysis (2001). Journal of the American Medical Association, 285(8): 1044-1055.

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

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

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)

The lymphangiography data

Description

Data on 17 studies of lymphangiography (LAG) for the diagnosis of lymph node metastasis in women with cervical cancer, one of three imaging techniques in the meta-analysis in Scheidler et al. (1997). Diagnosis of metastatic disease by LAG is based on the presence of nodal-filling defects.

Usage

data(LAG)

Format

A data frame with 17 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

The coronary CT angiography data in Menke and Kowalski (2016).

Description

Data on 30 studies from a systematic review for diagnostic accuracy studies of coronary computed tomography angiography for the detection of coronary artery disease.

Usage

data(MK2016)

Format

A data frame with 30 observations on the following 6 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives
NEP: the number of non-evaluable positives
NEN: the number of non-evaluable negatives

References

Menke, J. and Kowalski, J. (2016). Diagnostic accuracy and utility of coronary ct angiography with consideration of unevaluable results: A systematic review and multivariate bayesian random-effects meta-analysis with intention to diagnose. European Radiology, 26(2):451–458.

The magnetic resonance imaging data

Description

Data on 10 studies of magnetic resonance imaging (MRI) for the diagnosis of lymph node metastasis in women with cervical cancer, the last imaging technique in the meta-analysis in Scheidler et al. (1997). Diagnosis of metastatic disease by MRI relies on nodal enlargement.

Usage

data(MRI)

Format

A data frame with 10 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

The orale glucose tolerance data

Description

Data on 10 studies of the oral glucose tolerance test for the diagnosis of diabetes mellitus in patients during acute coronary syndrome hospitalization in Ye et al. (2012).

Usage

data(OGT)

Format

A data frame with 10 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

Ye, Y., Xie, H., Zhao, X., Zhang, S. (2012) The oral glucose tolerance test for the diagnosis of diabetes mellitus in patients during acute coronary syndrome hospitalization: a meta-analysis of diagnostic test accuracy. Cardiovascular Diabetology, 11(5):155.

The Pap test data

Description

These data are comprised of N=59 studies that published between January 1984 and March 1992. The diagnostic accuracy of the Pap test (i.e., index test) is evaluated by comparing with the histology test (i.e., reference test), which is not a perfect test (Fahey, et al., 1995).

Format

A data frame with 59 observations on the following 4 variables.

y11: the number of the test results where the Pap test outcome is positive and the histology test outcome is positive
y10: the number of the test results where the Pap test outcome is positive and the histology test outcome is negative
y01: the number of the test results where the Pap test outcome is negative and the histology test outcome is positive
y00: the number of the test results where the Pap test outcome is negative and the histology test outcome is negative

References

Fahey, M. T., Irwig, L., and Macaskill, P. (1995). Meta-analysis of pap test accuracy. American Journal of Epidemiology, 142:680–689.

Summary receiver operating characteristic curves for copula mixed effect models for bivariate meta-analysis of diagnostic test accuracy studies

Description

Summary receiver operating characteristic (SROC) curves are demonstrated for the proposed models through quantile regression techniques and different characterizations of the estimated bivariate random effects distribution

Usage

SROC.norm(param,dcop,qcondcop,tau2par,TP,FN,FP,TN,
          points=TRUE,curves=TRUE,
          NEP=rep(0,length(TP)),NEN=rep(0,length(TP)))
SROC.beta(param,dcop,qcondcop,tau2par,TP,FN,FP,TN,
          points=TRUE,curves=TRUE,
          NEP=rep(0,length(TP)),NEN=rep(0,length(TP)))
SROC(param.beta,param.normal,TP,FN,FP,TN,
          NEP=rep(0,length(TP)),NEN=rep(0,length(TP)))

Arguments

param

A vector with the sensitivities, specifities, variabilities and Kendall's tau value (the latter only for SROC.norm and SROC.beta)

param.beta

A vector with the sensitivity, specifity and variabilities of the countermonotonic CopulaREMADA with beta margins

param.normal

A vector with the sensitivity, specifity and variabilities of the countermonotonic CopulaREMADA with normal margins

dcop

function for copula density

qcondcop

function for the inverse of conditional copula cdf

tau2par

function for maping Kendall's tau to copula parameter

TP

the number of true positives

FN

the number of false negatives

FP

the number of false positives

TN

the number of true negatives

points

logical: print individual studies

curves

logical: print quantile regression curves

NEP

the number of non-evaluable positives in the presence of non-evaluable subjects

NEN

the number of non-evaluable negatives in the presence of non-evaluable subjects

Value

Summary receiver operating characteristic curves

References

Examples

nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n)

data(telomerase) 
attach(telomerase)
est.n=countermonotonicCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid)
est.b=countermonotonicCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid)
SROC(est.b$e,est.n$e,TP,FN,FP,TN)
detach(telomerase)

data(LAG)
attach(LAG)
c180est.b=CopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcondcln180,tau2par.cln180)
SROC.beta(c180est.b$e,dcln180,qcondcln180,tau2par.cln180,TP,FN,FP,TN)
detach(LAG)

data(MRI)
attach(MRI)
c270est.n=CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcondcln270,tau2par.cln270)
SROC.norm(c270est.n$e,dcln270,qcondcln270,tau2par.cln270,TP,FN,FP,TN)
detach(MRI)

data(MK2016)
attach(MK2016)
p=c(0.898745016,0.766105342,0.059168715,0.109217888)
g=c(0.090270947,0.079469009,0.367463579,0.154976269)
taus=c(0.82050793,-0.51867629,0.26457961)
SROC.beta(c(p[1:2],g[1:2],taus[1]),
          dcln180,qcondcln180,tau2par.cln180,
          TP,FN,FP,TN,points=TRUE,curves=TRUE,NEP,NEN)
detach(MK2016)

Maximum likelhood estimation for (truncated) vine copula mixed models for diagnostic test accurracy studies accounting for disease prevalence and non-evaluable outcomes

Description

Usage

VineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,
                               qcondcop12,qcondcop13,qcondcop23,
                               tau2par12,tau2par13,tau2par23,
                               NEP,NEN)
VineCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,
                               qcondcop12,qcondcop13,qcondcop23,
                               tau2par12,tau2par13,tau2par23,
                               NEP,NEN)
tVineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,
                               qcondcop12,qcondcop13,
                               tau2par12,tau2par13,
                               NEP,NEN)
tVineCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,
                               qcondcop12,qcondcop13,
                               tau2par12,tau2par13,
                               NEP,NEN)

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

gl

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

mgrid

a list containing three-dimensional arrays. Replicates of the quadrature points that produce a 3-dimensional full grid

qcondcop12

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

qcondcop13

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

qcondcop23

function for the inverse of conditional copula cdf at the (2,3|1) bivariate margin

tau2par12

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

tau2par13

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

tau2par23

function for maping Kendall's tau at the (2,3|1) bivariate margin to the conditional copula parameter

NEP

the number of non-evaluable positives in the presence of non-evaluable subjects

NEN

the number of non-evaluable negatives in the presence of non-evaluable subjects

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. (2020) An extended trivariate vine copula mixed model for meta-analysis of diagnostic studies in the presence of non-evaluable outcomes. The International Journal of Biostatistics, 16(2). doi:10.1515/ijb-2019-0107.

Examples


nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid=meshgrid(gl$n,gl$n,gl$n,nargout=3)

data(OGT)
attach(OGT)
out=tVineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,
qcondbvn,qcondbvn,tau2par.bvn,tau2par.bvn)
detach(OGT)
############################################
# In the precence of non-evaluable results #
data(coronary)
attach(coronary)
out=tVineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,
qcondbvn,qcondbvn,tau2par.bvn,tau2par.bvn,NEP,NEN)
detach(coronary)

The rheumatoid arthritis data

Description

Data obtained from a meta-analysis that aimed to determine whether anti-cyclic citrullinated peptide (anti-CCP) antibody identifies more accurately patients with rheumatoid arthritis than rheumatoid factor (RF) does. We include N=22 studies that assessed both RF and anti-CCP2 antibody for diagnosing rheumatoid arthritis.

Format

A data frame with 22 observations on the following 8 variables.

TP1: the number of true positives for RF
FN1: the number of false negatives for RF
FP1: the number of false positives for RF
TN1: the number of true negatives for RF
TP2: the number of true positives for anti-CCP2
FN2: the number of false negatives for anti-CCP2
FP2: the number of false positives for anti-CCP2
TN2: the number of true negatives for anti-CCP2

References

Nishimura, K., Sugiyama, D., Kogata, Y., et al. (2007) Meta-analysis: Diagnostic accuracy of anti-cyclic citrullinated peptide antibody and rheumatoid factor for rheumatoid arthritis. Annals of Internal Medicine, 146(11), 797–808.

Dimou, N.L., Adam, M. and Bagos, P.G. (2016) A multivariate method for meta-analysis and comparison of diagnostic tests. Statistics in Medicine, 35(20), 3509–3523.

The beta-D-Glucan-data

Description

Data on 8 cohort studies inthemeta-analysis in Karageorgopoulos et al. (2011). The interest there is to assess beta-D-Glucan as aserum or plasma marker for the presence of invasive fungal infections.

Usage

data(betaDG)

Format

A data frame with 8 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

Karageorgopoulos, D.E., Vouloumanou, E.K., Ntziora, F., et al. (2011) beta-D-Glucan assay for the diagnosis of invasive fungal infections: a meta-analysis. Clinical Infectious Diseases, 52(6), 750–770.

The coronary CT angiography data

Description

Data on 26 studies from a systematic review for diagnostic accuracy studies of coronary computed tomography angiography for the detection of coronary artery disease.

Usage

data(coronary)

Format

A data frame with 26 observations on the following 6 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives
NEP: the number of non-evaluable positives
NEN: the number of non-evaluable negatives

References

Schuetz, G. M., Schlattmann, P., and Dewey, M. (2012). Use of 3x2 tables with an intention to diagnose approach to assess clinical performance of diagnostic tests: Meta-analytical evaluation of coronary CT angiography studies. BMJ (Online), 345:e6717.

Simulation from a trivariate C-vine copula

Description

Simulation from a trivariate C-vine copula

Usage

cvinesim(N,param,qcondcop12,qcondcop13,qcondcop23,
                  tau2par12,tau2par13,tau2par23)

Arguments

N

sample size

param

Kendall's tau values for margins (1,2), (1,3), (23|1)

qcondcop12

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

qcondcop13

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

qcondcop23

function for the inverse of conditional copula cdf at the (2,3|1) bivariate margin

tau2par12

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

tau2par13

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

tau2par23

function for maping Kendall's tau at the (2,3|1) bivariate margin to the conditional copula parameter

Details

Choices are 'cop' in rcop are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for dcop for the abbreviations of the copula names.

Value

Nx3 matrix with values in (0,1)

References

Joe H (2011) Dependence comparisons of vine copulae with four or more variables. In: Kurowicka D, Joe H, editors. Dependence Modeling: Handbook on Vine Copulae. Singapore: World Scientific; 2011. p. 139–164

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Bivariate copula densities

Description

Bivariate copula densities for parametric families.

Usage

dbvn(u,v,cpar)
dfrk(u,v,cpar)
dcln(u,v,cpar)
dcln90(u,v,cpar)
dcln270(u,v,cpar)

Arguments

u

value in interval 0,1; could be a vector

v

value in interval 0,1; could be a vector

cpar

copula parameter: scalar.

Details

Choices are 'cop' in dcop are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

The copula names are abbreviations for:

bvn = bivariate normal or Gaussian

frk = Frank

cln = Clayton or Mardia-Takahasi-Cook-Johnson

Value

pdf value(s).

References

Joe H (1997) Multivariate Models and Dependence Concepts. Chapman & Hall

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Simulation from a six-variate 1-truncated D-vine copula

Description

Simulation from a six-variate 1-truncated D-vine copula.

Usage

dvine6dsim(nsim,tau,qcond,tau2par)

Arguments

nsim

sample size

tau

Kendall's tau values

qcond

function for the inverse conditional copula cdf

tau2par

function for maping Kendall's taus to copula parameters

Details

Choices are 'cop' in rcop are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for qcondcop for the abbreviations of the copula names.

Value

Nx6 matrix with values in (0,1)

References

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Simulation from a (truncated) quadrivariate D-vine copula

Description

Simulation from a (truncated) quadrivariate D-vine copula. Lower-order trees (if any) are composed with BVN copulas.

Usage

dvinesim(nsim,param,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)
dtrvinesim(nsim,trparam,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)

Arguments

nsim

sample size

param

Kendall's tau values for margins (1,2), (2,3), (3,4), (1,3|2), (2,4|3), (1,4|23)

trparam

Kendall's tau values for margins (1,2), (2,3), (3,4)

qcond1

function for the inverse conditional copula cdf at the (1,2) and (3,4) bivariate margins

pcond1

function for the conditional copula cdf at the (1,2) and and (3,4) bivariate margins

tau2par1

function for maping Kendall's tau at the (1,2) and (3,4) bivariate margins to copula parameter

qcond2

function for the inverse conditional copula cdf at the (2,3) bivariate margin

pcond2

function for the conditional copula cdf at the (2,3) bivariate margin

tau2par2

function for maping Kendall's tau at the (2,3) bivariate margin to copula parameter

Details

Choices are 'cop' in rcop are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for dcop for the abbreviations of the copula names.

Value

Nx4 matrix with values in (0,1)

References

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Nikoloulopoulos, A.K. (2018) A D-vine copula mixed model for joint meta-analysis and comparison of diagnostic tests. Statistical Methods in Medical Research, in press. doi:10.1177/0962280218796685.

Nikoloulopoulos, A.K. (2018) A multinomial quadrivariate D-vine copula mixed model for diagnostic studies meta-analysis accounting for non-evaluable subjects. ArXiv e-prints, arXiv:1812.05915. https://arxiv.org/abs/1812.05915.

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

Description

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

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

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.

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)

Maximum likelihood estimation of 5-variate 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Usage

imperfect.fivevariateVineCopulaREMADA.norm.comprehensive(y11,y10,y01,y00,
                               gl,mgrid,qcond,tau2par,start)
                               
imperfect.fivevariateVineCopulaREMADA.beta.comprehensive(y11,y10,y01,y00,
                               gl,mgrid,qcond,tau2par,start)

Arguments

y11

the number of the test results where the index test outcome is positive and the reference test outcome is positive

y10

the number of the test results where the index test outcome is positive and the reference test outcome is negative

y01

the number of the test results where the index test outcome is negative and the reference test outcome is positive

y00

the number of the test results where the index test outcome is negative and the reference test outcome is negative

gl

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

mgrid

a list containing five-dimensional arrays. Replicates of the quadrature points that produce a 5-dimensional full grid

qcond

function for the inverse of conditional copula cdf; choices are qconbvn and qcondfrk

tau2par

function for maping Kendall's tau to copula parameter; choices are tau2par.bvn and tau2par.frk

start

starting values for the parameters

Value

A list containing the following components:

LogLikelihood

the maximized log-likelihood

Estimates

the MLE

SE

the standard errors

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples


data(Pap)
attach(Pap)

nq=15
gl=gauss.quad.prob(nq,"uniform")
data(mgrid5d15)
mgrid=mgrid5d15

tau2par=tau2par.bvn
qcond=qcondbvn

start=c(rep(0.6,5),rep(0.5,5),rep(0.01,4))
est.norm=imperfect.fivevariateVineCopulaREMADA.norm.comprehensive(y11,y10,y01,
y00,gl,mgrid,qcond,tau2par,start)

detach(Pap)

Maximum likelihood estimation of quadrivariate 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Usage

imperfect.quadrivariateVineCopulaREMADA.norm.comprehensive(y11,y10,y01,y00,
gl,mgrid,qcond,tau2par,select.random,start)
                               
imperfect.quadrivariateVineCopulaREMADA.beta.comprehensive(y11,y10,y01,y00,
gl,mgrid,qcond,tau2par,select.random,start)

Arguments

y11

the number of the test results where the index test outcome is positive and the reference test outcome is positive

y10

the number of the test results where the index test outcome is positive and the reference test outcome is negative

y01

the number of the test results where the index test outcome is negative and the reference test outcome is positive

y00

the number of the test results where the index test outcome is negative and the reference test outcome is negative

gl

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

select.random

vector (t_{1},t_{2},t_3,t_4), where 1\leq t_1<t_2<t_3<t_4\leq 5, that indicates the random effects

mgrid

a list containing four-dimensional arrays. Replicates of the quadrature points that produce a 4-dimensional full grid

qcond

function for the inverse of conditional copula cdf; choices are qconbvn and qcondfrk

tau2par

function for maping Kendall's tau to copula parameter; choices are tau2par.bvn and tau2par.frk

start

starting values for the parameters

Value

A list containing the following components:

LogLikelihood

the maximized log-likelihood

Estimates

the MLE

SE

the standard errors

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples

data(Pap)
attach(Pap)

nq=30
gl=gauss.quad.prob(nq,"uniform")
data(mgrid30)
mgrid=mgrid30

tau2par=tau2par.bvn
qcond=qcondbvn

select.random=1:4
start=c(rep(0.6,5),rep(0.5,4),c(0.1,-0.1,0.1))
est.norm=imperfect.quadrivariateVineCopulaREMADA.norm.comprehensive(y11,y10,y01,
y00,gl,mgrid,qcond,tau2par,select.random,start)

detach(Pap)

Maximum likelihood estimation of trivariate 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Usage

imperfect.trivariateVineCopulaREMADA.norm.comprehensive(y11,
y10,y01,y00,gl,mgrid,qcond,tau2par,select.random,start)
                               
imperfect.trivariateVineCopulaREMADA.norm(y11,
y10,y01,y00,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start)

imperfect.trivariateVineCopulaREMADA.beta.comprehensive(y11,
y10,y01,y00,gl,mgrid,qcond,tau2par,select.random,start)
                               
imperfect.trivariateVineCopulaREMADA.beta(y11,
y10,y01,y00,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start)

Arguments

y11

the number of the test results where the index test outcome is positive and the reference test outcome is positive

y10

the number of the test results where the index test outcome is positive and the reference test outcome is negative

y01

the number of the test results where the index test outcome is negative and the reference test outcome is positive

y00

the number of the test results where the index test outcome is negative and the reference test outcome is negative

gl

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

mgrid

a list containing three-dimensional arrays. Replicates of the quadrature points that produce a 3-dimensional full grid

select.random

vector (t_{1},t_{2},t_3), where 1\leq t_1<t_2<t_3\leq 5, that indicates the random effects

qcond

function for the inverse of conditional copula cdf; choices are qconbvn and qcondfrk

tau2par

function for maping Kendall's tau to copula parameter; choices are tau2par.bvn and tau2par.frk

qcond1

function for the inverse of conditional copula cdf for the (t_{1},t_{2}) bivariate margin; choices are qcondcln and qcondcln180

tau2par1

function for maping Kendall's tau to copula parameter for the (t_{1},t_{2}) bivariate margin; choices are tau2par.cln and tau2par.cln180

qcond2

function for the inverse of conditional copula cdf for the (t_{2},t_{3}) bivariate margin; choices are qcondcln90 and qcondcln270

tau2par2

function for maping Kendall's tau to copula parameter for the (t_{2},t_{3}) bivariate margin; choices are tau2par.cln90 and tau2par.cln270

start

starting values for the parameters

Value

A list containing the following components:

LogLikelihood

the maximized log-likelihood

Estimates

the MLE

SE

the standard errors

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples


data(Pap)
attach(Pap)

nq=30
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n,gl$n,nargout=3)

tau2par=tau2par.bvn
qcond=qcondbvn

select.random=c(1,2,4)
start=c(rep(0.6,5),rep(0.5,3),c(0.01,-0.01))
est.norm=imperfect.trivariateVineCopulaREMADA.norm.comprehensive(y11,y10,
y01,y00,gl,mgrid,qcond,tau2par,select.random,start)

tau2par1=tau2par.cln180
qcond1=qcondcln180
tau2par2=tau2par.cln270
qcond2=qcondcln270
est.norm.cln=imperfect.trivariateVineCopulaREMADA.norm(y11,y10,y01,
y00,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start)

start=c(rep(0.6,5),rep(0.05,3),c(0.1,-0.1))
est.beta=imperfect.trivariateVineCopulaREMADA.beta.comprehensive(y11,y10,y01,y00,
gl,mgrid,qcond,tau2par,select.random,start)

est.beta.cln=imperfect.trivariateVineCopulaREMADA.beta(y11,y10,y01,y00,
gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start)

detach(Pap)

Maximum likelihood estimation of bivariate copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Usage

imperfectCopulaREMADA.norm(y11,y10,y01,y00,
gl,mgrid,qcond,tau2par,select.random,start)
                               
imperfectCopulaREMADA.beta(y11,y10,y01,y00,
gl,mgrid,qcond,tau2par,select.random,start)

Arguments

y11

the number of the test results where the index test outcome is positive and the reference test outcome is positive

y10

the number of the test results where the index test outcome is positive and the reference test outcome is negative

y01

the number of the test results where the index test outcome is negative and the reference test outcome is positive

y00

the number of the test results where the index test outcome is negative and the reference test outcome is negative

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

select.random

vector (t_{1},t_{2}), where 1\leq t_1<t_2\leq 5, that indicates the random effects

qcond

function for the inverse of conditional copula cdf

tau2par

function for maping Kendall's tau to copula parameter

start

starting values for the parameters

Value

A list containing the following components:

LogLikelihood

the maximized log-likelihood

Estimates

the MLE

SE

the standard errors

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples

data(Pap)
attach(Pap)

nq=30
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n)

tau2par=tau2par.bvn
qcond=qcondbvn

select.random=c(1,2)

start=c(rep(0.6,5),rep(0.5,2),-0.1)
est.norm=imperfectCopulaREMADA.norm(y11,y10,y01,y00,gl,mgrid,
qcond,tau2par,select.random,start)

detach(Pap)

Maximum likelihood estimation of univariate mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Usage

imperfectREMADA.norm(y11,y10,y01,y00,gl,select.random,start)
                               
imperfectREMADA.beta(y11,y10,y01,y00,gl,select.random,start)

Arguments

y11

the number of the test results where the index test outcome is positive and the reference test outcome is positive

y10

the number of the test results where the index test outcome is positive and the reference test outcome is negative

y01

the number of the test results where the index test outcome is negative and the reference test outcome is positive

y00

the number of the test results where the index test outcome is negative and the reference test outcome is negative

gl

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

select.random

a scalar from 1 to 5 that indicates the random effect

start

starting values for the parameters

Value

A list containing the following components:

LogLikelihood

the maximized log-likelihood

Estimates

the MLE

SE

the standard errors

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples

data(Pap)
attach(Pap)

nq=30
gl=gauss.quad.prob(nq,"uniform")
start=c(rep(0.6,5),0.5)
select.random=1 
est.norm=imperfectREMADA.norm(y11,y10,y01,y00,gl,
select.random,start)
start=c(rep(0.6,5),0.1)
est.beta=imperfectREMADA.beta(y11,y10,y01,y00,gl,
select.random,start)

detach(Pap)

A list containing four-dimensional arrays

Description

A list containing four-dimensional arrays. Replicates of the quadrature points that produce a 4-dimensional full grid.

Examples

data(mgrid15)
dim(mgrid15$x)
dim(mgrid15$y)
dim(mgrid15$z)
dim(mgrid15$w)

data(mgrid30)
dim(mgrid30$x)
dim(mgrid30$y)
dim(mgrid30$z)
dim(mgrid30$w)

data(mgrid50)
dim(mgrid50$x)
dim(mgrid50$y)
dim(mgrid50$z)
dim(mgrid50$w)

A list containing five-dimensional arrays

Description

A list containing five-dimensional arrays. Replicates of the quadrature points that produce a 5-dimensional full grid.

Examples

data(mgrid5d15)
dim(mgrid5d15$x1)
dim(mgrid5d15$x2)
dim(mgrid5d15$x3)
dim(mgrid5d15$x4)
dim(mgrid5d15$x5)

data(mgrid5d30)
dim(mgrid5d30$x1)
dim(mgrid5d30$x2)
dim(mgrid5d30$x3)
dim(mgrid5d30$x4)
dim(mgrid5d30$x5)

A list containing six-dimensional arrays

Description

A list containing six-dimensional arrays. Replicates of the quadrature points that produce an 6-dimensional full grid.

Examples

data(mgrid6d)
dim(mgrid$x1)
dim(mgrid$x2)
dim(mgrid$x3)
dim(mgrid$x4)
dim(mgrid$x5)
dim(mgrid$x6)

Maximum likelhood estimation for multinomial six-variate 1-truncated D-vine copula mixed models for meta-analysis of two diagnostic tests accounting for within and between studies dependence

Description

Usage

multinom6dVineCopulaREMADA.norm(y001,y011,y101,y111,y000,y010,y100,y110,
          gl,mgrid,qcond1,qcond2,qcond3,qcond4,qcond5,
          tau2par1,tau2par2,tau2par3,tau2par4,tau2par5,
          sel1,sel2,sel3)
multinom6dVineCopulaREMADA.beta(y001,y011,y101,y111,y000,y010,y100,y110,
          gl,mgrid,qcond1,qcond2,qcond3,qcond4,qcond5,
          tau2par1,tau2par2,tau2par3,tau2par4,tau2par5,
          sel1,sel2,sel3)

Arguments

y001

the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is negative

y011

the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is positive

y101

the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is negative

y111

the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is positive

y000

the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is negative

y010

the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is positive

y100

the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is negative

y110

the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is positive

gl

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

mgrid

a list containing six-dimensional arrays. Replicates of the quadrature points that produce a 6-dimensional full grid

qcond1

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

qcond2

function for the inverse conditional copula cdf at the (2,3) bivariate margin

qcond3

function for the inverse conditional copula cdf at the (3,4) bivariate margin

qcond4

function for the inverse conditional copula cdf at the (4,5) bivariate margin

qcond5

function for the inverse conditional copula cdf at the (5,6) bivariate margin

tau2par1

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

tau2par2

function for maping Kendall's tau at the (2,3) bivariate margin to copula parameter

tau2par3

function for maping Kendall's tau at the (3,4) bivariate margin to copula parameter

tau2par4

function for maping Kendall's tau at the (4,5) bivariate margin to copula parameter

tau2par5

function for maping Kendall's tau at the (5,6) bivariate margin to copula parameter

sel1

Indicates the position of bivariate copulas with positive dependence only such as the Clayton and the Clayton rotated by 180 degrees

sel2

Indicates the position of bivariate copulas with negative dependence only such as the Clayton rotated by 90 degrees and the Clayton rotated by 270 degrees

sel3

Indicates the position of bivariate copulas with comprehensive dependence such as the BVN and Frank copulas

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

Examples


data(Down)
attach(Down)
y111=down_n_11 
y110=nodown_n_11
y101=down_n_10
y100=nodown_n_10 
y001=down_n_00 
y000=nodown_n_00 
y010=nodown_n_01
y011=down_n_01

nq=15
gl=gauss.quad.prob(nq,"uniform")
data(mgrid6d)

tau2par1=tau2par.cln90
qcond1=qcondcln90
tau2par3=tau2par4=tau2par5=tau2par.cln
qcond3=qcond4=qcond5=qcondcln
tau2par2=tau2par.bvn
qcond2=qcondbvn

sel1=3:5; sel2=1; sel3=2

est=multinom6dVineCopulaREMADA.norm(y001,y011,y101,y111,
y000,y010,y100,y110,gl,mgrid,qcond1,qcond2,qcond3,qcond4,qcond5,
tau2par1,tau2par2,tau2par3,tau2par4,tau2par5,sel1,sel2,sel3)

detach(Down)

Maximum likelhood estimation for multinomial quadrivariate (truncated) D-vine copula mixed models for diagnostic test accurracy studies accounting for non-evaluable outcomes

Description

Usage

multinomVineCopulaREMADA.norm(TP,FN,FP,TN,NEP,NEN,
                                 gl,mgrid,qcond1,pcond1,tau2par1,
                                 qcond2,pcond2,tau2par2)
multinomVineCopulaREMADA.beta(TP,FN,FP,TN,NEP,NEN,
                                 gl,mgrid,qcond1,pcond1,tau2par1,
                                 qcond2,pcond2,tau2par2)
tmultinomVineCopulaREMADA.norm(TP,FN,FP,TN,NEP,NEN,
                                 gl,mgrid,
                                 qcond1,pcond1,tau2par1,
                                 qcond2,pcond2,tau2par2)
tmultinomVineCopulaREMADA.beta(TP,FN,FP,TN,NEP,NEN,
                                 gl,mgrid,
                                 qcond1,pcond1,tau2par1,
                                 qcond2,pcond2,tau2par2)

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

NEP

the number of non-evaluable positives in the presence of non-evaluable subjects

NEN

the number of non-evaluable negatives in the presence of non-evaluable subjects

gl

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

mgrid

a list containing 4-dimensional arrays.

qcond1

function for the inverse conditional copula cdf at the (1,2) and (3,4) bivariate margin

pcond1

function for the conditional copula cdf at the (1,2) and (3,4) bivariate margin

tau2par1

function for maping Kendall's tau at the (1,2) and (3,4) bivariate margin to copula parameter

qcond2

function for the inverse conditional copula cdf at the (2,3) bivariate margin

pcond2

function for the conditional copula cdf at the (2,3) bivariate margin

tau2par2

function for maping Kendall's tau at the (2,3) bivariate margin to 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. (2020) A multinomial quadrivariate D-vine copula mixed model for diagnostic studies meta-analysis in the presence of non-evaluable subjects. Statistical Methods in Medical Research, 29 (10), 2988–3005. doi:10.1177/0962280220913898.

Examples


nq=15
gl=gauss.quad.prob(nq,"uniform")
data(mgrid15)

data(MK2016)
attach(MK2016)

out=tmultinomVineCopulaREMADA.beta(TP,FN,FP,TN,NEP,NEN,
gl,mgrid15,qcondcln180,pcondcln180,tau2par.cln180,
qcondcln90,pcondcln90,tau2par.cln90)

detach(MK2016)

Bivariate copula conditional distribution functions

Description

Bivariate copula conditional distribution functions

Usage

pcondbvn(v,u,cpar)
pcondfrk(v,u,cpar)
pcondcln(v,u,cpar)
pcondcln90(v,u,cpar)
pcondcln270(v,u,cpar)

Arguments

v

conditioning value in interval 0,1; could be a vector

u

value in interval 0,1; could be a vector

cpar

copula parameter: scalar.

Details

Choices appending 'cop' are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for dcop for the abbreviations of the copula names.

Value

inverse conditional cdf value(s)

References

Joe H (1997) Multivariate Models and Dependence Concepts. Chapman & Hall

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Bivariate copula conditional quantile functions

Description

Bivariate copula conditional quantile functions

Usage

qcondbvn(p,u,cpar)
qcondfrk(p,u,cpar)
qcondcln(p,u,cpar)
qcondcln90(p,u,cpar)
qcondcln270(p,u,cpar)

Arguments

u

conditioning value in interval 0,1; could be a vector

p

quantile in interval 0,1; could be a vector

cpar

copula parameter: scalar.

Details

Choices appending 'cop' are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for dcop for the abbreviations of the copula names.

Value

inverse conditional cdf value(s)

References

Joe H (1997) Multivariate Models and Dependence Concepts. Chapman & Hall

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Maximum likelihood estimation of quadrivariate D-vine copula mixed models for joint meta-analysis and comparison of two diagnostic tests

Description

Usage

quadVineCopulaREMADA.norm(TP1,FN1,FP1,TN1,TP2,FN2,FP2,TN2,
                               gl,mgrid,qcond1,pcond1,tau2par1,
                               qcond2,pcond2,tau2par2)
                               
quadVineCopulaREMADA.beta(TP1,FN1,FP1,TN1,TP2,FN2,FP2,TN2,
                               gl,mgrid,qcond1,pcond1,tau2par1,
                               qcond2,pcond2,tau2par2) 
quadVineCopulaREMADA.norm.beta(TP1,FN1,FP1,TN1,TP2,FN2,FP2,TN2,
                               gl,mgrid,qcond1,pcond1,tau2par1,
                               qcond2,pcond2,tau2par2)

Arguments

TP1

the number of true positives for test 1

FN1

the number of false negatives for test 1

FP1

the number of false positives for test 1

TN1

the number of true negatives for test 1

TP2

the number of true positives for test 2

FN2

the number of false negatives for test 2

FP2

the number of false positives for test 2

TN2

the number of true negatives for test 2

gl

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

mgrid

a list containing four-dimensional arrays. Replicates of the quadrature points that produce a 4-dimensional full grid

qcond1

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

pcond1

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

tau2par1

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

qcond2

function for the inverse conditional copula cdf at the (3,4) bivariate margin

pcond2

function for the conditional copula cdf at the (3,4) bivariate margin

tau2par2

function for maping Kendall's tau at the (3,4) bivariate margin to 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

Examples


nq=15
gl=gauss.quad.prob(nq,"uniform")
data(mgrid15)

data(arthritis)
attach(arthritis)

qcond1=qcondcln270
pcond1=pcondcln270
tau2par1=tau2par.cln270

qcond2=qcondfrk
pcond2=pcondfrk
tau2par2=tau2par.frk

out<-quadVineCopulaREMADA.norm(TP1,FN1,FP1,TN1,TP2,FN2,FP2,TN2,
gl,mgrid15,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)

detach(arthritis)

Simulation from copula mixed models for diagnostic test accuaracy studies

Description

To simulate the data we have used the following steps:

1. Simulate the study size n from a shifted gamma distribution with parameters \alpha=1.2,\beta=0.01,lag=30 and round off to the nearest integer.

2. Simulate (u_1,u_2) from a parametric family of copulas 'cop'.

3. Convert to beta realizations or normal realizations.

4. Draw the number of diseased n_1 from a B(n,0.43) distribution.

5. Set n_2=n-n_1, y_j=n_jx_j and then round y_j for j=1,2.

Usage

rCopulaREMADA.norm(N,p,si,tau,rcop,tau2par)
rCopulaREMADA.beta(N,p,g,tau,rcop,tau2par)

Arguments

N

sample size

p

Pair (\pi_1,\pi_2) of sensitivity/specificity

si

Pair (\sigma_1,\sigma_2) of variability; normal margins

g

Pair (\gamma_1,\gamma_2) of variability; beta margins

tau

Kendall's tau value

rcop

function for copula generation

tau2par

function for mapping from Kendall's tau to copula parameter

Value

A list containing the following simulated components:

TP

the number of true positives

FN

the number of false negatives

FP

the number of false positives

TN

the number of true negatives

References

Examples

nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n)

N=20
tau=-0.5
p=c(0.7,0.9)
g=c(0.2,0.1)
simDat=rCopulaREMADA.beta(N,p,g,tau,rcln270,tau2par.cln270)
TP=simDat$TP
TN=simDat$TN
FP=simDat$FP
FN=simDat$FN
c270est.b=CopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcondcln270,tau2par.cln270)

si=c(2,1)
tau=0.5
simDat=rCopulaREMADA.norm(N,p,si,tau,rcln,tau2par.cln)
TP=simDat$TP
TN=simDat$TN
FP=simDat$FP
FN=simDat$FN
cest.n=CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcondcln,tau2par.cln)

Simulation from 1-factor copula mixed models for joint meta-analysis of `T` diagnostic tests

Description

Simulation from 1-factor copula mixed models for joint meta-analysis of T diagnostic tests

Usage

rFactorCopulaREMADA.norm(N,p,si,taus,qcond1,tau2par1,qcond2,tau2par2)
rFactorCopulaREMADA.beta(N,p,g,taus,qcond1,tau2par1,qcond2,tau2par2)

Arguments

N

number of studies

p

vector of sensitivities and specificities

si

vector of variabilities; normal margins

g

vector of variabilities; beta margins

taus

Kendall's tau values

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 with the simulated data in matrices with T columns and N rows.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

Examples

N=50

qcond1=qcondcln
tau2par1=tau2par.cln
qcond2=qcondcln270
tau2par2=tau2par.cln270

p=c(0.8,0.7,0.8,0.7,0.8,0.7)
mu=log(p/(1-p))
si=rep(1,6)
taus=c(0.6,0.7,0.5,-0.3,-0.4,-0.2)

out=rFactorCopulaREMADA.norm(N,p,si,taus,qcond1,tau2par1,qcond2,tau2par2)
  
TP=out$TP
FN=out$FN
TN=out$TN
FP=out$FP

Simulation from trivariate vine copula mixed models for diagnostic test accuaracy studies accounting for disease prevalence and non-evaluable results

Description

Simulation from trivariate vine copula mixed models for diagnostic test accuaracy studies accounting for disease prevalence and non-evaluable results

Usage

rVineCopulaREMADA.beta(N,p,g,taus,omega1,omega0,qcondcop12,qcondcop13,
qcondcop23,tau2par12,tau2par13,tau2par23)
rVineCopulaREMADA.norm(N,p,si,taus,omega1,omega0,qcondcop12,qcondcop13,
qcondcop23,tau2par12,tau2par13,tau2par23)

Arguments

N

sample size

p

Vector (\pi_1,\pi_2,\pi_3) of sensitivity/specificity/prevalence

si

Vector (\sigma_1,\sigma_2,\sigma_3) of variability; normal margins

g

Vector (\gamma_1,\gamma_2,\gamma_3) of variability; beta margins

taus

Kendall's tau values

omega1

the probability for non-evaluable positives

omega0

the probability for non-evaluable negatives

qcondcop12

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

qcondcop13

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

qcondcop23

function for the inverse of conditional copula cdf at the (2,3|1) bivariate margin

tau2par12

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

tau2par13

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

tau2par23

function for maping Kendall's tau at the (2,3|1) bivariate margin to the conditional copula parameter

Value

Simuated data with 6 columns and N rows.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives
NEP: the number of non-evaluable positives
NEN: the number of non-evaluable negatives

References

Nikoloulopoulos, A.K. (2018) A vine copula mixed model for trivariate meta-analysis of diagnostic studies accounting for disease prevalence and non-evaluable subjects. ArXiv e-prints, arXiv:1812.03685. https://arxiv.org/abs/1812.03685.

Examples

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
# in the absence of non-evaluable results
omega1=0
omega0=0
rVineCopulaREMADA.beta(50,p,g,taus,omega1,omega0,
qcondcop12,qcondcop13,qcondcop23,tau2par12,
tau2par13,tau2par23)
# in the presence of non-evaluable results
omega1=0.1
omega0=0.2
rVineCopulaREMADA.beta(50,p,g,taus,omega1,omega0,
qcondcop12,qcondcop13,qcondcop23,tau2par12,
tau2par13,tau2par23)

Simulation from parametric bivariate copula families

Description

Simulation from parametric bivariate copula families

Usage

rbvn(N,cpar)
rfrk(N,cpar)
rcln(N,cpar)
rcln90(N,cpar)
rcln270(N,cpar)

Arguments

N

sample size

cpar

copula parameter: scalar

Details

Choices are 'cop' in rcop are bvn, frk, cln, cln90 (rotated by 90 degrees cln), cln180 (rotated by 180 degrees cln), cln270 (rotated by 270 degrees cln).

See help page for dcop for the abbreviations of the copula names.

Value

nx2 matrix with values in (0,1)

References

Joe H (1997) Multivariate Models and Dependence Concepts. Chapman & Hall

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

Simulation from trivariate 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Description

Simulation from trivariate 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard

Usage

rimperfect.trivariateVineCopulaREMADA.norm(N,p,si,taus,select.random,qcond1,
tau2par1,qcond2,tau2par2)
rimperfect.trivariateVineCopulaREMADA.beta(N,p,g,taus,select.random,qcond1,
tau2par1,qcond2,tau2par2)

Arguments

N

sample size

p

Vector (\pi_{1},\pi_{2},\pi_{3},\pi_{4},\pi_{5}), where \pi_1 is the meta-analytic parameter for the prevalence, \pi_2 and \pi_3 are the meta-analytic parameters for the sensitivity of the index and the reference test, respectively, and \pi_4 and \pi_5 are the meta-analytic parameters for the specificity of the index and the reference test, respectively.

si

Vector (\sigma_{1},\sigma_{2},\sigma_{3}), where \sigma_t,\,t=1,\ldots,3 denote the between-study heterogeneities (normal margins)

g

Vector (\gamma_{1},\gamma_{2},\gamma_{3}) where\gamma_t,\,t=1,\ldots,3 denote the between-study heterogeneities (beta margins)

taus

Kendall's tau values

select.random

vector (t_{1},t_{2},t_3), where 1\leq t_1<t_2<t_3\leq 5

qcond1

function for the inverse of conditional copula cdf for the (t_{1},t_{2}) bivariate margin; choices are qcondbvn, qcondfrk, qcondcln, qcondcln90, qcondcln180 and qcondcln270

tau2par1

function for maping Kendall's tau to copula parameter for the (t_{1},t_{2}) bivariate margin; choices are tau2par.bvn, tau2par.frk, tau2par.cln, tau2par.cln90, tau2par.cln180 and tau2par.cln270

qcond2

function for the inverse of conditional copula cdf for the (t_{2},t_{3}) bivariate margin; choices are qcondbvn, qcondfrk, qcondcln, qcondcln90, qcondcln180 and qcondcln270

tau2par2

function for maping Kendall's tau to copula parameter for the (t_{2},t_{3}) bivariate margin; choices are tau2par.bvn, tau2par.frk, tau2par.cln, tau2par.cln90, tau2par.cln180 and tau2par.cln270

Value

Simulated data with 4 columns and N rows.

y11: the number of the test results where the index test outcome is positive and the reference test outcome is positive
y10: the number of the test results where the index test outcome is positive and the reference test outcome is negative
y01: the number of the test results where the index test outcome is negative and the reference test outcome is positive
y00: the number of the test results where the index test outcome is negative and the reference test outcome is negative

References

Nikoloulopoulos, A.K. (2025) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Biometrics, 81(2), ujaf037. doi:10.1093/biomtc/ujaf037.

Examples

N=59
p=c(0.631,0.653,0.902,0.843,0.987)
si=c(1.513,1.341,1.341)
taus=c(0.3,-0.3)
select.random=c(1,2,4)

out=rimperfect.trivariateVineCopulaREMADA.norm(N,p,si,taus,select.random,
qcondcln180,tau2par.cln180,qcondcln270,tau2par.cln270)

g=c(0.290,0.244,0.190)
out=rimperfect.trivariateVineCopulaREMADA.beta(N,p,g,taus,select.random,
qcondcln180,tau2par.cln180,qcondcln270,tau2par.cln270)

Simulation from multinomial six-variate 1-truncated D-vine copula mixed models for meta-analysis of two diagnostic tests accounting for within and between studies dependence

Description

Simulation from multinomial six-variate 1-truncated D-vine copula mixed models for meta-analysis of two diagnostic tests accounting for within and between studies dependence

Usage

rmultinom6dVineCopulaREMADA.norm(N,p,si,taus,qcond,tau2par)
rmultinom6dVineCopulaREMADA.beta(N,p,g,taus,qcond,tau2par)

Arguments

N

sample size

p

Vector (\pi_{101},\pi_{011},\pi_{111},\pi_{100},\pi_{010},\pi_{110}) of the meta-analytic parameters of interest for each combination of test results in diseased and non-diseased participants in a 4\times 2 table

si

Vector (\sigma_{101},\sigma_{011},\sigma_{111},\sigma_{100},\sigma_{010},\sigma_{110}) of variability parameters; normal margins

g

Vector (\gamma_{101},\gamma_{011},\gamma_{111},\gamma_{100},\gamma_{010},\gamma_{110}) of variability parameters; beta margins

taus

Kendall's tau values

qcond

function for the inverse conditional copula cdf

tau2par

function for maping Kendall's taus to copula parameters

Value

Simulated data with 8 columns and N rows.

y001: the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is negative
y011: the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is positive
y101: the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is negative
y111: the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is positive
y000: the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is negative
y010: the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is positive
y100: the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is negative
y110: the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is positive

References

Examples

N=11
p=c(0.03667409,  0.09299767,  0.29450436,  0.01733081,  0.04923809,  0.02984361)
si=c(1.69868880, 0.54292079,  0.58489574,  0.92918177,  0.48998484,  0.57004098)
taus=c(-0.52475006,  0.55768873, 0.18454559,  0.02233204,  0.57570506)


tau2par=tau2par.bvn
qcond=qcondbvn

out=rmultinom6dVineCopulaREMADA.norm(N,p,si,taus,qcond,tau2par)
 
  
y101=out[,1]
y011=out[,2]
y111=out[,3]
y001=out[,4]
y100=out[,5]
y010=out[,6]
y110=out[,7]
y000=out[,8]

Simulation from multinomial quadrivariate (truncated) D-vine copula mixed models for diagnostic test accurracy studies accounting for non-evaluable outcomes

Description

Simulation from multinomial quadrivariate (truncated) D-vine copula mixed models for diagnostic test accurracy studies accounting for non-evaluable outcomes

Usage

rmultinomVineCopulaREMADA.norm(N,p,si,taus,qcond1,
                                    pcond1,tau2par1,qcond2,
                                    pcond2,tau2par2)
rmultinomVineCopulaREMADA.beta(N,p,g,taus,qcond1,
                                    pcond1,tau2par1,qcond2,
                                    pcond2,tau2par2)

Arguments

N

sample size

p

Vector (\pi_1,\pi_2,\pi_3) of sensitivity/specificity/prevalence

si

Vector (\sigma_1,\sigma_2,\sigma_3) of variability; normal margins

g

Vector (\gamma_1,\gamma_2,\gamma_3) of variability; beta margins

taus

Kendall's tau values

qcond1

function for the inverse conditional copula cdf at the (1,2) and (3,4) bivariate margin

pcond1

function for the conditional copula cdf at the (1,2) and (3,4) bivariate margin

tau2par1

function for maping Kendall's tau at the (1,2) and (3,4) bivariate margin to copula parameter

qcond2

function for the inverse conditional copula cdf at the (2,3) bivariate margin

pcond2

function for the conditional copula cdf at the (2,3) bivariate margin

tau2par2

function for maping Kendall's tau at the (2,3) bivariate margin to copula parameter

Value

Simulated data with 6 columns and N rows.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives
NEP: the number of non-evaluable positives
NEN: the number of non-evaluable negatives

References

Examples

N=30
p=c(0.898745016,0.766105342,0.059168715,0.109217888)
g=c(0.090270947,0.079469009,0.367463579,0.154976269)
taus=c(	0.82050793,-0.51867629,0.26457961)

qcond1=qcondcln180
pcond1=pcondcln180
tau2par1=tau2par.cln180

qcond2=qcondcln90
pcond2=pcondcln90
tau2par2=tau2par.cln90

out=rmultinomVineCopulaREMADA.beta(N,p,g,taus,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)
  
TP=out[,1]
NEP=out[,2]
FN=out[,3]
TN=out[,4]
NEN=out[,5]
FP=out[,6]

Mapping of Kendall's tau and copula parameter

Description

Bivariate copulas: mapping of Kendall's tau and copula parameter.

Usage

tau2par.bvn(tau)
tau2par.frk(tau)
tau2par.cln(tau)
tau2par.cln90(tau)
tau2par.cln180(tau)
tau2par.cln270(tau)

Arguments

tau

Kendall's tau for the copula

Details

For abbreviations of names of copula families (after the dot in function names), see dcop help page.

Value

copula parameter

References

Joe H (1997) Multivariate Models and Dependence Concepts. Chapman & Hall

Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC.

Joe H (2014) CopulaModel: Dependence Modeling with Copulas. Software for book: Dependence Modeling with Copulas, Chapman & Hall/CRC, 2014.

The telomerase data

Description

In Glas et al. (2003) the telomerase marker for the diagnosis of bladder cancer is evaluated using 10 studies. The interest was to define if this non-invasive and cheap marker could replace the standard of cystoscopy or histopathology.

Usage

data(telomerase)

Format

A data frame with 10 observations on the following 4 variables.

TP: the number of true positives
FN: the number of false negatives
FP: the number of false positives
TN: the number of true negatives

References

Glas A.S., Roos D., Deutekom M., Zwinderman A.H., Bossuyt P.M., Kurth K.H. (2003) Tumor markers in the diagnosis of primary bladder cancer. A systematic review. Journal of Urology, 169(6), 1975–82.

Vuong's test for the comparison of non-nested vine copula mixed models for diagnostic test accuaracy studies

Description

Vuong (1989)'s test for the comparison of non-nested vine copula mixed models for diagnostic test accuaracy studies. It shows if a vine copula mixed model provides better fit than the standard GLMM. We compute the Vuong's test with Model 1 being the vine copula mixed model with BVN copula and normal margins, i.e., the standard GLMM.

Usage

vine.vuong.beta(qcondcop12,qcondcop13,qcondcop23,
tau2par12,tau2par13,tau2par23,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN)
vine.vuong.norm(qcondcop12,qcondcop13,qcondcop23,
tau2par12,tau2par13,tau2par23,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN) 
tvine.vuong.beta(qcondcop12,qcondcop13,
tau2par12,tau2par13,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN)
tvine.vuong.norm(qcondcop12,qcondcop13,
tau2par12,tau2par13,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN) 
tvine2.vuong.beta(qcondcop12,qcondcop13,
tau2par12,tau2par13,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN)
tvine2.vuong.norm(qcondcop12,qcondcop13,
tau2par12,tau2par13,param1,param2,TP,FN,FP,TN,gl,mgrid,NEP,NEN)

Arguments

qcondcop12

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

qcondcop13

function for the inverse of conditional copula cdf at the (1,3) bivariate margin for Model 2

qcondcop23

function for the inverse of conditional copula cdf at the (2,3|1) bivariate margin for Model 2

tau2par12

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

tau2par13

function for maping Kendall's tau at the (1,3) bivariate margin to copula parameter for Model 2

tau2par23

function for maping Kendall's tau at the (2,3|1) bivariate margin to the conditional copula parameter for Model 2

param1

parameters for the Model 1. i.e., the GLMM

param2

parameters for the Model 2

TP

the number of true positives

FN

the number of false negatives

FP

the number of false positives

TN

the number of true negatives

gl

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

mgrid

a list containing three-dimensional arrays

NEP

the number of non-evaluable positives in the presence of non-evaluable subjects

NEN

the number of non-evaluable negatives in the presence of non-evaluable subjects

Value

A list containing the following components:

z

the test statistic

p-value

the p-value

References

Vuong Q.H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57, 307–333.

Examples

nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid=meshgrid(gl$n,gl$n,gl$n,nargout=3)

data(betaDG)
attach(betaDG)
#nest.n2=VineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,
#qcondbvn,qcondbvn,qcondbvn,
#tau2par.bvn,tau2par.bvn,tau2par.bvn)
nest.n2.est= #nest.n2$e
c(0.87186926,  0.13696066,  0.70614956,  0.69152133,  
0.51780203,  0.70883558, -0.41354870,0.07701287, -0.12111253)
#c090est.b2=VineCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,
#qcondcln90,qcondcln,qcondcln90,tau2par.cln90,tau2par.cln,tau2par.cln90)
c090est.b2.est= #c090est.b2$e
c(0.85528463,  0.14667571,  0.68321231,  0.04897466,
0.02776290,  0.08561436, -0.34639172, 0.04621924, -0.21627977)
c090vuong.b2=vine.vuong.beta(qcondcln90,qcondcln,qcondcln90,
tau2par.cln90,tau2par.cln,tau2par.cln90,
nest.n2.est,c090est.b2.est,TP,FN,FP,TN,gl,mgrid)
c090vuong.b2
detach(betaDG)

Vuong's test for the comparison of non-nested copula mixed models for diagnostic test accuaracy studies

Description

Vuong (1989)'s test for the comparison of non-nested copula mixed models for diagnostic test accuaracy studies. It shows if a copula mixed model provides better fit than the standard GLMM. We compute the Vuong's test with Model 1 being the copula mixed model with BVN copula and normal margins, i.e., the standard GLMM.

Usage

vuong.norm(qcond,tau2par,param1,param2,TP,FN,FP,TN,gl,mgrid)
vuong.beta(qcond,tau2par,param1,param2,TP,FN,FP,TN,gl,mgrid)
countermonotonicity.vuong(param1,param2,TP,FN,FP,TN,gl,mgrid)

Arguments

qcond

function for conditional copula cdf for Model 2

tau2par

function for maping Kendall's tau to copula parameter for Model 2

param1

parameters for the Model 1. i.e., the GLMM

param2

parameters for the Model 2

TP

the number of true positives

FN

the number of false negatives

FP

the number of false positives

TN

the number of true negatives

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

Value

A list containing the following components:

z

the test statistic

p-value

the p-value

References

Vuong Q.H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57:307–333.

Examples

nq=15
gl=gauss.quad.prob(nq,"uniform")
mgrid<- meshgrid(gl$n,gl$n)

data(MRI)
attach(MRI)
c270est.b=CopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcondcln270,tau2par.cln270)
nest.n=CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcondbvn,tau2par.bvn)
c90est.n=CopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcondcln90,tau2par.cln90)
vuong.beta(qcondcln270,tau2par.cln270,nest.n$e,c270est.b$e,TP,FN,FP,TN,gl,mgrid)
vuong.norm(qcondcln90,tau2par.cln90,nest.n$e,c90est.n$e,TP,FN,FP,TN,gl,mgrid)
detach(MRI)

data(CT)
attach(CT)
est.n=countermonotonicCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid)
est.b=countermonotonicCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid)
countermonotonicity.vuong(est.n$e,est.b$e,TP,FN,FP,TN,gl,mgrid)
detach(CT)

Copula Mixed Models for Multivariate Meta-Analysis of Diagnostic Test Accuracy Studies

Description

Details

Author(s)

References

The computing tomography data

Description

Usage

Format

References

Maximum likelhood estimation for copula mixed models for diagnostic test accurracy studies

Description

Usage

Arguments

Value

References

See Also

Examples

The down syndrome data

Description

Format

References

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

Description

Usage

Arguments

Value

References

Examples

The lymphangiography data

Description

Usage

Format

References

The coronary CT angiography data in Menke and Kowalski (2016).

Description

Usage

Format

References

The magnetic resonance imaging data

Description

Usage

Format

References

The orale glucose tolerance data

Description

Usage

Format

References

The Pap test data

Description

Format

References

Summary receiver operating characteristic curves for copula mixed effect models for bivariate meta-analysis of diagnostic test accuracy studies

Description

Usage

Arguments

Value

References

See Also

Examples

Maximum likelhood estimation for (truncated) vine copula mixed models for diagnostic test accurracy studies accounting for disease prevalence and non-evaluable outcomes

Description

Usage

Arguments

Value

References

See Also

Examples

The rheumatoid arthritis data

Description

Format

References

The beta-D-Glucan-data

Description

Usage

Format

References

The coronary CT angiography data

Description

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