| Title: | Copula Mixed Models for Multivariate Meta-Analysis of Diagnostic Test Accuracy Studies |
|---|---|
| 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, 2024). |
| Authors: | Aristidis K. Nikoloulopoulos [aut, cre]
|
| Maintainer: | Aristidis K. Nikoloulopoulos <[email protected]> |
| License: | GPL (>= 3.5.0) |
| Version: | 1.7.3 |
| Built: | 2024-12-17 07:01:50 UTC |
| Source: | CRAN |
Fits copula mixed models for multivariate meta-analysis of diagnostic test accuracy studies proposed in Nikoloulopoulos (2015, 2017, 2018a, 2018b, 2019, 2020a, 2020b, 2022, 2024a, 2024b).
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, 2024a).
The 1-truncated D-vine copula mixed model for for meta-analysis of diagnostic test accuracy studies without a gold standard (Nikoloulopoulos, 2024b).
Aristidis K. Nikoloulopoulos.
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. (2024a) Joint meta-analysis of two diagnostic tests accounting for within and between studies dependence. Statistical Methods in Medical Research. doi:10.1177/09622802241269645.
Nikoloulopoulos, A.K. (2024b) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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 studies that assessed both RF and anti-CCP2 antibody for diagnosing rheumatoid arthritis.
A data frame with 22 observations on the following 8 variables.
the number of true positives for RF
the number of false negatives for RF
the number of false positives for RF
the number of true negatives for RF
the number of true positives for anti-CCP2
the number of false negatives for anti-CCP2
the number of false positives for anti-CCP2
the number of true negatives for anti-CCP2
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.
Data on 8 cohort studies inthemeta-analysis in Karageorgopoulos et al. (2011). The interest there is to assess -D-Glucan as aserum or plasma marker for the presence of invasive fungal infections.
data(betaDG)data(betaDG)
A data frame with 8 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
Karageorgopoulos, D.E., Vouloumanou, E.K., Ntziora, F., et al.
(2011) -D-Glucan assay for the diagnosis of invasive fungal infections: a meta-analysis. Clinical Infectious Diseases, 52(6), 750–770.
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 is adequate with good precision to at least at four decimal places.
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)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)
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 |
mgrid |
a list containing two matrices with the rows of the output matrix x are copies of the vector |
qcond |
function for the inverse of conditional copula cdf |
tau2par |
function for maping Kendall's tau to copula parameter |
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
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.
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)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)
Data on 26 studies from a systematic review for diagnostic accuracy studies of coronary computed tomography angiography for the detection of coronary artery disease.
data(coronary)data(coronary)
A data frame with 26 observations on the following 6 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
the number of non-evaluable positives
the number of non-evaluable negatives
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.
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.
data(CT)data(CT)
A data frame with 17 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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.
Simulation from a trivariate C-vine copula
cvinesim(N,param,qcondcop12,qcondcop13,qcondcop23, tau2par12,tau2par13,tau2par23)cvinesim(N,param,qcondcop12,qcondcop13,qcondcop23, tau2par12,tau2par13,tau2par23)
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 |
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.
Nx3 matrix with values in (0,1)
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 for parametric families.
dbvn(u,v,cpar) dfrk(u,v,cpar) dcln(u,v,cpar) dcln90(u,v,cpar) dcln270(u,v,cpar)dbvn(u,v,cpar) dfrk(u,v,cpar) dcln(u,v,cpar) dcln90(u,v,cpar) dcln270(u,v,cpar)
u |
value in interval 0,1; could be a vector |
v |
value in interval 0,1; could be a vector |
cpar |
copula parameter: scalar. |
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
pdf value(s).
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.
Data on 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).
A data frame with 11 observations on the following 8 variables.
the number of the test results in the diseased where the shortened humerus outcome is negative and the shortened femur outcome is negative
the number of the test results in the diseased where the shortened humerus outcome is negative and the shortened femur outcome is positive
the number of the test results in the diseased where the shortened humerus outcome is positive and the shortened femur outcome is negative
the number of the test results in the diseased where the shortened humerus outcome is positive and the shortened femur outcome is positive
the number of the test results in the non-diseased where the shortened humerus outcome is negative and the shortened femur outcome is negative
the number of the test results in the non-diseased where the shortened humerus outcome is negative and the shortened femur outcome is positive
the number of the test results in the non-diseased where the shortened humerus outcome is positive and the shortened femur outcome is negative
the number of the test results in the non-diseased where the shortened humerus outcome is positive and the shortened femur outcome is positive
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.
Simulation from a six-variate 1-truncated D-vine copula.
dvine6dsim(nsim,tau,qcond,tau2par)dvine6dsim(nsim,tau,qcond,tau2par)
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 |
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.
Nx6 matrix with values in (0,1)
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.
Nikoloulopoulos, A.K. (2024) Joint meta-analysis of two diagnostic tests accounting for within and between studies dependence. Statistical Methods in Medical Research. doi:10.1177/09622802241269645
Simulation from a (truncated) quadrivariate D-vine copula. Lower-order trees (if any) are composed with BVN copulas.
dvinesim(nsim,param,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2) dtrvinesim(nsim,trparam,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)dvinesim(nsim,param,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2) dtrvinesim(nsim,trparam,qcond1,pcond1,tau2par1,qcond2,pcond2,tau2par2)
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 |
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.
Nx4 matrix with values in (0,1)
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.
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.
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.
FactorCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2) FactorCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2)FactorCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2) FactorCopulaREMADA.beta(TP,FN,FP,TN,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2)
TP |
an |
FN |
an |
FP |
an |
TN |
an |
gl |
a list containing the components of Gauss-Legendre nodes |
mgrid |
a list containing two matrices with the rows of the output matrix x are copies of the vector |
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 |
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
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.
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$hessian))) detach(arthritis)data(arthritis) attach(arthritis) TP=cbind(TP1,TP2) TN=cbind(TN1,TN2) FP=cbind(FP1,FP2) FN=cbind(FN1,FN2) nq=25 gl=gauss.quad.prob(nq,"uniform") mgrid=meshgrid(gl$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 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.
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)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)
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 |
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 |
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 |
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
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.
VineCopulaREMADA, CopulaREMADA
# 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$n) qcond=qcondcln tau2par=tau2par.cln est=hybridCopulaREMADA.beta(TP,FN,FP,TN,type,gl,mgrid1,mgrid2, qcondcop12,qcondcop13,tau2par12,tau2par13,qcond,tau2par)# simulate the data from N=25 cohort studies N=25 p=c(0.8,0.7,0.4) g=c(0.1,0.1,0.05) taus=c(-0.5,-0.3,-0.0001) qcondcop12=qcondcop23=qcondcop13=qcondcln90 tau2par12=tau2par23=tau2par13=tau2par.cln90 simdat1=rVineCopulaREMADA.beta(N,p,g,taus,0,0, qcondcop12,qcondcop13,qcondcop23,tau2par12,tau2par13,tau2par23) # simulate data from the N=25 case-control studies tau=0.5 p=p[-3] g=g[-3] simdat2=rCopulaREMADA.beta(N,p,g,tau,rcln,tau2par.cln) # combine the data TP=c(simdat1$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)
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.
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)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)
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 |
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 |
tau2par |
function for maping Kendall's tau to copula parameter; choices are |
start |
starting values for the parameters |
A list containing the following components:
LogLikelihood |
the maximized log-likelihood |
Estimates |
the MLE |
SE |
the standard errors |
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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)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)
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.
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)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)
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 |
select.random |
vector |
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 |
tau2par |
function for maping Kendall's tau to copula parameter; choices are |
start |
starting values for the parameters |
A list containing the following components:
LogLikelihood |
the maximized log-likelihood |
Estimates |
the MLE |
SE |
the standard errors |
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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)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)
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.
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)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)
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 |
mgrid |
a list containing three-dimensional arrays. Replicates of the quadrature points that produce a 3-dimensional full grid |
select.random |
vector |
qcond |
function for the inverse of conditional copula cdf; choices are |
tau2par |
function for maping Kendall's tau to copula parameter; choices are |
qcond1 |
function for the inverse of conditional copula cdf for the |
tau2par1 |
function for maping Kendall's tau to copula parameter for the |
qcond2 |
function for the inverse of conditional copula cdf for the |
tau2par2 |
function for maping Kendall's tau to copula parameter for the |
start |
starting values for the parameters |
A list containing the following components:
LogLikelihood |
the maximized log-likelihood |
Estimates |
the MLE |
SE |
the standard errors |
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
data(Pap) attach(Pap) nq=30 gl=gauss.quad.prob(nq,"uniform") mgrid<- meshgrid(gl$n,gl$n,gl$n,nargout=3) tau2par=tau2par.bvn qcond=qcondbvn select.random=c(1,2,4) start=c(rep(0.6,5),rep(0.5,3),c(0.01,-0.01)) est.norm=imperfect.trivariateVineCopulaREMADA.norm.comprehensive(y11,y10, y01,y00,gl,mgrid,qcond,tau2par,select.random,start) tau2par1=tau2par.cln180 qcond1=qcondcln180 tau2par2=tau2par.cln270 qcond2=qcondcln270 est.norm.cln=imperfect.trivariateVineCopulaREMADA.norm(y11,y10,y01, y00,gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start) start=c(rep(0.6,5),rep(0.05,3),c(0.1,-0.1)) est.beta=imperfect.trivariateVineCopulaREMADA.beta.comprehensive(y11,y10,y01,y00, gl,mgrid,qcond,tau2par,select.random,start) est.beta.cln=imperfect.trivariateVineCopulaREMADA.beta(y11,y10,y01,y00, gl,mgrid,qcond1,tau2par1,qcond2,tau2par2,select.random,start) detach(Pap)data(Pap) attach(Pap) nq=30 gl=gauss.quad.prob(nq,"uniform") mgrid<- meshgrid(gl$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)
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.
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)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)
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 |
mgrid |
a list containing two matrices with the rows of the output matrix x are copies of the vector |
select.random |
vector |
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 |
A list containing the following components:
LogLikelihood |
the maximized log-likelihood |
Estimates |
the MLE |
SE |
the standard errors |
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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)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)
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.
imperfectREMADA.norm(y11,y10,y01,y00,gl,select.random,start) imperfectREMADA.beta(y11,y10,y01,y00,gl,select.random,start)imperfectREMADA.norm(y11,y10,y01,y00,gl,select.random,start) imperfectREMADA.beta(y11,y10,y01,y00,gl,select.random,start)
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 |
select.random |
a scalar from 1 to 5 that indicates the random effect |
start |
starting values for the parameters |
A list containing the following components:
LogLikelihood |
the maximized log-likelihood |
Estimates |
the MLE |
SE |
the standard errors |
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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)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)
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.
data(LAG)data(LAG)
A data frame with 17 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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.
A list containing four-dimensional arrays. Replicates of the quadrature points that produce a 4-dimensional full grid.
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)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. Replicates of the quadrature points that produce a 5-dimensional full grid.
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)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. Replicates of the quadrature points that produce an 6-dimensional full grid.
data(mgrid6d) dim(mgrid$x1) dim(mgrid$x2) dim(mgrid$x3) dim(mgrid$x4) dim(mgrid$x5) dim(mgrid$x6)data(mgrid6d) dim(mgrid$x1) dim(mgrid$x2) dim(mgrid$x3) dim(mgrid$x4) dim(mgrid$x5) dim(mgrid$x6)
Data on 30 studies from a systematic review for diagnostic accuracy studies of coronary computed tomography angiography for the detection of coronary artery disease.
data(MK2016)data(MK2016)
A data frame with 30 observations on the following 6 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
the number of non-evaluable positives
the number of non-evaluable negatives
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.
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.
data(MRI)data(MRI)
A data frame with 10 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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.
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.
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)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)
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 |
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 |
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
Nikoloulopoulos, A.K. (2024) Joint meta-analysis of two diagnostic tests accounting for within and between studies dependence. Statistical Methods in Medical Research. doi:10.1177/09622802241269645
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)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)
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.
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)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)
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 |
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 |
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
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.
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)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)
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).
data(OGT)data(OGT)
A data frame with 10 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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.
These data are comprised of 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).
A data frame with 59 observations on the following 4 variables.
the number of the test results where the Pap test outcome is positive and the histology test outcome is positive
the number of the test results where the Pap test outcome is positive and the histology test outcome is negative
the number of the test results where the Pap test outcome is negative and the histology test outcome is positive
the number of the test results where the Pap test outcome is negative and the histology test outcome is negative
Fahey, M. T., Irwig, L., and Macaskill, P. (1995). Meta-analysis of pap test accuracy. American Journal of Epidemiology, 142:680–689.
Bivariate copula conditional distribution functions
pcondbvn(v,u,cpar) pcondfrk(v,u,cpar) pcondcln(v,u,cpar) pcondcln90(v,u,cpar) pcondcln270(v,u,cpar)pcondbvn(v,u,cpar) pcondfrk(v,u,cpar) pcondcln(v,u,cpar) pcondcln90(v,u,cpar) pcondcln270(v,u,cpar)
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. |
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.
inverse conditional cdf value(s)
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
qcondbvn(p,u,cpar) qcondfrk(p,u,cpar) qcondcln(p,u,cpar) qcondcln90(p,u,cpar) qcondcln270(p,u,cpar)qcondbvn(p,u,cpar) qcondfrk(p,u,cpar) qcondcln(p,u,cpar) qcondcln90(p,u,cpar) qcondcln270(p,u,cpar)
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. |
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.
inverse conditional cdf value(s)
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 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.
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)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)
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 |
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 |
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
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.
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)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 parametric bivariate copula families
rbvn(N,cpar) rfrk(N,cpar) rcln(N,cpar) rcln90(N,cpar) rcln270(N,cpar)rbvn(N,cpar) rfrk(N,cpar) rcln(N,cpar) rcln90(N,cpar) rcln270(N,cpar)
N |
sample size |
cpar |
copula parameter: scalar |
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.
nx2 matrix with values in (0,1)
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.
To simulate the data we have used the following steps:
1. Simulate the study size from a shifted gamma distribution with parameters and round off to the nearest integer.
2. Simulate from a parametric family of copulas 'cop'.
3. Convert to beta realizations or normal realizations.
4. Draw the number of diseased from a distribution.
5. Set and then round for .
rCopulaREMADA.norm(N,p,si,tau,rcop,tau2par) rCopulaREMADA.beta(N,p,g,tau,rcop,tau2par)rCopulaREMADA.norm(N,p,si,tau,rcop,tau2par) rCopulaREMADA.beta(N,p,g,tau,rcop,tau2par)
N |
sample size |
p |
Pair |
si |
Pair |
g |
Pair |
tau |
Kendall's tau value |
rcop |
function for copula generation |
tau2par |
function for mapping from Kendall's tau to copula parameter |
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 |
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.
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)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 diagnostic tests
rFactorCopulaREMADA.norm(N,p,si,taus,qcond1,tau2par1,qcond2,tau2par2) rFactorCopulaREMADA.beta(N,p,g,taus,qcond1,tau2par1,qcond2,tau2par2)rFactorCopulaREMADA.norm(N,p,si,taus,qcond1,tau2par1,qcond2,tau2par2) rFactorCopulaREMADA.beta(N,p,g,taus,qcond1,tau2par1,qcond2,tau2par2)
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 |
A list with the simulated data in matrices with columns and rows.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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.
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$TN FP=out$FPN=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 1-truncated D-vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard
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)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)
N |
sample size |
p |
Vector |
si |
Vector |
g |
Vector |
taus |
Kendall's tau values |
select.random |
vector |
qcond1 |
function for the inverse of conditional copula cdf for the |
tau2par1 |
function for maping Kendall's tau to copula parameter for the |
qcond2 |
function for the inverse of conditional copula cdf for the |
tau2par2 |
function for maping Kendall's tau to copula parameter for the |
Simulated data with 4 columns and rows.
the number of the test results where the index test outcome is positive and the reference test outcome is positive
the number of the test results where the index test outcome is positive and the reference test outcome is negative
the number of the test results where the index test outcome is negative and the reference test outcome is positive
the number of the test results where the index test outcome is negative and the reference test outcome is negative
Nikoloulopoulos, A.K. (2024) Vine copula mixed models for meta-analysis of diagnostic accuracy studies without a gold standard. Submitted.
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)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
rmultinom6dVineCopulaREMADA.norm(N,p,si,taus,qcond,tau2par) rmultinom6dVineCopulaREMADA.beta(N,p,g,taus,qcond,tau2par)rmultinom6dVineCopulaREMADA.norm(N,p,si,taus,qcond,tau2par) rmultinom6dVineCopulaREMADA.beta(N,p,g,taus,qcond,tau2par)
N |
sample size |
p |
Vector |
si |
Vector |
g |
Vector |
taus |
Kendall's tau values |
qcond |
function for the inverse conditional copula cdf |
tau2par |
function for maping Kendall's taus to copula parameters |
Simulated data with 8 columns and rows.
the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is negative
the number of the test results in the diseased where the test 1 outcome is negative and the test 2 outcome is positive
the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is negative
the number of the test results in the diseased where the test 1 outcome is positive and the test 2 outcome is positive
the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is negative
the number of the test results in the non-diseased where the test 1 outcome is negative and the test 2 outcome is positive
the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is negative
the number of the test results in the non-diseased where the test 1 outcome is positive and the test 2 outcome is positive
Nikoloulopoulos, A.K. (2024) Joint meta-analysis of two diagnostic tests accounting for within and between studies dependence. Statistical Methods in Medical Research. doi:10.1177/09622802241269645
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]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
rmultinomVineCopulaREMADA.norm(N,p,si,taus,qcond1, pcond1,tau2par1,qcond2, pcond2,tau2par2) rmultinomVineCopulaREMADA.beta(N,p,g,taus,qcond1, pcond1,tau2par1,qcond2, pcond2,tau2par2)rmultinomVineCopulaREMADA.norm(N,p,si,taus,qcond1, pcond1,tau2par1,qcond2, pcond2,tau2par2) rmultinomVineCopulaREMADA.beta(N,p,g,taus,qcond1, pcond1,tau2par1,qcond2, pcond2,tau2par2)
N |
sample size |
p |
Vector |
si |
Vector |
g |
Vector |
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 |
Simulated data with 6 columns and rows.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
the number of non-evaluable positives
the number of non-evaluable negatives
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.
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]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]
Simulation from trivariate vine copula mixed models for diagnostic test accuaracy studies accounting for disease prevalence and non-evaluable results
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)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)
N |
sample size |
p |
Vector |
si |
Vector |
g |
Vector |
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 |
Simuated data with 6 columns and rows.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
the number of non-evaluable positives
the number of non-evaluable negatives
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. (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.
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)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)
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
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)))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)))
param |
A vector with the sensitivities, specifities, variabilities and Kendall's tau value (the latter only for |
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 |
Summary receiver operating characteristic curves
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.
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)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)
Bivariate copulas: mapping of Kendall's tau and copula parameter.
tau2par.bvn(tau) tau2par.frk(tau) tau2par.cln(tau) tau2par.cln90(tau) tau2par.cln180(tau) tau2par.cln270(tau)tau2par.bvn(tau) tau2par.frk(tau) tau2par.cln(tau) tau2par.cln90(tau) tau2par.cln180(tau) tau2par.cln270(tau)
tau |
Kendall's tau for the copula |
For abbreviations of names of copula families (after the dot in function
names), see dcop help page.
copula parameter
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.
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.
data(telomerase)data(telomerase)
A data frame with 10 observations on the following 4 variables.
the number of true positives
the number of false negatives
the number of false positives
the number of true negatives
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 (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.
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)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)
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 |
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 |
A list containing the following components:
z |
the test statistic |
p-value |
the |
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. (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.
Vuong Q.H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57, 307–333.
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$e c(0.85528463, 0.14667571, 0.68321231, 0.04897466, 0.02776290, 0.08561436, -0.34639172, 0.04621924, -0.21627977) c090vuong.b2=vine.vuong.beta(qcondcln90,qcondcln,qcondcln90, tau2par.cln90,tau2par.cln,tau2par.cln90, nest.n2.est,c090est.b2.est,TP,FN,FP,TN,gl,mgrid) c090vuong.b2 detach(betaDG)nq=15 gl=gauss.quad.prob(nq,"uniform") mgrid=meshgrid(gl$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)
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.
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)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)
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 |
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 |
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
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. (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.
nq=15 gl=gauss.quad.prob(nq,"uniform") mgrid=meshgrid(gl$n,gl$n,gl$n,nargout=3) data(OGT) attach(OGT) out=tVineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid, qcondbvn,qcondbvn,tau2par.bvn,tau2par.bvn) detach(OGT) ############################################ # In the precence of non-evaluable results # data(coronary) attach(coronary) out=tVineCopulaREMADA.norm(TP,FN,FP,TN,gl,mgrid, qcondbvn,qcondbvn,tau2par.bvn,tau2par.bvn,NEP,NEN) detach(coronary)nq=15 gl=gauss.quad.prob(nq,"uniform") mgrid=meshgrid(gl$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)
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.
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)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)
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 |
mgrid |
a list containing two matrices with the rows of the output matrix X are copies of the vector |
A list containing the following components:
z |
the test statistic |
p-value |
the |
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.
Vuong Q.H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57:307–333.
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)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)