CommonMean.Copula {CommonMean.Copula}

Estimate bivariate common mean vector under copula models

Description

Estimate the common mean vector under copula models with known correlation. A maximum likelihood estimation procedure is employed. See Shih et al. (2019) and Shih et al. (2021) for details under the Farlie-Gumbel-Morgenstern (FGM) and general copulas, respectively.

Usage

CommonMean.Copula(Y1, Y2, Sigma1, Sigma2, rho, copula = "Clayton")

Arguments

Y1

Outcome 1

Y2

Outcome 2

Sigma1

Standard deviation of outcome 1.

Sigma2

Standard deviation of outcome 2.

rho

Correlation coefficient between outcomes.

copula

The copula to be used with possible options "Clayton", "Gumbel", "Frank", "FGM", and "normal".

Details

We apply "optim" routine to maximize the log-likelihood function. In addition, boundary corrected correlations will be used (Shih et al., 2019).

Value

Outcome 1

Outcome 1.

Outcome 2

Outcome 2.

Correlation

Correlation coefficient between outcomes.

Sample size

Sample size.

Copula

Selected copula.

Copula parameter

Copula parameter.

Corrected correlation

Boundary corrected correlations.

CommonMean 1

Estimation results of outcome 1.

CommonMean 2

Estimation results of outcome 2.

V

Covariance matrix of the common mean vector estimate.

Log-likelihood values

Fitted log-likelihood values.

Note

When rho is 1 or -1, there are some computational issues since the copula parameter may correspond to infinite or negative infinite under some copulas. For the Clayton copula, if rho > 0.95, it will be approximated by 0.95. For the Frank copula, if rho > 0.95 or rho < -0.95, it will be approximated by 0.95 or -0.95, respectively.

References

Shih J-H, Konno Y, Chang Y-T, Emura T (2019) Estimation of a common mean vector in bivariate meta-analysis under the FGM copula, Statistics 53(3): 673-95.

Shih J-H, Konno Y, Emura T (2021-) Copula-based estimation methods for a common mean vector for bivariate meta-analyses, under review.

Examples

library(CommonMean.Copula)
Y1 = c(35,25,30,50,60) # outcome 1
Y2 = c(30,30,50,65,40) # outcome 2
Sigma1 = c(1.3,1.4,1.5,2.0,1.8) # SE of outcome 1
Sigma2 = c(1.7,1.9,2.5,2.2,1.8) # SE of outcome 2
rho = c(0.4,0.7,0.6,0.7,0.6) # correlation between two outcomes
CommonMean.Copula(Y1,Y2,Sigma1,Sigma2,rho) # input