Copula-Based Estimation Methods for a Common Mean Vector for Bivariate Meta-Analyses

Abstract

Traditional bivariate meta-analyses adopt the bivariate normal model. As the bivariate normal distribution produces symmetric dependence, it is not flexible enough to describe the true dependence structure of real meta-analyses. As an alternative to the bivariate normal model, recent papers have adopted “copula” models for bivariate meta-analyses. Copulas consist of both symmetric copulas (e.g., the normal copula) and asymmetric copulas (e.g., the Clayton copula). While copula models are promising, there are only a few studies on copula-based bivariate meta-analysis. Therefore, the goal of this article is to fully develop the methodologies and theories of the copula-based bivariate meta-analysis, specifically for estimating the common mean vector. This work is regarded as a generalization of our previous methodological/theoretical studies under the FGM copula to a broad class of copulas. In addition, we develop a new R package, “CommonMean.Copula”, to implement the proposed methods. Simulations are performed to check the proposed methods. Two real dataset are analyzed for illustration, demonstrating the insufficiency of the bivariate normal model.

Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies

“…There exist a number of copulas [31], and new copulas keep emerging [35,36]. Copulas are applied to a variety of fields, including reliability [37], ecology [38], meta-analysis [39,40], and econometrics [41]. However, to reduce the complexity of estimation, the copula has to be as simple as possible.…”

“…If θ → 0 , the Clayton copula reduces to C(u, v) = uv, the independent risks model of [26]. The Clayton copula is one of the most frequently used copulas in competing risks models [22,24] and other models [33,[40][41][42] due to its remarkable mathematical tractability. Other copulas might be considered.…”

Testing Equality of Multiple Population Means under Contaminated Normal Model Using the Density Power Divergence

“…Under the independent heterogeneous setup, the above conditions are required to stabilize the matrices J and K for the existence of the asymptotic distribution (cf. [ 40 , 41 , 42 ]). These assumptions are satisfied by the true density defined in Assumption 1.…”

“…The independent and non-identically distributed samples leading to heterogeneity in variances technically impose a computational burden (cf. [ 42 ]). Hence, a positive definite matrix J in assumption (A2) is required to stabilize the asymptotic variance of the MDPDE.…”

“…Then Differentiating Equation ( 3 ), it can be written as an M-estimator as follows where Let be the true value of , then Taking a Taylor series expansion of Equation ( A2 ), we get Under regularity condition (A1)–(A3), it can be easily shown that the reminder term Now, we will show that Therefore, from Equation ( A5 ), we will prove the theorem as First Part: So Equation ( 6 ) gives Second Part: From Equation ( A4 ), we get Now Therefore, Finally, the asymptotic normality will be proved using the central limit theorem for the independent but not identical random variables using assumption (A5). A bound can be shown following Section 2.7 of [ 49 ] or Section 5 of [ 50 ] (also see [ 42 ]).…”