Performs a multivariate meta-analysis using the hybrid random-effects model when the within-study correlations are unknown.
mvma.hybrid(ys, vars, data, method = "reml", tol = 1e-10)This function returns a list containing the following elements:
The estimated overall effect sizes of the p endpoints.
The estimated between-study variances of the p endpoints.
The estimated marginal correlation matrix.
The covariance matrix of the estimated overall effect sizes.
The method used to produce the estimates.
an n x p numeric matrix containing the observed effect sizes. The n rows represent studies, and the p columns represent the multivariate endpoints. NA is allowed for missing endpoints.
an n x p numeric matrix containing the observed within-study variances. The n rows represent studies, and the p columns represent the multivariate endpoints. NA is allowed for missing endpoints.
an optional data frame containing the multivariate meta-analysis dataset. If data is specified, the previous arguments, ys and vars, should be specified as their corresponding column names in data.
a character string specifying the method for estimating the overall effect sizes. It should be "ml" (random-effects model using the maximum likelihood method) or "reml" (random-effects model using the restricted maximum likelihood method, the default).
a small number specifying the convergence tolerance for the estimates by maximizing (restricted) likelihood. The default is 1e-10.
Suppose \(n\) studies are collected in a multivariate meta-analysis on a total of \(p\) endpoints. Denote the \(p\)-dimensional vector of effect sizes as \(\boldsymbol{y}_i\), and their within-study variances form a diagonal matrix \(\mathbf{D}_i\). However, the within-study correlations are unknown. Then, the random-effects hybrid model is as follows (Riley et al., 2008; Lin and Chu, 2018): $$\boldsymbol{y}_i \sim N (\boldsymbol{\mu}, (\mathbf{D}_i + \mathbf{T})^{1/2} \mathbf{R} (\mathbf{D}_i + \mathbf{T})^{1/2}),$$ where \(\boldsymbol{\mu}\) represents the overall effect sizes across studies, \(\mathbf{T} = diag(\tau_1^2, \ldots, \tau_p^2)\) consists of the between-study variances, and \(\mathbf{R}\) is the marginal correlation matrix. Although the within-study correlations are unknown, this model accounts for both within- and between-study correlations by using the marginal correlation matrix.
Lin L, Chu H (2018), "Bayesian multivariate meta-analysis of multiple factors." Research Synthesis Methods, 9(2), 261--272. <tools:::Rd_expr_doi("10.1002/jrsm.1293")>
Riley RD, Thompson JR, Abrams KR (2008), "An alternative model for bivariate random-effects meta-analysis when the within-study correlations are unknown." Biostatistics, 9(1), 172--186. <tools:::Rd_expr_doi("10.1093/biostatistics/kxm023")>
mvma, mvma.bayesian, mvma.hybrid.bayesian
# \donttest{
data("dat.fib")
y <- dat.fib$y
sd <- dat.fib$sd
mvma.hybrid(y = y, vars = sd^2)
# }
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