It estimates the variance components of random-effects in univariate and multivariate meta-analysis with restricted (residual) maximum likelihood (REML) estimation method.
reml(y, v, x, data, RE.constraints = NULL, RE.startvalues = 0.1,
RE.lbound = 1e-10, intervals.type = c("z", "LB"),
model.name="Variance component with REML",
suppressWarnings = TRUE, silent = TRUE, run = TRUE, ...)A vector of effect size for univariate meta-analysis or a \(k\) x \(p\) matrix of effect sizes for multivariate meta-analysis where \(k\) is the number of studies and \(p\) is the number of effect sizes.
A vector of the sampling variance of the effect size for univariate
meta-analysis or a \(k\) x \(p*\) matrix of the sampling
covariance matrix of the effect sizes for multivariate meta-analysis
where \(p* = p(p+1)/2 \). It is arranged by column
major as used by vech.
A predictor or a \(k\) x \(m\) matrix of predictors where \(m\) is the number of predictors.
An optional data frame containing the variables in the model.
A \(p\) x \(p\) matrix
specifying the variance components of the random effects. If the input
is not a matrix, it is converted into a matrix by
as.matrix(). The default is that all
covariance/variance components are free. The format of this matrix
follows as.mxMatrix. Elements of the variance
components can be constrained equally by using the same labels. If a zero matrix is
specified, it becomes a fixed-effects meta-analysis.
A vector of \(p\) starting values on the diagonals of the variance component of the random effects. If only one scalar is given, it will be repeated across the diagonals. Starting values for the off-diagonals of the variance component are all 0. A \(p\) x \(p\) symmetric matrix of starting values is also accepted.
A vector of \(p\) lower bounds on the
diagonals of the variance component of the random effects. If only one
scalar is given, it will be repeated across the diagonals. Lower
bounds for the off-diagonals of the variance component are set at NA. A \(p\) x
\(p\) symmetric matrix of the lower bounds is also accepted.
Either z (default if missing) or
LB. If it is z, it calculates the 95% Wald confidence
intervals (CIs) based on the z statistic. If it is LB, it
calculates the 95% likelihood-based CIs on the
parameter estimates. Note that the z values and their
associated p values are based on the z statistic. They are not
related to the likelihood-based CIs.
A string for the model name in mxModel.
Logical. If TRUE, warnings are
suppressed. It is passed to mxRun.
Logical. An argument to be passed to mxRun
Logical. If FALSE, only return the mx model without running the analysis.
Further arguments to be passed to mxRun
An object of class reml with a list of
Object returned by match.call
A data matrix of y, v and x
No. of effect sizes
No. of predictors
A vector indicating missing data. Studies will be removed before the analysis if they are TRUE
A fitted object returned from mxRun
Restricted (residual) maximum likelihood obtains the parameter estimates on the transformed data that do not include the fixed-effects parameters. A transformation matrix \(M=I-X(X'X)^{-1}X\) is created based on the design matrix \(X\) which is just a column vector when there is no predictor in x. The last \(N\) redundant rows of \(M\) is removed where \(N\) is the rank of \(X\). After pre-multiplying by \(M\) on y, the parameters of fixed-effects are removed from the model. Thus, only the parameters of random-effects are estimated.
An alternative but equivalent approach is to minimize the
-2*log-likelihood function: $$
\log(\det|V+T^2|)+\log(\det|X'(V+T^2)^{-1}X|)+(y-X\hat{\alpha})'(V+T^2)^{-1}(y-X\hat{\alpha})$$
where \(V\) is the known conditional sampling covariance matrix
of \(y\), \(T^2\) is the variance component of the random
effects, and \(\hat{\alpha}=(X'(V+T^2)^{-1}X)^{-1}
X'(V+T^2)^{-1}y\). reml()
minimizes the above likelihood function to obtain the parameter estimates.
Cheung, M. W.-L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling, 20(1), 157-167.
Mehta, P. D., & Neale, M. C. (2005). People Are Variables Too: Multilevel Structural Equations Modeling. Psychological Methods, 10(3), 259-284.
Searle, S. R., Casella, G., & McCulloch, C. E. (1992). Variance components. New York: Wiley.
Viechtbauer, W. (2005). Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30(3), 261-293.