It estimates the variance components of random-effects in three-level univariate meta-analysis with restricted (residual) maximum likelihood (REML) estimation method.
reml3(y, v, cluster, x, data, RE2.startvalue=0.1, RE2.lbound=1e-10,
RE3.startvalue=RE2.startvalue, RE3.lbound=RE2.lbound, RE.equal=FALSE,
intervals.type=c("z", "LB"), model.name="Variance component with REML",
suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)A vector of \(k\) studies of effect size.
A vector of \(k\) studies of sampling variance.
A vector of \(k\) characters or numbers indicating the clusters.
A predictor or a \(k\) x \(m\) matrix of level-2 and level-3 predictors where \(m\) is the number of predictors.
An optional data frame containing the variables in the model.
Starting value for the level-2 variance.
Lower bound for the level-2 variance.
Starting value for the level-3 variance.
Lower bound for the level-3 variance.
Logical. Whether the variance components at level-2 and level-3 are constrained equally.
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. 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
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 the 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 combining
level-2 and level-3 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.
Cheung, M. W.-L. (2014). Modeling dependent effect sizes with three-level meta-analyses: A structural equation modeling approach. Psychological Methods, 19, 211-229.
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.