It estimates the variance components of random-effects in three-level univariate meta-analysis with restricted (residual) maximum likelihood (REML) estimation method.
## Depreciated in the future
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, ...)
reml3L(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, ...)
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
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
Mike W.-L. Cheung <mikewlcheung@nus.edu.sg>
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.
meta3L
, reml
, Cooper03
, Bornmann07