metaSEM (version 0.9.4)

reml3: Estimate Variance Components in Three-Level Univariate Meta-Analysis with Restricted (Residual) Maximum Likelihood Estimation

Description

It estimates the variance components of random-effects in three-level univariate meta-analysis with restricted (residual) maximum likelihood (REML) estimation method.

Usage

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, ...)

Arguments

y

A vector of $k$ studies of effect size.

v

A vector of $k$ studies of sampling variance.

cluster

A vector of $k$ characters or numbers indicating the clusters.

x

A predictor or a $k$ x $m$ matrix of level-2 and level-3 predictors where $m$ is the number of predictors.

data

An optional data frame containing the variables in the model.

RE2.startvalue

Starting value for the level-2 variance.

RE2.lbound

Lower bound for the level-2 variance.

RE3.startvalue

Starting value for the level-3 variance.

RE3.lbound

Lower bound for the level-3 variance.

RE.equal

Logical. Whether the variance components at level-2 and level-3 are constrained equally.

intervals.type

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 paramet

model.name

A string for the model name in mxModel.

suppressWarnings

Logical. If TRUE, warnings are suppressed. Argument to be passed to mxRun.

silent

Logical. Argument to be passed to mxRun

run

Logical. If FALSE, only return the mx model without running the analysis.

...

Futher arguments to be passed to mxRun

Value

• An object of class reml with a list of
• call
  Object returned by match.call
• data
  A data matrix of y, v and x
• mx.fit
  A fitted object returned from mxRun

Details

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 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.

References

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.

See Also

meta3, reml, Cooper03, Bornmann07