metaSEM: reml3 – R documentation

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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 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.
`model.name`	A string for the model name in `mxModel`.
`suppressWarnings`	Logical. If `TRUE`, warnings are suppressed. It is passed to `mxRun`.
`silent`	Logical. Argument to be passed to `mxRun`
`run`	Logical. If `FALSE`, only return the mx model without running the analysis.
`...`	Further arguments to be passed to `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 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-Xα)'(V+T^2)^{-1}(y-X*α)

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{α}=(t(X)(V+T^2)^{-1}X)^{-1}t(X)(V+T^2)^{-1}y. reml() minimizes the above likelihood function to obtain the parameter estimates.

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

Note

reml is more computationally intensive than meta. Moreover, reml is more likely to encounter errors during optimization. Since a likelihood function is directly employed to obtain the parameter estimates, there is no number of studies and number of observed statistics returned by mxRun. Ad-hoc steps are used to modify mx.fit@runstate$objectives[[1]]@numObs and mx.fit@runstate$objectives[[1]]@numStats.

Author(s)

Mike W.-L. Cheung <mikewlcheung@nus.edu.sg>

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.