meta3: Three-Level Univariate Meta-Analysis with Maximum Likelihood Estimation

Description

It conducts three-level univariate meta-analysis with maximum likelihood estimation method. Mixed-effects meta-analysis can be conducted by including study characteristics as predictors. Equality constraints on the intercepts, regression coefficients and variance components on the level-2 and on the level-3 can be easily imposed by setting the same labels on the parameter estimates.

Usage

meta3(y, v, cluster, x, data, intercept.constraints = NULL,
      coef.constraints = NULL , RE2.constraints = NULL,
      RE2.lbound = 1e-10, RE3.constraints = NULL, RE3.lbound = 1e-10,
      intervals.type = c("z", "LB"), I2="I2q",
      R2=TRUE, model.name = "Meta analysis with ML",
      suppressWarnings = TRUE, silent = TRUE, run = TRUE, ...)
meta3X(y, v, cluster, x2, x3, av2, av3, data, intercept.constraints=NULL,
       coef.constraints=NULL, RE2.constraints=NULL, RE2.lbound=1e-10,
       RE3.constraints=NULL, RE3.lbound=1e-10, intervals.type=c("z", "LB"),
       R2=TRUE, model.name="Meta analysis with ML",
       suppressWarnings=TRUE, silent = TRUE, run = TRUE, ...)

Arguments

A vector of $k$ studies of effect size.

A vector of $k$ studies of sampling variance.

cluster

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.

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

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

av2

A predictor or a $k$ x $m$ matrix of level-2 auxiliary variables where $m$ is the number of variables.

av3

A predictor or a $k$ x $m$ matrix of level-3 auxiliary variables where $m$ is the number of variables.

data

An optional data frame containing the variables in the model.

intercept.constraints

A $1$ x $1$ matrix specifying whether the intercept of the effect size is fixed or constrained. The format of this matrix follows as.mxMatrix. The intercept can be constrained with other para

coef.constraints

A $1$ x $m$ matrix specifying how the level-2 and level-3 predictors predict the effect sizes. If the input is not a matrix, it is converted into a matrix by as.matrix(). The default is that all $m$ predictors predict the effect size. The

RE2.constraints

A scalar or a $1$ x $1$ matrix specifying the variance componets of the random effects. The default is that the variance components is free. The format of this matrix follows as.mxMatrix. Elem

RE2.lbound

A scalar or a $1$ x $1$ matrix of lower bound on the level-2 variance component of the random effects.

RE3.constraints

A scalar of a $1$ x $1$ matrix specifying the variance componets of the random effects at level-3. The default is that the variance components is free. The format of this matrix follows as.mxMatrix

RE3.lbound

A scalar or a $1$ x $1$ matrix of lower bound on the level-3 variance component of the random effects.

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

I2

Possible options are "I2q", "I2hm",
	"I2am" and "ICC". They represent the I2 calculated by using a
	typical within-study sampling variance from the Q statistic, the
	harmonic mean, the arith

R2

Logical. If TRUE and there are predictors, R2 is
	calculated.

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

`Details`

$$y_{ij} = \beta_0 + \mathbf{\beta'}*\mathbf{x}_{ij} + u_{(2)ij} + u_{(3)j} + e_{ij}$$ where $y_{ij}$ is the effect size for the ith study in the jth cluster,
  $\beta_0$ is the intercept, $\mathbf{\beta}$ is the
  regression coefficients, $\mathbf{x}_{ij}$ is a vector of predictors, $u_{(2)ij} \sim N(0, \tau^2_2)$ and $u_{(3)j} \sim N(0, \tau^2_3)$ are the level-2 and level-3 heterogeneity variances,
	respectively, and $e_{ij} \sim N(0, v_{ij})$ is the conditional known sampling variance.
 meta3() does not differentiate between level-2 or level-3
 variables in x since both variables are treated as a design
 matrix. When there are missing values in x, the data will be
 deleted. meta3X() treats the predictors x2 and x3
 as level-2 and level-3 variables. Thus, their means and covariance
 matrix will be estimated. Missing values in x2 and x3
 will be handled by (full information) maximum likelihood (FIML) in meta3X(). Moreover,
 auxiliary variables av2 at level-2 and av3 at level-3 may
 be included to improve the estimation. Although meta3X() is more
 flexible in handling missing covariates, it is more likely to encounter
 estimation problems.

`References`

Cheung, M. W.-L. (2014). Modeling dependent effect sizes with three-level meta-analyses: A structural equation modeling approach. Psychological Methods, 19, 211-229.
  Enders, C. K. (2010). Applied missing data analysis. New York:
  Guilford Press.
  Graham, J. (2003). Adding missing-data-relevant variables to
 FIML-based structural equation models. Structural Equation
   Modeling: A Multidisciplinary Journal, 10(1), 80-100.
 Konstantopoulos, S. (2011). Fixed effects and variance components
  estimation in three-level meta-analysis. Research Synthesis
  Methods, 2, 61-76.

`See Also`

reml3, Cooper03, Bornmann07