Description Usage Arguments Details Author(s) References See Also
It conducts threelevel univariate metaanalysis with maximum likelihood estimation method. Mixedeffects metaanalysis can be conducted by including study characteristics as predictors. Equality constraints on the intercepts, regression coefficients and variance components on the level2 and on the level3 can be easily imposed by setting the same labels on the parameter estimates.
1 2 3 4 5 6 7 8 9 10 11  meta3(y, v, cluster, x, data, intercept.constraints = NULL,
coef.constraints = NULL , RE2.constraints = NULL,
RE2.lbound = 1e10, RE3.constraints = NULL, RE3.lbound = 1e10,
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=1e10,
RE3.constraints=NULL, RE3.lbound=1e10, intervals.type=c("z", "LB"),
R2=TRUE, model.name="Meta analysis with ML",
suppressWarnings=TRUE, silent = TRUE, run = TRUE, ...)

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 level2 and level3 predictors where m is the number of predictors. 
x2 
A predictor or a k x m matrix of level2 predictors where m is the number of predictors. 
x3 
A predictor or a k x m matrix of level3 predictors where m is the number of predictors. 
av2 
A predictor or a k x m matrix of level2 auxiliary variables where m is the number of variables. 
av3 
A predictor or a k x m matrix of level3 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

coef.constraints 
A 1 x m matrix
specifying how the level2 and level3 predictors predict the effect sizes. If the input
is not a matrix, it is converted into a matrix by

RE2.constraints 
A scalar or a 1 x 1 matrix
specifying the variance components of the random effects. The default
is that the variance components are free. The format of this matrix
follows 
RE2.lbound 
A scalar or a 1 x 1 matrix of lower bound on the level2 variance component of the random effects. 
RE3.constraints 
A scalar of a 1 x 1 matrix
specifying the variance components of the random effects at
level3. The default is that the variance components are free. The format of this matrix
follows 
RE3.lbound 
A scalar or a 1 x 1 matrix of lower bound on the level3 variance component of the random effects. 
intervals.type 
Either 
I2 
Possible options are 
R2 
Logical. If 
model.name 
A string for the model name in 
suppressWarnings 
Logical. If 
silent 
Logical. Argument to be passed to 
run 
Logical. If 
... 
Further arguments to be passed to

y_{ij} = β_0 + \mathbf{β'}*\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, β_0 is the intercept, \mathbf{β} is the regression coefficients, \mathbf{x}_{ij} is a vector of predictors, u_{(2)ij}~ N(0, tau^2_2) and u_{(3)j}~ N(0, tau^2_3) are the level2 and level3 heterogeneity variances, respectively, and e_{ij}~ N(0, v_{ij}) is the conditional known sampling variance.
meta3()
does not differentiate between level2 or level3
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 level2 and level3 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 level2 and av3
at level3 may
be included to improve the estimation. Although meta3X()
is more
flexible in handling missing covariates, it is more likely to encounter
estimation problems.
Mike W.L. Cheung <[email protected]>
Cheung, M. W.L. (2014). Modeling dependent effect sizes with threelevel metaanalyses: A structural equation modeling approach. Psychological Methods, 19, 211229.
Enders, C. K. (2010). Applied missing data analysis. New York: Guilford Press.
Graham, J. (2003). Adding missingdatarelevant variables to FIMLbased structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 10(1), 80100.
Konstantopoulos, S. (2011). Fixed effects and variance components estimation in threelevel metaanalysis. Research Synthesis Methods, 2, 6176.
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