Package 'metaSEM'

Description

A collection of functions for conducting meta-analysis using a structural equation modeling (SEM) approach via the 'OpenMx' and 'lavaan' packages. It also implements various procedures to perform meta-analytic structural equation modeling on the correlation and covariance matrices.

Details

Author(s)

Mike W.-L. Cheung <[email protected]>

Maintainer: Mike W.-L. Cheung <[email protected]>

References

Cheung, M. W.-L. (2008). A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling. Psychological Methods, 13 (3), 182-202. https://doi.org/10.1037/a0013163

Cheung, M. W.-L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16 (2), 267-294. https://doi.org/10.1080/10705510902751291

Cheung, M. W.-L. (2010). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling, 17 (3), 481-509. https://doi.org/10.1080/10705511.2010.489367

Cheung, M. W.-L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling, 20 (1), 157-167. https://doi.org/10.1080/10705511.2013.742404

Cheung, M. W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20 (3), 429-454. https://doi.org/10.1080/10705511.2013.797827

Cheung, M. W.-L. (2014). Modeling dependent effect sizes with three-level meta-analyses: A structural equation modeling approach. Psychological Methods, 19 (2), 211-229. https://doi.org/10.1037/a0032968

Cheung, M. W.-L. (2014). Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods, 46 (1), 29-40. https://doi.org/10.3758/s13428-013-0361-y

Cheung, M. W.-L. (2015). metaSEM: An R package for meta-analysis using structural equation modeling. Frontiers in Psychology, 5 (1521). https://doi.org/10.3389/fpsyg.2014.01521

Cheung, M. W.-L. (2015). Meta-Analysis: A Structural Equation Modeling Approach. Chichester, West Sussex: John Wiley & Sons, Inc.

Cheung, M. W.-L. (2018). Issues in solving the problem of effect size heterogeneity in meta-analytic structural equation modeling: A commentary and simulation study on Yu, Downes, Carter, and O'Boyle (2016). Journal of Applied Psychology, 103 (7), 787-803. https://doi.org/10.1037/apl0000284

Cheung, M. W.-L. (2018). Computing multivariate effect sizes and their sampling covariance matrices with structural equation modeling: Theory, examples, and computer simulations. Frontiers in Psychology, 9 (1387). https://doi.org/10.3389/fpsyg.2018.01387

Cheung, M. W.-L. (2019). Some reflections on combining meta-analysis and structural equation modeling. Research Synthesis Methods, 10 (1), 15-22. https://doi.org/10.1002/jrsm.1321

Cheung, M. W.-L. (2021). Meta-analytic structural equation modeling. In Oxford Research Encyclopedia of Business and Management. Oxford University Press. https://doi.org/10.1093/acrefore/9780190224851.013.225

Cheung, M. W.-L., & Chan, W. (2004). Testing dependent correlation coefficients via structural equation modeling. Organizational Research Methods, 7 (2), 206-223. https://doi.org/10.1177/1094428104264024

Cheung, M. W.-L., & Chan, W. (2005). Meta-analytic structural equation modeling: A two-stage approach. Psychological Methods, 10 (1), 40-64. https://doi.org/10.1037/1082-989X.10.1.40

Cheung, M. W.-L., & Chan, W. (2009). A two-stage approach to synthesizing covariance matrices in meta-analytic structural equation modeling. Structural Equation Modeling, 16 (1), 28-53. https://doi.org/10.1080/10705510802561295

Cheung, M. W.-L., & Cheung, S.-F. (2016). Random-effects models for meta-analytic structural equation modeling: Review, issues, and illustrations. Research Synthesis Methods, 7 (2), 140-155. https://doi.org/10.1002/jrsm.1166

Jak, S., & Cheung, M. W.-L. (2018). Testing moderator hypotheses in meta-analytic structural equation modeling using subgroup analysis. Behavior Research Methods, 50 (4), 1359-1373. https://doi.org/10.3758/s13428-018-1046-3

Jak, S., & Cheung, M. W.-L. (2020). Meta-analytic structural equation modeling with moderating effects on SEM parameters. Psychological Methods, 25 (4), 430-455. https://doi.org/10.1037/met0000245

Description

This study reports sixteen studies on the effect sizes (correlation coefficients) between CMSE and emotional exhaustion (EE), depersonalization (DP), and (lowered) personal accomplishment (PA) reported by Aloe et al. (2014).

Usage

data("Aloe14")

Format

A data frame with 16 observations on the following 14 variables.

Study

a factor with levels Betoret Brouwers & Tomic Bumen Chang Durr Evers et al. Friedman Gold Huk Kress Kumarakulasingam Martin et al. Ozdemir Skaalvik and Skaalvik Williams

Year

Year of publication

EE

Emotional exhaustion

DP

Depersonalization

PA

(Lowered) personal accomplishment

V_EE

Sampling variance of emotional exhaustion

V_DP

Sampling variance of depersonalization

V_PA

Sampling variance of (lowered) personal accomplishment

C_EE_DP

Sampling covariance between EE and DP

C_EE_PA

Sampling covariance between EE and PA

C_DP_PA

Sampling covariance between DP and PA

Publication_type

Either Dissertation or Journal

Percentage_females

Percentage of females in the study

Years_experience

Average years of experience

Source

Aloe, A. M., Amo, L. C., & Shanahan, M. E. (2014). Classroom management self-efficacy and burnout: A multivariate meta-analysis. Educational Psychology Review, 26(1), 101-126. doi:10.1007/s10648-013-9244-0

Examples

data(Aloe14)

## Random-effects meta-analysis
meta1 <- meta(cbind(EE,DP,PA),
              cbind(V_EE, C_EE_DP, C_EE_PA, V_DP, C_DP_PA, V_PA),
              data=Aloe14)
## Remove error code
meta1 <- rerun(meta1)

summary(meta1)

## Extract the coefficients for the variance component of the random effects
coef1 <- coef(meta1, select="random")

## Convert it into a symmetric matrix by row major
my.cov <- vec2symMat(coef1, byrow=TRUE)

## Convert it into a correlation matrix
cov2cor(my.cov)

## Plot the multivariate effect sizes
plot(meta1)

Description

It compares nested models with the likelihood ratio statistic from various objects. It is a wrapper of mxCompare.

Usage

## S3 method for class 'wls'
anova(object, ..., all=FALSE)
## S3 method for class 'meta'
anova(object, ..., all=FALSE)
## S3 method for class 'meta3LFIML'
anova(object, ..., all=FALSE)
## S3 method for class 'reml'
anova(object, ..., all=FALSE)
## S3 method for class 'osmasem'
anova(object, ..., all=FALSE)
## S3 method for class 'osmasem2'
anova(object, ..., all=FALSE)
## S3 method for class 'mxsem'
anova(object, ..., all=FALSE)

Arguments

Value

A table of comparisons between the models in base and comparison.

Note

When the objects are class wls, the degrees of freedom in the base and comparison models are incorrect, while the degrees of freedom of the difference between them is correct. If users want to obtain the correct degrees of freedom in the base and comparison models, they may individually apply the summary function on the base and comparison models.

Author(s)

Mike W.-L. Cheung <[email protected]>

Examples

## Test the significance of a predictor with likelihood ratio test
## Model0: No predictor
model0 <- meta(y=yi, v=vi, data=Hox02, model.name="No predictor")

## Model1: With a predictor
model1 <- meta(y=yi, v=vi, x=weeks, data=Hox02, model.name="One predictor")

## Compare these two models
anova(model1, model0)

Description

It converts a character matrix into MxAlgebra object.

Usage

as.mxAlgebra(x, startvalues=NULL, lbound=NULL, ubound=NULL, name="X")

Arguments

Details

Suppose the name argument is "X", the output is a list of the following elements.

Value

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

as.mxMatrix, mxAlgebra

Examples

## a, b, and c are free parameters
(A1 <- matrix(c(1, "a*b", "a^b", "exp(c)"), ncol=2, nrow=2))
##      [,1]  [,2]    
## [1,] "1"   "a^b"   
## [2,] "a*b" "exp(c)"

A <- as.mxAlgebra(A1, startvalues=list(a=1, b=2),
                  lbound=list(a=0), ubound=list(b=1, c=2),
                  name="A")

## An object of mxAlgebra
A$mxalgebra
## mxAlgebra 'A' 
## $formula:  rbind(cbind(A1_1, A1_2), cbind(A2_1, A2_2)) 
## $result: (not yet computed) <0 x 0 matrix>
## dimnames: NULL

## A matrix of parameters
A$parameters
## FullMatrix 'Avars' 

## $labels
##      [,1]
## [1,] "a" 
## [2,] "b" 
## [3,] "c" 

## $values
##      [,1]
## [1,] 1   
## [2,] 2   
## [3,] 0   

## $free
##      [,1]
## [1,] TRUE
## [2,] TRUE
## [3,] TRUE

## $lbound
##      [,1]
## [1,]    0
## [2,]   NA
## [3,]   NA

## $ubound
##      [,1]
## [1,]   NA
## [2,]    1
## [3,]    2

## A list of matrices of elements for the mxAlgebra
A$list
## $A1_1
## mxAlgebra 'A1_1' 
## $formula:  1 
## $result: (not yet computed) <0 x 0 matrix>
## dimnames: NULL

## $A2_1
## mxAlgebra 'A2_1' 
## $formula:  a * b 
## $result: (not yet computed) <0 x 0 matrix>
## dimnames: NULL

## $A1_2
## mxAlgebra 'A1_2' 
## $formula:  a^b 
## $result: (not yet computed) <0 x 0 matrix>
## dimnames: NULL

## $A2_2
## mxAlgebra 'A2_2' 
## $formula:  exp(c) 
## $result: (not yet computed) <0 x 0 matrix>
## dimnames: NULL

Description

It converts a matrix into MxMatrix-class via mxMatrix.

Usage

as.mxMatrix(x, name, ...)

Arguments

Details

If there are non-numeric values in x, they are treated as the labels of the parameters. If a "*" is present, the numeric value on the left-hand side will be treated as the starting value for a free parameter. If an "@" is present, the numeric value on the left-hand side will be considered as the value for a fixed parameter. If it is a matrix of numeric values, there are no free parameters in the output matrix.

Value

A MxMatrix-class object with the same dimensions as x

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

mxMatrix, create.mxMatrix, create.Fmatrix, checkRAM, lavaan2RAM, as.symMatrix

Examples

## a and b are free parameters with starting values and labels
(a1 <- matrix(c(1:4, "5*a", 6, "7*b", 8, 9), ncol=3, nrow=3))
#      [,1] [,2]  [,3] 
# [1,] "1"  "4"   "7*b"
# [2,] "2"  "5*a" "8"  
# [3,] "3"  "6"   "9"  

a1 <- as.mxMatrix(a1)

## a and b are fixed parameters without any labels, name="new2"
(a2 <- matrix(1:9, ncol=3, nrow=3))
#      [,1] [,2] [,3]
# [1,]    1    4    7
# [2,]    2    5    8
# [3,]    3    6    9

new2 <- as.mxMatrix(a2, name="new2")

## Free parameters without starting values
(a3 <- matrix(c(1:4, "*a", 6, "*b", 8, 9), ncol=3, nrow=3))
#      [,1] [,2] [,3]
# [1,] "1"  "4"  "*b"
# [2,] "2"  "*a" "8" 
# [3,] "3"  "6"  "9" 

a3 <- as.mxMatrix(a3, lbound=0)

## A free parameter without label
(a4 <- matrix(c(1:4, "5*", 6, "7*b", 8, 9), ncol=3, nrow=3))
#      [,1] [,2] [,3] 
# [1,] "1"  "4"  "7*b"
# [2,] "2"  "5*" "8"  
# [3,] "3"  "6"  "9"  

a4 <- as.mxMatrix(a4)

## Convert a scalar into mxMatrix object
## "name" is required as "3*a" is not a valid name.
(a5 <- as.mxMatrix("3*a", name="a5"))

## Free and fixed parameters
(a6 <- matrix(c(1, "2*a", "3@b", 4), ncol=2, nrow=2))

as.mxMatrix(a6)

Description

It converts a character matrix with starting values to a character matrix without the starting values.

Usage

as.symMatrix(x)

Arguments

Details

If there are non-numeric values in x, they are treated as the labels of the free parameters. If a "*" is present, the numeric value on the left-hand side will be treated as the starting value for a free parameter or a fixed value for a fixed parameter. If it is a matrix of numeric values, there are no free parameters in the output matrix. This function removes the starting values and "*" in the matrices.

Value

A character matrix.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

as.mxMatrix

Examples

## a and b are free parameters with starting values and labels
(a1 <- matrix(c(1:4, "5*a", 6, "7*b", 8, 9), ncol=3, nrow=3))
#      [,1] [,2]  [,3] 
# [1,] "1"  "4"   "7*b"
# [2,] "2"  "5*a" "8"  
# [3,] "3"  "6"   "9"  

(as.symMatrix(a1))
#      [,1] [,2] [,3]
# [1,] "1"  "4"  "b" 
# [2,] "2"  "a"  "8" 
# [3,] "3"  "6"  "9"

Description

It computes the asymptotic sampling covariance matrix of a correlation/covariance matrix under the assumption of multivariate normality.

Usage

asyCov(x, n, cor.analysis = TRUE, as.matrix = TRUE,
       acov=c("weighted", "individual", "unweighted"), ...)
asyCovOld(x, n, cor.analysis = TRUE, dropNA = FALSE, as.matrix = TRUE,
       acov=c("individual", "unweighted", "weighted"),
       suppressWarnings = TRUE,  silent = TRUE, run = TRUE, ...)

Arguments

Value

An asymptotic covariance matrix of the vectorized correlation/covariance matrix or a list of these matrices. If as.matrix=TRUE and x is a list of matrices, the output is a stacked matrix.

Note

Before 1.2.6, asyCov used an SEM approach based on Cheung and Chan (2004). After 1.2.6, asyCov was rewritten based on Olkin and Siotani (1976) for correlation matrix and Yuan and Bentler (2007, p. 371) for covariance matrix. Arguments such as dropNA, suppressWarnings, silent, and run were dropped. The original version was renamed to asyCovOld for compatibility.

Author(s)

Mike W.-L. Cheung <[email protected]>

References

Cheung, M. W.-L., & Chan, W. (2004). Testing dependent correlation coefficients via structural equation modeling. Organizational Research Methods, 7, 206-223.

Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. In S. Ideka (Ed.), Essays in probability and statistics (pp. 235-251). Shinko Tsusho.

Yuan, K.-H., & Bentler, P. M. (2007). Robust procedures in structural equation modeling. In S.-Y. Lee (Ed.), Handbook of Latent Variable and Related Models (pp. 367-397). Elsevier/North-Holland.

Examples

C1 <- matrix(c(1,0.5,0.4,0.5,1,0.2,0.4,0.2,1), ncol=3)  
asyCov(C1, n=100)

## Data with missing values
C2 <- matrix(c(1,0.4,NA,0.4,1,NA,NA,NA,NA), ncol=3)  
C3 <- matrix(c(1,0.2,0.2,1), ncol=2)

## Output is a stacked matrix of asymptotic covariance matrices
asyCov(list(C1,C2), n=c(100,50), as.matrix=TRUE)

## Output is a stacked matrix of asymptotic covariance matrices
asyCov(list(C3,C3), n=c(100,50), as.matrix=TRUE)

## Output is a list of asymptotic covariance matrices using the old version
asyCovOld(list(C1,C2,C3), n=c(100,50,50), dropNA=TRUE, as.matrix=FALSE)

Description

This dataset includes 13 studies on the effectiveness of the Bacillus Calmette-Guerin (BCG) vaccine for preventing tuberculosis (see van Houwelingen, Arends, & Stijnen (2002) for details).

Usage

data(BCG)

Details

A list of data with the following structure:

Trial

Number of the trials

Author

Authors of the original studies

Year

Year of publication

VD

Vaccinated group with disease

VWD

Vaccinated group without the disease

NVD

Not vaccinated group with disease

NVWD

Not vaccinated group without the disease

Latitude

Geographic latitude of the place where the study was done

Allocation

Method of treatment allocation

ln_OR

Natural logarithm of the odds ratio: log((VD/VWD)/(NVD/NVWD))

v_ln_OR

Sampling variance of ln_OR: 1/VD+1/VWD+1/NVD+1/NVWD

ln_Odd_V

Natural logarithm of the odds of the vaccinated group: log(VD/VWD)

ln_Odd_NV

Natural logarithm of the odds of the not vaccinated group: log(NVD/NVWD)

v_ln_Odd_V

Sampling variance of ln_Odd_V: 1/VD+1/VWD

cov_V_NV

Sampling covariance between ln_Odd_V and ln_Odd_NV: It is always 0

v_ln_Odd_NV

Sampling variance of ln_Odd_NV: 1/NVD+1/NVWD

Source

Colditz, G. A., Brewer, T. F., Berkey, C. S., Wilson, M. E., Burdick, E., Fineberg, H. V., & Mosteller, F. (1994). Efficacy of BCG vaccine in the prevention of tuberculosis: Meta-analysis of the published literature. Journal of the American Medical Association, 271, 698–702.

References

Berkey, C. S., Hoaglin, D. C., Mosteller, F., & Colditz, G. A. (1995). A random-effects regression model for meta-analysis. Statistics in Medicine, 14, 395–411.

van Houwelingen, H. C., Arends, L. R., & Stijnen, T. (2002). Advanced methods in meta-analysis: Multivariate approach and meta-regression. Statistics in Medicine, 21, 589–624.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48. https://www.jstatsoft.org/v36/i03/.

Examples

data(BCG)

## Univariate meta-analysis on the log of the odds ratio
summary( meta(y=ln_OR, v=v_ln_OR, data=BCG,
              x=cbind(scale(Latitude,scale=FALSE),
              scale(Year,scale=FALSE))) )

## Multivariate meta-analysis on the log of the odds
## The conditional sampling covariance is 0
bcg <- meta(y=cbind(ln_Odd_V, ln_Odd_NV), data=BCG,
            v=cbind(v_ln_Odd_V, cov_V_NV, v_ln_Odd_NV))
summary(bcg)

plot(bcg)

Description

It creates a block diagonal matrix from a list of numeric or character matrices.

Usage

bdiagMat(x)

Arguments

Value

A numeric or character block diagonal matrix

Author(s)

Mike W.-L. Cheung <[email protected]>

References

It was based on a function posted by Scott Chasalow at http://www.math.yorku.ca/Who/Faculty/Monette/pub/stmp/0827.html.

See Also

bdiagRep, matrix2bdiag

Examples

## Block diagonal matrix of numbers
bdiagMat( list(matrix(1:4,nrow=2,ncol=2),
               matrix(5:6,nrow=1,ncol=2)) )
#      [,1] [,2] [,3] [,4]
# [1,]    1    3    0    0
# [2,]    2    4    0    0
# [3,]    0    0    5    6

## Block diagonal matrix of characters
bdiagMat( list(matrix(letters[1:4],nrow=2,ncol=2),
               matrix(letters[5:6],nrow=1,ncol=2)) )
#      [,1] [,2] [,3] [,4]
# [1,] "a"  "c"  "0"  "0" 
# [2,] "b"  "d"  "0"  "0" 
# [3,] "0"  "0"  "e"  "f"

Description

It creates a block diagonal matrix by repeating the input matrix several times.

Usage

bdiagRep(x, times)

Arguments

Value

A numeric or character block diagonal matrix

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

bdiagMat, matrix2bdiag

Examples

## Block diagonal matrix of numerics
bdiagRep( matrix(1:4,nrow=2,ncol=2), 2 )
#      [,1] [,2] [,3] [,4]
# [1,]    1    3    0    0
# [2,]    2    4    0    0
# [3,]    0    0    1    3
# [4,]    0    0    2    4

## Block diagonal matrix of characters
bdiagRep( matrix(letters[1:4],nrow=2,ncol=2), 2 )
#      [,1] [,2] [,3] [,4]
# [1,] "a"  "c"  "0"  "0" 
# [2,] "b"  "d"  "0"  "0" 
# [3,] "0"  "0"  "a"  "c" 
# [4,] "0"  "0"  "b"  "d"

Description

This dataset includes ten studies on the relationships between CSAI subscales and sports behavior. The original data were used in Craft et al. (2003), whereas a subset of them was illustrated in Becker (2009).

Usage

data("Becker09")

Details

A list of data with the following structure:

data

A list of 4x4 correlation matrices. The variables are Performance, Cognitive, Somatic, and Self_confidence

n

A vector of sample sizes

Type_of_sport

Samples based on Individual or Team

Source

Craft, L. L., Magyar, T. M., Becker, B. J., & Feltz, D. L. (2003). The relationship between the Competitive State Anxiety Inventory-2 and sport performance: a meta-analysis. Journal of Sport and Exercise Psychology, 25(1), 44-65.

References

Becker, B. J. (2009). Model-based meta-analysis. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 377-395). New York: Russell Sage Foundation.

Examples

data(Becker09)

#### Fixed-effects model
## First stage analysis
fixed1 <- tssem1(Becker09$data, Becker09$n, method="FEM")
summary(fixed1)

## Prepare a regression model using create.mxMatrix()
A1 <- create.mxMatrix(c(0, "0.1*Cog2Per", "0.1*SO2Per", "0.1*SC2Per",
                        0, 0, 0, 0,
                        0, 0, 0, 0,
                        0, "0.1*Cog2SC", "0.1*SO2SC",0),
                      type="Full", byrow=TRUE, ncol=4, nrow=4,
                      as.mxMatrix=FALSE)

## This step is not necessary but it is useful for inspecting the model.
dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- c("Performance", "Cognitive",
                                            "Somatic", "Self_confidence")

## Display A1
A1

S1 <- create.mxMatrix(c("0.1*var_Per",
                        0, 1,
                        0, "0.1*cor", 1,
                        0, 0, 0, "0.1*var_SC"), byrow=TRUE, type="Symm",
                      as.mxMatrix=FALSE)

## This step is not necessary but it is useful for inspecting the model.
dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- c("Performance", "Cognitive",
                                            "Somatic", "Self_confidence")

## Display S1
S1

################################################################################
## Alternative model specification in lavaan model syntax
model <- "## Regression paths
          Performance ~ Cog2Per*Cognitive + SO2Per*Somatic + SC2Per*Self_confidence
          Self_confidence ~ Cog2SC*Cognitive + SO2SC*Somatic
          ## Fix the variances of Cog and SO at 1
          Cognitive ~~ 1*Cognitive
          Somatic ~~ 1*Somatic
          ## Label the correlation between Cog and SO
          Cognitive ~~ cor*Somatic
          ## Label the error variances of Per and SC
          Performance ~~ var_Per*Performance
          Self_confidence ~~ var_SC*Self_confidence"

## Display the model
plot(model, layout="spring")

RAM <- lavaan2RAM(model, obs.variables=c("Performance", "Cognitive",
                                         "Somatic", "Self_confidence"))
RAM

A1 <- RAM$A
S1 <- RAM$S
################################################################################

## Second stage analysis
fixed2 <- tssem2(fixed1, Amatrix=A1, Smatrix=S1, diag.constraints=TRUE,
                 intervals.type="LB", model.name="TSSEM2 Becker09",
                 mx.algebras=list( Cog=mxAlgebra(Cog2SC*SC2Per, name="Cog"),
                                   SO=mxAlgebra(SO2SC*SC2Per, name="SO"),
                                   Cog_SO=mxAlgebra(Cog2SC*SC2Per+SO2SC*SC2Per,
                                   name="Cog_SO")) )
summary(fixed2)

## Display the model with the parameter estimates
plot(fixed2, layout="spring")

#### Fixed-effects model: with type of sport as cluster
## First stage analysis
cluster1 <- tssem1(Becker09$data, Becker09$n, method="FEM",
                   cluster=Becker09$Type_of_sport)
summary(cluster1)

## Second stage analysis
cluster2 <- tssem2(cluster1, Amatrix=A1, Smatrix=S1, diag.constraints=TRUE,
                 intervals.type="LB", model.name="TSSEM2 Becker09",
                 mx.algebras=list( Cog=mxAlgebra(Cog2SC*SC2Per, name="Cog"),
                                   SO=mxAlgebra(SO2SC*SC2Per, name="SO"),
                                   Cog_SO=mxAlgebra(Cog2SC*SC2Per+SO2SC*SC2Per,
                                   name="Cog_SO")) )
summary(cluster2)

## Convert the model to semPlotModel object with 2 plots
## Use the short forms of the variable names
my.plots <- lapply(X=cluster2, FUN=meta2semPlot, manNames=c("Per","Cog","SO","SC") )

## Load the library
library("semPlot")

## Setup two plots
layout(t(1:2))
## The labels are overlapped. We may modify it by using layout="spring"
semPaths(my.plots[[1]], whatLabels="est", nCharNodes=10, color="orange",
         layout="spring", edge.label.cex=0.8)
title("Individual sport")
semPaths(my.plots[[2]], whatLabels="est", nCharNodes=10, color="skyblue",
         layout="spring", edge.label.cex=0.8)
title("Team sport")


#### Random-effects model
## First stage analysis
random1 <- tssem1(Becker09$data, Becker09$n, method="REM", RE.type="Diag")
summary(random1)

## Second stage analysis
random2 <- tssem2(random1, Amatrix=A1, Smatrix=S1, diag.constraints=TRUE,
                  intervals.type="LB", model.name="TSSEM2 Becker09",
                  mx.algebras=list( Cog=mxAlgebra(Cog2SC*SC2Per, name="Cog"),
                                    SO=mxAlgebra(SO2SC*SC2Per, name="SO"),
                                    Cog_SO=mxAlgebra(Cog2SC*SC2Per+SO2SC*SC2Per,
                                    name="Cog_SO")) )
summary(random2)

## Display the model
plot(random2, what="path", layout="spring")

## Display the model with the parameter estimates
plot(random2, layout="spring", color="yellow")

#### Univariate r approach
#### First stage of the analysis
uni1 <- uniR1(Becker09$data, Becker09$n)
uni1

#### Second stage of analysis using OpenMx
model2 <- "## Regression paths
           Performance ~ Cog2Per*Cognitive + SO2Per*Somatic + SC2Per*Self_confidence
           Self_confidence ~ Cog2SC*Cognitive + SO2SC*Somatic
           ## Provide starting values for Cog and SO
           Cognitive ~~ start(1)*Cognitive
           Somatic ~~ start(1)*Somatic
           ## Label the correlation between Cog and SO
           Cognitive ~~ cor*Somatic
           ## Label the error variances of Per and SC
           Performance ~~ var_Per*Performance
           Self_confidence ~~ var_SC*Self_confidence"

RAM2 <- lavaan2RAM(model2, obs.variables=c("Performance", "Cognitive",
                                           "Somatic", "Self_confidence"))
RAM2

uni2mx <- uniR2mx(uni1, RAM=RAM2)
summary(uni2mx)

#### Second stage of analysis Using lavaan
model3 <- "## Regression paths
           Performance ~ Cognitive + Somatic + Self_confidence
           Self_confidence ~ Cognitive + Somatic"

uni2lavaan <- uniR2lavaan(uni1, model3)
lavaan::summary(uni2lavaan)

Description

The data set includes studies on sex differences in conformity using the fictitious norm group paradigm reported by Becker (1983).

Usage

data(Becker83)

Details

The variables are:

study

study number

di

Standardized mean difference

vi

Sampling variance of the effect size

percentage

Percentage of male authors

items

Number of items

Source

Becker, B. J. (1983, April). Influence again: A comparison of methods for meta-analysis. Paper presented at the annual meeting of the American Educational Research Association, Montreal.

Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando, FL: Academic Press.

References

Cheung, M. W.-L. (2010). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling, 17, 481-509.

Examples

data(Becker83)

## Random-effects meta-analysis
summary( meta(y=di, v=vi, data=Becker83) )

## Mixed-effects meta-analysis with log(items) as the predictor
summary( meta(y=di, v=vi, x=log(items), data=Becker83) )

Description

This data set includes six studies of correlation matrices reported by Becker (1992; 1995).

Usage

data(Becker92)

Details

A list of data with the following structure:

data

A list of 6 studies of correlation matrices. The variables are Math (math aptitude), Spatial (spatial ability), and Verbal (verbal ability)

n

A vector of sample sizes

Source

Becker, B. J. (1992). Using results from replicated studies to estimate linear models. Journal of Educational Statistics, 17(4), 341-362. doi:10.3102/10769986017004341

Becker, B. J. (1995). Corrections to "Using Results from Replicated Studies to Estimate Linear Models." Journal of Educational and Behavioral Statistics, 20(1), 100-102. doi:10.2307/1165390

Examples

data(Becker92)

#### Fixed-effects model
## First stage analysis
## Replicate Becker's (1992) analysis using 4 studies only

fixed1 <- tssem1(Becker92$data[1:4], Becker92$n[1:4], method="FEM")
summary(fixed1)

## ## Prepare a regression model using create.mxMatrix()
## A1 <- create.mxMatrix(c(0,0,0,"0.2*Spatial2Math",
##                         0,0,"0.2*Verbal2Math",0,0), type="Full",
##                         ncol=3, nrow=3, as.mxMatrix=FALSE)

## var.names <- c("Math_aptitude","Spatial","Verbal") 

## ## This step is not necessary but it is useful for inspecting the model.
## dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- var.names

## ## Display A1
## A1

## S1 <- create.mxMatrix(c("0.2*ErrorVarMath",0,0,1,"0.2*CorSpatialVerbal",1),
##                         type="Symm", as.mxMatrix=FALSE)

## ## This step is not necessary but it is useful for inspecting the model.
## dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- var.names

## ## Display S1
## S1

################################################################################
## Alternative model specification in lavaan model syntax
model <- "## Regression paths
          Math ~ Spatial2Math*Spatial + Verbal2Math*Verbal
          Spatial ~~ CorSpatialVerbal*Verbal
          ## Fix the variances of Spatial and Verbal at 1
          Spatial ~~ 1*Spatial
          Verbal ~~ 1*Verbal
          ## Label the error variance of Math
          Math ~~ ErrorVarMath*Math + start(0.2)*Math"

## Display the model
plot(model)

RAM <- lavaan2RAM(model, obs.variables=c("Math", "Spatial", "Verbal"))
RAM

################################################################################

## Fixed-effects model: Second stage analysis
## Two equivalent versions to calculate the R2 and its 95% LBCI
fixed2 <- tssem2(fixed1, RAM=RAM, intervals.type="LB",
                 mx.algebras=list(R1=mxAlgebra(Spatial2Math^2+Verbal2Math^2
                        +2*CorSpatialVerbal*Spatial2Math*Verbal2Math, name="R1"),
                        R2=mxAlgebra(One-Smatrix[1,1], name="R2"),
                        One=mxMatrix("Iden", ncol=1, nrow=1, name="One")))
summary(fixed2)

## Display the model with the parameter estimates
plot(fixed2)

#### Random-effects model
## First stage analysis
## No random effects for off-diagonal elements
random1 <- tssem1(Becker92$data, Becker92$n, method="REM", RE.type="Diag")
summary(random1)

## Random-effects model: Second stage analysis
random2 <- tssem2(random1, RAM=RAM)
summary(random2)

## Display the model with the parameter estimates
plot(random2, color="yellow")

#### Similar to conventional fixed-effects GLS approach
## First stage analysis
## No random effects
## Replicate Becker's (1992) analysis using 4 studies only
gls1 <- tssem1(Becker92$data[1:4], Becker92$n[1:4], method="REM", RE.type="Zero",
               model.name="Fixed effects GLS Stage 1")
summary(gls1)

## Fixed-effects GLS model: Second stage analysis
gls2 <- tssem2(gls1, RAM=RAM, model.name="Fixed effects GLS Stage 2")
summary(gls2)

Description

This data set includes five studies of ten correlation matrices reported by Becker and Schram (1994).

Usage

data(Becker94)

Details

A list of data with the following structure:

data

A list of 10 correlation matrices. The variables are Math (math aptitude), Spatial (spatial ability), and Verbal (verbal ability)

n

A vector of sample sizes

gender

Females or Males samples

Source

Becker, B. J., & Schram, C. M. (1994). Examining explanatory models through research synthesis. In H. Cooper & L. V. Hedges (Eds.), The handbook of research synthesis (pp. 357-381). New York: Russell Sage Foundation.

Examples

data(Becker94)

#### Fixed-effects model
## First stage analysis
fixed1 <- tssem1(Becker94$data, Becker94$n, method="FEM")
summary(fixed1)

## Prepare a regression model using create.mxMatrix()
## A1 <- create.mxMatrix(c(0,0,0,"0.2*Spatial2Math",
##                         0,0,"0.2*Verbal2Math",0,0), type="Full",
##                       ncol=3, nrow=3, name="A1")
## S1 <- create.mxMatrix(c("0.2*ErrorVarMath",0,0,1,
##                         "0.2*CorBetweenSpatialVerbal",1),
##                       type="Symm", name="S1")

## An alternative method to create a regression model with the lavaan syntax
model <- "## Regression model
          Math ~ Spatial2Math*Spatial + Verbal2Math*Verbal
          ## Error variance of Math
          Math ~~ ErrorVarMath*Math
          ## Variances of Spatial and Verbal fixed at 1.0
          Spatial ~~ 1*Spatial
          Verbal ~~ 1*Verbal
          ## Correlation between Spatial and Verbal
          Spatial ~~ CorBetweenSpatialVerbal*Verbal"

## Display the model
plot(model)

RAM <- lavaan2RAM(model, obs.variables=c("Math", "Spatial", "Verbal"))
RAM

## Second stage analysis
## A1 <- RAM$A
## S1 <- RAM$S
## fixed2 <- tssem2(fixed1, Amatrix=A1, Smatrix=S1, intervals.type="LB")

fixed2 <- tssem2(fixed1, RAM=RAM, intervals.type="LB")
summary(fixed2)

## Display the model with the parameter estimates
plot(fixed2)
    
#### Fixed-effects model: with gender as cluster
## First stage analysis
cluster1 <- tssem1(Becker94$data, Becker94$n, method="FEM", cluster=Becker94$gender)
summary(cluster1)

## Second stage analysis  
cluster2 <- tssem2(cluster1, RAM=RAM, intervals.type="LB")
summary(cluster2)

#### Conventional fixed-effects GLS approach
## First stage analysis
## No random effects
## Replicate Becker's (1992) analysis using 4 studies only
gls1 <- tssem1(Becker92$data[1:4], Becker92$n[1:4], method="REM", RE.type="Zero",
               model.name="Fixed effects GLS Stage 1")
summary(gls1)

## Fixed-effects GLS model: Second stage analysis
gls2 <- tssem2(gls1, RAM=RAM, intervals.type="LB",
               model.name="Fixed effects GLS Stage 2")
summary(gls2)

Description

The data set includes five published trials, reported by Berkey et al. (1998), comparing surgical and non-surgical treatments for medium-severity periodontal disease, one year after treatment.

Usage

data(Berkey98)

Details

The variables are:

trial

Trial number

pub_year

Publication year

no_of_patients

Number of patients

PD

Patient improvements (mm) in probing depth

AL

Patient improvements (mm) in attachment level

var_PD

Sampling variance of PD

cov_PD_AL

Sampling covariance between PD and AD

var_AL

Sampling variance of AL

Source

Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F, & Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. Statistics in Medicine, 17, 2537-2550.

Examples

data(Berkey98)

#### ML estimation method
## Multivariate meta-analysis
x <- meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL), data=Berkey98)
x <- rerun(x)
summary(x)
plot(x)

## Plot individual studies proportional to the weights
plot(x, study.weight.plot=TRUE)

## Include forest plot from the metafor package
library(metafor)
plot(x, diag.panel=TRUE, main="Multivariate meta-analysis",
     axis.label=c("PD", "AL"))
     forest( rma(yi=PD, vi=var_PD, data=Berkey98) )
     title("Forest plot of PD")
     forest( rma(yi=AL, vi=var_AL, data=Berkey98) )
     title("Forest plot of AL")

## Multivariate meta-analysis with "publication year-1979" as the predictor
summary( meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL),
              x=scale(pub_year, center=1979), data=Berkey98,
              RE.lbound=NA) )

## Multivariate meta-analysis with equality constraint on the regression coefficients
summary( meta(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL),
              x=scale(pub_year, center=1979), data=Berkey98,
              coef.constraints=matrix(c("0.3*Eq_slope", "0.3*Eq_slope"),
              nrow=2)) )

#### REML estimation method
## Multivariate meta-analysis
summary( reml(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL),
              data=Berkey98,
              model.name="Multivariate meta analysis with REML") )

## Multivariate meta-analysis with "publication year-1979" as the predictor
## Diagonal structure for the variance component
summary( reml(y=cbind(PD, AL), v=cbind(var_PD, cov_PD_AL, var_AL),
              RE.constraints=Diag(c("1e-5*Tau2_1_1", "1e-5*Tau2_2_2")),
              x=scale(pub_year, center=1979), data=Berkey98) )

Description

The data set includes correlation matrices of leader-member exchange in transformational leadership reported by Boer et al. (2016).

Usage

data(Boer16)

Details

A list of data with the following structure:

data

A list of correlation matrices. The variables are LMX (leader-member exchange), TFL (transformational leadership), JS (job satisfaction), OC (organizational commitment), and LE (leader effectiveness)

n

A vector of sample sizes

RelLMX

The reliability of LMX

RelTFL

The reliability of TFL

Source

Boer, D., Deinert, A., Homan, A. C., & Voelpel, S. C. (2016). Revisiting the mediating role of leader-member exchange in transformational leadership: the differential impact model. European Journal of Work and Organizational Psychology, 25(6), 883-899.

Examples

## Stage 1 analysis
rand1 <- tssem1(Boer16$data, Boer16$n, method="REM", RE.type="Diag",
                acov="weighted")
summary(rand1)

## Stage 2 analysis
model2a <- 'JS+OC+LE ~ LMX+TFL
            LMX ~ TFL
            ## Variance of TFL is fixed at 1
            TFL ~~ 1*TFL
            ## Correlated residuals
            JS ~~ OC
            JS ~~ LE
            OC ~~ LE'

## Display the model
plot(model2a)    
    
RAM2a <- lavaan2RAM(model2a, obs.variables = c("LMX", "TFL", "JS", "OC", "LE"),
                    A.notation="on", S.notation="with")

rand2a <- tssem2(rand1, Amatrix=RAM2a$A, Smatrix=RAM2a$S)
summary(rand2a)

## Display the model with the parameter estimates
plot(rand2a, layout="spring")

Description

It generates correlation matrices with the parametric bootstrap on the univariate R (uniR) object.

Usage

bootuniR1(x, Rep, nonPD.pop=c("replace", "nearPD", "accept"))

Arguments

Details

This function implements the parametric bootstrap approach suggested by Yu et al. (2016). It is included in this package for research interests. Please refer to Cheung (2018) for the issues associated with this parametric bootstrap approach.

Value

An object of the generated correlation matrices.

Author(s)

Mike W.-L. Cheung <[email protected]>

References

Cheung, M. W.-L. (2018). Issues in solving the problem of effect size heterogeneity in meta-analytic structural equation modeling: A commentary and simulation study on Yu, Downes, Carter, and O'Boyle (2016). Journal of Applied Psychology, 103, 787-803.

Yu, J. (Joya), Downes, P. E., Carter, K. M., & O'Boyle, E. H. (2016). The problem of effect size heterogeneity in meta-analytic structural equation modeling. Journal of Applied Psychology, 101, 1457-1473.

See Also

rCor, bootuniR2, Nohe15

Description

It fits structural equation models on the bootstrapped correlation matrices.

Usage

bootuniR2(model, data, n, ...)

Arguments

Details

This function fits the lavaan model with the bootstrapped correlation matrices. It implements the parametric bootstrap approach suggested by Yu et al. (2016). It is included in this package for research interests. Please refer to Cheung (2018) for the issues associated with this parametric bootstrap approach.

Value

A list of the fitted object from sem.

Author(s)

Mike W.-L. Cheung <[email protected]>

References

Cheung, M. W.-L. (2018). Issues in solving the problem of effect size heterogeneity in meta-analytic structural equation modeling: A commentary and simulation study on Yu, Downes, Carter, and O'Boyle (2016). Journal of Applied Psychology, 103, 787-803.

Yu, J. (Joya), Downes, P. E., Carter, K. M., & O'Boyle, E. H. (2016). The problem of effect size heterogeneity in meta-analytic structural equation modeling. Journal of Applied Psychology, 101, 1457-1473.

See Also

bootuniR2, tssemParaVar, Nohe15

Description

A dataset from Bornmann et al. (2007) for three-level meta-analysis.

Usage

data(Bornmann07)

Details

The variables are:

ID

ID of the study

Study

Study name

Cluster

Cluster for effect sizes

logOR

Effect size: log odds ratio

v

Sampling variance of logOR

Year

Year of publication

Type

Type of proposal: either Grant or Fellowship

Discipline

Discipline of the proposal: either Physical sciences, Life sciences/biology, Social sciences/humanities or Multidisciplinary)

Country

Country of the proposal: either the United States, Canada, Australia, United Kingdom or Europe

Source

Bornmann, L., Mutz, R., & Daniel, H.-D. (2007). Gender differences in grant peer review: A meta-analysis. Journal of Informetrics, 1(3), 226-238. doi:10.1016/j.joi.2007.03.001

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.

Marsh, H. W., Bornmann, L., Mutz, R., Daniel, H.-D., & O'Mara, A. (2009). Gender Effects in the Peer Reviews of Grant Proposals: A Comprehensive Meta-Analysis Comparing Traditional and Multilevel Approaches. Review of Educational Research, 79(3), 1290-1326. doi:10.3102/0034654309334143

Examples

data(Bornmann07)

#### ML estimation method
## No predictor
summary( meta3L(y=logOR, v=v, cluster=Cluster, data=Bornmann07) )

## Type as a predictor
## Grant: 0
## Fellowship: 1
summary( meta3L(y=logOR, v=v, x=(as.numeric(Type)-1),
               cluster=Cluster, data=Bornmann07) )

## Centered Year as a predictor
summary( meta3L(y=logOR, v=v, x=scale(Year, scale=FALSE),
               cluster=Cluster, data=Bornmann07) )

#### REML estimation method
## No predictor
summary( reml3L(y=logOR, v=v, cluster=Cluster, data=Bornmann07) )

## Type as a predictor
## Grants: 0
## Fellowship: 1
summary( reml3L(y=logOR, v=v, x=(as.numeric(Type)-1),
                cluster=Cluster, data=Bornmann07) )

## Centered Year as a predictor
summary( reml3L(y=logOR, v=v, x=scale(Year, scale=FALSE),
                cluster=Cluster, data=Bornmann07) )

## Handling missing covariates with FIML
## MCAR
## Set seed for replication
set.seed(1000000)

## Copy Bornmann07 to my.df
my.df <- Bornmann07
## "Fellowship": 1; "Grant": 0
my.df$Type_MCAR <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)

## Create 17 out of 66 missingness with MCAR
my.df$Type_MCAR[sample(1:66, 17)] <- NA
summary(meta3LFIML(y=logOR, v=v, cluster=Cluster, x2=Type_MCAR, data=my.df))

## MAR
Type_MAR <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)

## Create 27 out of 66 missingness with MAR for cases Year<1996
index_MAR <- ifelse(Bornmann07$Year<1996, yes=TRUE, no=FALSE)
Type_MAR[index_MAR] <- NA

## Include auxiliary variable
summary(meta3LFIML(y=logOR, v=v, cluster=Cluster, x2=Type_MAR, av2=Year, data=my.df))

Description

It calculates effect sizes with Delta Method by formulating the effect sizes as functions of SEM in lavaan.

Usage

calEffSizes(model, data=NULL, n, Cov, Mean=NULL, group=NULL, lavaan.output=FALSE, ...)

Arguments

Value

Effect sizes and their sampling covariance matrix or a lavaan fitted object.

Note

The input matrices are treated as covariance matrices unless there are explicit constraints in the model.

Author(s)

Mike W.-L. Cheung <[email protected]>

References

Cheung, M. W.-L. (2015). Meta-analysis: A structural equation modeling approach. Chichester, West Sussex: John Wiley & Sons, Inc.

Cheung, M. W.-L. (2018). Computing multivariate effect sizes and their sampling covariance matrices with structural equation modeling: Theory, examples, and computer simulations. Frontiers in Psychology, 9(1387). https://doi.org/10.3389/fpsyg.2018.01387

See Also

smdMES, smdMTS

Examples

## Select ATT, Bi, and BEH 
obs.vars <- c("BEH", "BI", "ATT")

## Select one study from Cooke16 for illustration
my.cor <- Cooke16$data[[4]][obs.vars, obs.vars]
my.n  <- Cooke16$n[4]

## Effect sizes: indirect effect and direct effect
model <- "BEH ~ c*ATT + b*BI
          BI ~ a*ATT
          ## Indirect effect
          Ind := a*b
          Dir := c"

calEffSizes(model=model, n=my.n, Cov=my.cor, lavaan.output=FALSE)

## Return the lavaan fitted model
fit <- calEffSizes(model=model, n=my.n, Cov=my.cor, lavaan.output=TRUE)
lavaan::summary(fit)
    
lavaan::parameterestimates(fit)

Description

A dataset of multiple treatment effects of standardized mean differences on misinformation and debunking effects.

Usage

data(Chan17)

Format

A data frame with 34 independent samples from 6 research reports.

Author

a character vector of study

g_misinfo

Hedges' g of misinformation comparing the misinformation experimental and control groups

g_debunk

Hedges' g of debunking comparing the debuking experimental and misinformation experimental groups

v_misinfo

sampling variance of g_misinfo

c_mis_deb

Sampling covariance between g_misinfo and g_debunk due to the overlap of the misinformation experimental group

v_debunk

sampling variance of g_debunk

PublicationYear

publication year

Published

published or unpublished

MeanAge

mean age of participants

PctFemale

percentage of female participants

Details

The sampling variances and covariances are calculated using Gleser and Olkin's (2009) method for multiple treatment effects (Equations 3.3 and 3.4). Since the sample sizes of the misinformation, debunking, and control groups are not given, it is assumed they are equal.

Source

Chan, M. S., Jones, C. R., Hall Jamieson, K., & Albarracin, D. (2017). Debunking: A meta-analysis of the psychological efficacy of messages countering misinformation. Psychological Science, 28(11), 1531-1546. https://doi.org/10.1177/0956797617714579

References

Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis. (2nd ed., pp. 357-376). Russell Sage Foundation.

Description

It provides simple checks on the correctness of the RAM formulation.

Usage

checkRAM(Amatrix, Smatrix, cor.analysis=TRUE)

Arguments

Value

It returns silently if no error has been detected; otherwise, it returns a warning message.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

as.mxMatrix, lavaan2RAM

Examples

## Digman97 example
model1 <- "## Factor loadings
           Alpha=~A+C+ES
           Beta=~E+I
           ## Factor correlation
           Alpha~~Beta"

RAM1 <- lavaan2RAM(model1, obs.variables=c("A","C","ES","E","I"),
                   A.notation="on", S.notation="with")
RAM1

## The model is okay.    
checkRAM(Amatrix=RAM1$A, Smatrix=RAM1$S)

## Hunter83 example    
model2 <- "## Regression paths
           Job_knowledge ~ A2J*Ability
           Work_sample ~ A2W*Ability + J2W*Job_knowledge
           Supervisor ~ J2S*Job_knowledge + W2S*Work_sample

           ## Fix the variance of Ability at 1
           Ability ~~ 1*Ability

           ## Label the error variances of the dependent variables
           Job_knowledge ~~ VarE_J*Job_knowledge
           Work_sample ~~ VarE_W*Work_sample
           Supervisor ~~ VarE_S*Supervisor"

RAM2 <- lavaan2RAM(model2, obs.variables=c("Ability","Job_knowledge",
                   "Work_sample","Supervisor"))

## The model is okay.     
checkRAM(Amatrix=RAM2$A, Smatrix=RAM2$S)

Description

This data set includes fifty studies of correlation matrices on the theory of planned theory reported by Cheung and Chan (2000).

Usage

data(Cheung00)

Details

A list of data with the following structure:

data

A list of 50 studies of correlation matrices. The variables are the attitude toward behavior att, subjective norm sn, behavioral intention bi, and behavior beh

n

A vector of sample sizes

Note

These studies were extracted from the original data set for illustration purpose. Some samples contained two or more correlation matrices, and only one of them was arbitrarily selected to avoid the problem of dependence. Moreover, studies with less than 3 correlation coefficients were also excluded.

Source

Cheung, S.-F., & Chan, D. K.-S. (2000). The role of perceived behavioral control in predicting human behavior: A meta-analytic review of studies on the theory of planned behavior. Unpublished manuscript, Chinese University of Hong Kong.

References

Cheung, M.W.-L., & Cheung, S.-F. (2016). Random-effects models for meta-analytic structural equation modeling: Review, issues, and illustrations. Research Synthesis Methods, 7, 140-155.

Examples

data(Cheung00)

## Variable labels
labels <- colnames(Cheung00$data[[1]])

## Full mediation model
S <- create.mxMatrix(c("1", 
                       ".2*cov_att_sn", "1", 
                       0, 0, ".2*e_bi", 
                       0, 0, 0, ".2*e_beh"), 
                     type="Symm", as.mxMatrix=FALSE, byrow=TRUE)
dimnames(S) <- list(labels, labels)
S

A <- matrix(c("0","0","0","0",
              "0","0","0","0",
              ".2*att2bi", ".2*sn2bi", "0", "0",
              "0", "0", ".2*bi2beh", "0"),
            byrow=TRUE, 4, 4)
dimnames(A) <- list(labels, labels)
A

#### Random-effects model

## Stage 1 analysis
random_1 <- tssem1(Cheung00$data, Cheung00$n, method="REM", RE.type="Diag",
                   acov="weighted")
summary(random_1)

## Stage 2 analysis
random_2 <- tssem2(random_1, Amatrix=A, Smatrix=S, intervals.type="LB",
                   diag.constraints=TRUE)
summary(random_2)

## Display the model
plot(random_2, what="path")

## Display the model with the parameter estimates
plot(random_2, color="yellow")

## Load the library
library("semPlot")

Description

Four studies were selected from the data set used by Cheung and Chan (2005; 2009). Some variables were randomly deleted to illustrate the analysis with missing data.

Usage

data(Cheung09)

Details

A list of data with the following structure:

data

A list of 4 studies of correlation matrices

n

A vector of sample sizes

References

Cheung, M. W.-L., & Chan, W. (2005). Meta-analytic structural equation modeling: A two-stage approach. Psychological Methods, 10, 40-64.

Cheung, M. W.-L., & Chan, W. (2009). A two-stage approach to synthesizing covariance matrices in meta-analytic structural equation modeling. Structural Equation Modeling, 16, 28-53.

Examples

data(Cheung09)

#### Fixed-effects model: Stage 1 analysis
fixed1 <- tssem1(Cheung09$data, Cheung09$n, method="FEM")
summary(fixed1)
  
## Prepare a model implied matrix
## Factor correlation matrix
Phi <- create.mxMatrix( c("0.3*corf2f1","0.3*corf3f1","0.3*corf3f2"),
                        type="Stand", as.mxMatrix=FALSE )
## Error variances
Psi <- create.mxMatrix( paste("0.2*e", 1:9, sep=""), type="Diag",
                        as.mxMatrix=FALSE )

## Create Smatrix
S1 <- bdiagMat(list(Psi, Phi))
## dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- c(paste("x",1:9,sep=""),
##                                             paste("f",1:3,sep=""))
## S1
S1 <- as.mxMatrix(S1)

## Factor loadings
Lambda <- create.mxMatrix( c(".3*f1x1",".3*f1x2",".3*f1x3",rep(0,9),
                             ".3*f2x4",".3*f2x5",".3*f2x6",".3*f2x7",
                             rep(0,9),".3*f3x8",".3*f3x9"), type="Full",
                             ncol=3, nrow=9, as.mxMatrix=FALSE )
Zero1 <- matrix(0, nrow=9, ncol=9)
Zero2 <- matrix(0, nrow=3, ncol=12)

## Create Amatrix
A1 <- rbind( cbind(Zero1, Lambda),
             Zero2 )
## dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- c(paste("x",1:9,sep=""),
##                                             paste("f",1:3,sep=""))
## A1
A1 <- as.mxMatrix(A1)

## Create Fmatrix
F1 <- create.Fmatrix(c(rep(1,9), rep(0,3)))
  
#### Fixed-effects model: Stage 2 analysis
fixed2 <- tssem2(fixed1, Amatrix=A1, Smatrix=S1, Fmatrix=F1,
                 intervals.type="LB")
summary(fixed2)

## Display the model
plot(fixed2, what="path")

## Display the model with the parameter estimates
plot(fixed2, latNames=c("f1", "f2", "f3"), edge.label.cex=0.8,
     color="yellow")

Description

It extracts the parameter estimates from objects of various classes.

Usage

## S3 method for class 'tssem1FEM'
coef(object, ...)
## S3 method for class 'tssem1FEM.cluster'
coef(object, ...)
## S3 method for class 'tssem1REM'
coef(object, select = c("all", "fixed", "random"), ...)
## S3 method for class 'wls'
coef(object, ...)
## S3 method for class 'wls.cluster'
coef(object, ...)
## S3 method for class 'meta'
coef(object, select = c("all", "fixed", "random"), ...)
## S3 method for class 'meta3LFIML'
coef(object, select = c("all", "fixed", "random", "allX"), ...)
## S3 method for class 'reml'
coef(object, ...)
## S3 method for class 'osmasem'
coef(object, select=c("fixed", "all", "random"), ...)
## S3 method for class 'osmasem2'
coef(object, select=c("fixed", "all", "random"), ...)
## S3 method for class 'mxsem'
coef(object, ...)

Arguments

Value

Parameter estimates for both fixed-effects (if any) and random-effects (if any)

Note

coef.sem is simply a wraper of omxGetParameters. Extra arguments will be passed to it

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

tssem1, wls, meta, reml, omxGetParameters, osmasem

Examples

## Random-effects meta-analysis
model1 <- meta(y=yi, v=vi, data=Hox02)
coef(model1)

## Fixed-effects only
coef(model1, select="fixed")

Description

The data set includes correlation matrices on using the theory of planned behavior to predict alcohol consumption reported by Cooke et al. (2016).

Usage

data(Cooke16)

Details

A list of data with the following structure:

data

A list of correlation matrices. The variables are SN (subjective norm), ATT (attitude), PBC (perceived behavior control), BI (behavioral intention), and BEH (behavior).

n

A vector of sample sizes.

MeanAge

Mean age of the participants except for Ajzen and Sheikh (2013), which is the median age, and Glassman, et al. (2010a) to Glassman, et al. (2010d), which are based on the range of 18 to 24.

Female

Percentage of female participants.

Source

Cooke, R., Dahdah, M., Norman, P., & French, D. P. (2016). How well does the theory of planned behaviour predict alcohol consumption? A systematic review and meta-analysis. Health Psychology Review, 10(2), 148-167.

References

Cheung, M. W.-L., & Hong, R. Y. (2017). Applications of meta-analytic structural equation modeling in health psychology: Examples, issues, and recommendations. Health Psychology Review, 11, 265-279.

Examples

## Check whether the correlation matrices are valid (positive definite)
Cooke16$data[is.pd(Cooke16$data)==FALSE]

## Since the correlation matrix in Study 3 is not positive definite,
## we exclude it in the following analyses
my.data <- Cooke16$data[-3]
my.n <- Cooke16$n[-3]

## Show the no. of studies per correlation
pattern.na(my.data, show.na = FALSE)

## Show the total sample sizes per correlation
pattern.n(my.data, my.n)

## Stage 1 analysis
## Random-effects model
random1 <- tssem1(my.data, my.n, method="REM", RE.type="Diag", acov="weighted")
summary(random1)

A1 <- create.mxMatrix(c(0,0,0,0,0,
                        0,0,0,0,0,
                        0,0,0,0,0,
                        "0.2*SN2BI","0.2*ATT2BI","0.2*PBC2BI",0,0,
                        0,0,"0.2*PBC2BEH","0.2*BI2BEH",0),
                        type="Full", ncol=5, nrow=5,
                        byrow=TRUE, as.mxMatrix=FALSE)

## This step is not necessary but it is useful for inspecting the model.
dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- colnames(Cooke16$data[[1]])

## Display A1
A1

S1 <- create.mxMatrix(c(1,
                        "0.1*ATT_SN", 1,
                        "0.1*PBC_SN", "0.1*PBC_ATT", 1,
                        0, 0, 0, "0.5*VarBI",
                        0, 0, 0, 0, "0.5*VarBEH"),
                      type = "Symm", ncol=5, nrow=5,
                      byrow=TRUE, as.mxMatrix=FALSE)

dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- colnames(Cooke16$data[[1]])
S1

## Stage 2 analysis
random2 <- tssem2(random1, Amatrix=A1, Smatrix=S1, diag.constraints=FALSE,
                  intervals.type="LB")
summary(random2)

## Display the model
plot(random2, what="path")    
    
## Display the model with the parameter estimates
plot(random2, color="yellow")

Description

Fifty-six effect sizes from 11 districts from Cooper et al. (2003) were reported by Konstantopoulos (2011).

Usage

data(Cooper03)

Details

The variables are:

District

District ID

Study

Study ID

y

Effect size

v

Sampling variance

Year

Year of publication

Source

Cooper, H., Valentine, J. C., Charlton, K., & Melson, A. (2003). The Effects of Modified School Calendars on Student Achievement and on School and Community Attitudes. Review of Educational Research, 73(1), 1-52. doi:10.3102/00346543073001001

References

Konstantopoulos, S. (2011). Fixed effects and variance components estimation in three-level meta-analysis. Research Synthesis Methods, 2, 61-76. doi:10.1002/jrsm.35

Examples

data(Cooper03)

#### ML estimation method
## No predictor
summary( model1 <- meta3L(y=y, v=v, cluster=District, data=Cooper03) )

## Show all heterogeneity indices and their 95% confidence intervals
summary( meta3L(y=y, v=v, cluster=District, data=Cooper03,
               intervals.type="LB", I2=c("I2q", "I2hm", "I2am", "ICC")) )

## Year as a predictor
summary( meta3L(y=y, v=v, cluster=District, x=scale(Year, scale=FALSE),
               data=Cooper03, model.name="Year as a predictor") )

## Equality of level-2 and level-3 heterogeneity
summary( model2 <- meta3L(y=y, v=v, cluster=District, data=Cooper03,
                         RE2.constraints="0.2*EqTau2",
                         RE3.constraints="0.2*EqTau2",
                         model.name="Equal Tau2") )

## Compare model2 vs. model1
anova(model1, model2)

#### REML estimation method
## No predictor
summary( reml3L(y=y, v=v, cluster=District, data=Cooper03) )

## Level-2 and level-3 variances are constrained equally 
summary( reml3L(y=y, v=v, cluster=District, data=Cooper03,
               RE.equal=TRUE, model.name="Equal Tau2") )

## Year as a predictor
summary( reml3L(y=y, v=v, cluster=District, x=scale(Year, scale=FALSE),
               data=Cooper03, intervals.type="LB") )

## Handling missing covariates with FIML
## Create 20/56 MCAR data in Year
set.seed(10000)
Year_MCAR <- Cooper03$Year
Year_MCAR[sample(56, 20)] <- NA
summary( meta3LFIML(y=y, v=v, cluster=District, x2=scale(Year_MCAR, scale=FALSE),
                    data=Cooper03, model.name="NA in Year_MCAR") )

Description

It converts the correlation or covariance matrices into a dataframe of correlations or covariances with their asymptotic sampling covariance matrices. It uses the asyCov at the backend.

Usage

Cor2DataFrame(x, n, v.na.replace=TRUE, cor.analysis=TRUE,
                acov=c("weighted", "individual", "unweighted"),
                Means, row.names.unique=FALSE, append.vars=TRUE,
                asyCovOld=FALSE, ...)

Arguments

Value

A list of components: (1) a data frame of correlations or covariances with their sampling covariance matrices; (2) a vector of sample sizes; (3) labels of the correlations; and (3) labels of their sampling covariance matrices.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

asyCov, osmasem, create.vechsR, create.Tau2, create.V

Examples

## Provide a list of correlation matrices and a vector of sample sizes as the inputs
my.df1 <- Cor2DataFrame(Nohe15A1$data, Nohe15A1$n)

## Add Lag time as a variable
my.df1$data <- data.frame(my.df1$data, Lag=Nohe15A1$Lag, check.names=FALSE)

## Data
my.df1$data

## Sample sizes
my.df1$n

## ylabels
my.df1$ylabels

## vlabels
my.df1$vlabels

#### Simplified version to do it
my.df2 <- Cor2DataFrame(Nohe15A1)

Description

It creates an F matrix to select observed variables for wls function.

Usage

create.Fmatrix(x, name, as.mxMatrix=TRUE, ...)

Arguments

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

as.mxMatrix, create.mxMatrix, wls

Examples

## Select the first 3 variables while the other 2 variables are latent.
create.Fmatrix(c(1,1,1,0,0))
# FullMatrix 'Fmatrix' 
#
# @labels: No labels assigned.
#
# @values
#      [,1] [,2] [,3] [,4] [,5]
# [1,]    1    0    0    0    0
# [2,]    0    1    0    0    0
# [3,]    0    0    1    0    0
#
# @free: No free parameters.
#
# @lbound: No lower bounds assigned.
#
# @ubound: No upper bounds assigned.

create.Fmatrix(c(1,1,1,0,0), as.mxMatrix=FALSE)
#      [,1] [,2] [,3] [,4] [,5]
# [1,]    1    0    0    0    0
# [2,]    0    1    0    0    0
# [3,]    0    0    1    0    0

Description

It creates a moderator matrix used in OSMASEM.

Usage

create.modMatrix(RAM, output=c("A", "S"), mod)

Arguments

Value

A character matrix.

Author(s)

Mike W.-L. Cheung <[email protected]>

Examples

## A multiple regression model
model <- "y ~ x1 + x2
          x1 ~~ 1*x1
          x2 ~~ 1*x2
          x1 ~~ x2"

## RAM specification
RAM <- lavaan2RAM(model, obs.variables=c("y", "x1", "x2"))

## Create a moderator matrix on A with "meanAge as the moderator.
A1 <-  create.modMatrix(RAM=RAM, output="A", mod="meanAge")
A1

## Create a moderator matrix on S with "meanAge as the moderator.
S1 <-  create.modMatrix(RAM=RAM, output="S", mod="meanAge")
S1

Description

It converts a vector into MxMatrix-class via mxMatrix.

Usage

create.mxMatrix(x, type=c("Full","Symm","Diag","Stand"), ncol=NA,
nrow=NA, as.mxMatrix=TRUE, byrow=FALSE, ...)

Arguments

Details

If there are non-numeric values in x, they are treated as the labels of the free parameters. If an "*" is present, the numeric value on the left-hand side will be treated as the starting value for a free parameter or a fixed value for a fixed parameter. If it is a matrix of numeric values, there are no free parameters in the output matrix. nrow and ncol will be calculated from the length of x unless type="Full" is specified.

Value

A MxMatrix-class object with the same dimensions as x

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

mxMatrix, create.mxMatrix, create.Fmatrix

Examples

## a and b are free parameters with starting values and labels
(a1 <- c(1:4, "5*a", 6, "7*b", 8, 9))

(mat1 <- create.mxMatrix(a1, ncol=3, nrow=3, name="mat1"))

## Arrange the elements by row
(mat2 <- create.mxMatrix(a1, ncol=3, nrow=3, as.mxMatrix=FALSE, byrow=TRUE))

(a3 <- c(1:3, "4*f4", "5*f5", "6*f6"))

(mat3 <- create.mxMatrix(a3, type="Symm", name="mat3"))

## Create character matrix
(mat4 <- create.mxMatrix(a3, type="Symm", as.mxMatrix=FALSE))

## Arrange the elements by row
(mat5 <- create.mxMatrix(a3, type="Symm", as.mxMatrix=FALSE, byrow=TRUE))

(mat6 <- create.mxMatrix(a3, type="Diag", lbound=6:1, name="mat6"))

Description

It creates variance component of the heterogeneity of the random effects by decomposing the variance component into matrices of correlation and standard deviations.

Usage

create.Tau2(RAM, no.var, Tau1.labels=seq(no.var),
            RE.type = c("Diag", "Symm", "Zero", "User"),
            level=c("single", "between", "within"),
            RE.User=NULL, Transform = c("expLog", "sqSD"),
            RE.startvalues=0.05)

Arguments

Value

A list of MxMatrix-class. The variance component is computed in Tau2.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

osmasem, create.V, create.vechsR

Examples

T0 <- create.Tau2(no.var=4, RE.type="Diag", Transform="expLog", RE.startvalues=0.05)
T0

T1 <- create.Tau2(no.var=4, Tau1.labels=c("a", "b", "c", "d"))
T1

Description

It creates a V-known matrix of the sampling covariance matrix using definition variables.

Usage

create.V(x, type = c("Symm", "Diag", "Full"), as.mxMatrix = TRUE)

Arguments

Value

A list of MxMatrix-class. The V-known sampling covariance matrix is computed in V.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

osmasem, create.Tau2, create.vechsR

Examples

my.df <- Cor2DataFrame(Nohe15A1)

## Create known sampling variance covariance matrix
V0 <- create.V(my.df$vlabels)
V0

Description

It creates implicit diagonal constraints on the model implied correlation matrix by treating the error variances as functions of other parameters.

Usage

create.vechsR(A0, S0, F0 = NULL, Ax = NULL, Sx = NULL, A.lbound=NULL, A.ubound=NULL)

Arguments

Value

A list of MxMatrix-class. The model implied correlation matrix is computed in impliedR and vechsR.

Note

Since A0 are the intercepts and Ax are the regression coefficients. The parameters in Ax must be a subset of those in A0.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

osmasem, create.Tau2, create.V

Examples

## Proposed model
model1 <- 'W2 ~ w2w*W1 + s2w*S1
           S2 ~ w2s*W1 + s2s*S1
           W1 ~~ w1WITHs1*S1
           W2 ~~ w2WITHs2*S2
           W1 ~~ 1*W1
           S1 ~~ 1*S1
           W2 ~~ Errw2*W2
           S2 ~~ Errs2*S2'

## Convert into RAM    
RAM1 <- lavaan2RAM(model1, obs.variables=c("W1", "S1", "W2", "S2"))

## No moderator    
M0 <- create.vechsR(A0=RAM1$A, S0=RAM1$S, F0=NULL, Ax=NULL, Sx=NULL)

## Lag (definition variable) as a moderator on the paths in the Amatrix    
Ax <- matrix(c(0,0,0,0,
               0,0,0,0,
               "0*data.Lag","0*data.Lag",0,0,
               "0*data.Lag","0*data.Lag",0,0),
             nrow=4, ncol=4, byrow=TRUE)
                
M1 <- create.vechsR(A0=RAM1$A, S0=RAM1$S, F0=NULL, Ax=Ax, Sx=NULL)    

## Lag (definition variable) as a moderator on the correlation in the Smatrix
Sx <- matrix(c(0,"0*data.Lag",0,0,
               "0*data.Lag",0,0,0,
               0,0,0,"0*data.Lag",
               0,0,"0*data.Lag",0),
             nrow=4, ncol=4, byrow=TRUE)

M2 <- create.vechsR(A0=RAM1$A, S0=RAM1$S, F0=NULL, Ax=NULL, Sx=Sx)

Description

Extract or replace the diagonal of a matrix, or construct a diagonal matrix with the same behaviors as diag prior to R-3.0.0.

Usage

Diag(x, ...)
Diag(x) <- value

Arguments

Details

Started from R-3.0.0, diag(x) returns a numeric matrix with NA in the diagonals when x is a character vector. Although this follows what the manual says, this breaks the metaSEM. The Diag has the same functions as diag except that Diag(x) works for a character vector of x by returning a square matrix of character "0" with x as the diagonals.

Note

See http://r.789695.n4.nabble.com/Behaviors-of-diag-with-character-vector-in-R-3-0-0-td4663735.html for the discussion.

Author(s)

Mike W.-L. Cheung <[email protected]>

See Also

diag

Examples

v <- c("a", "b")
Diag(v)

Description

The data set includes fourteen studies of the factor correlation matrices of the Five-Factor Model of personality reported by Digman (1997).

Usage

data(Digman97)

Details

A list of data with the following structure:

data

A list of 14 studies of correlation matrices. The variables are Agreeableness (A), Conscientiousness (C), Emotional Stability (ES), Extraversion (E) and Intellect (I)

n

A vector of sample sizes

cluster

Types of participants of the studies

Source

Digman, J.M. (1997). Higher-order factors of the Big Five. Journal of Personality and Social Psychology, 73, 1246-1256.

References

Cheung, M. W.-L., & Chan, W. (2005). Classifying correlation matrices into relatively homogeneous subgroups: A cluster analytic approach. Educational and Psychological Measurement, 65, 954-979.

Examples

Digman97

##### Fixed-effects TSSEM
fixed1 <- tssem1(Digman97$data, Digman97$n, method="FEM")
summary(fixed1)

## Factor covariance among latent factors
Phi <- matrix(c(1,"0.3*cor","0.3*cor",1), ncol=2, nrow=2)

## Error covariance matrix
Psi <- Diag(c("0.2*e1","0.2*e2","0.2*e3","0.2*e4","0.2*e5"))

## S matrix
S1 <- bdiagMat(list(Psi, Phi))

## This step is not necessary but it is useful for inspecting the model.
dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- c("A","C","ES","E","I","Alpha","Beta")

## Display S1
S1

## A matrix
Lambda <-
matrix(c(".3*Alpha_A",".3*Alpha_C",".3*Alpha_ES",rep(0,5),".3*Beta_E",".3*Beta_I"),
       ncol=2, nrow=5)
A1 <- rbind( cbind(matrix(0,ncol=5,nrow=5), Lambda),
             matrix(0, ncol=7, nrow=2) )

## This step is not necessary but it is useful for inspecting the model.
dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- c("A","C","ES","E","I","Alpha","Beta")

## Display A1
A1

## F matrix to select the observed variables
F1 <- create.Fmatrix(c(1,1,1,1,1,0,0), as.mxMatrix=FALSE)

## Display F1
F1

################################################################################
## Alternative model specification in lavaan model syntax
model <- "## Factor loadings
          Alpha=~A+C+ES
          Beta=~E+I
          ## Factor correlation
          Alpha~~Beta"

## Display the model
plot(model)

RAM <- lavaan2RAM(model, obs.variables=c("A","C","ES","E","I"),
                  A.notation="on", S.notation="with")
RAM

A1 <- RAM$A
S1 <- RAM$S
F1 <- RAM$F

################################################################################
fixed2 <- tssem2(fixed1, Amatrix=A1, Smatrix=S1, Fmatrix=F1,
                 model.name="TSSEM2 Digman97")
summary(fixed2)

## Display the model with the parameter estimates
plot(fixed2)

#### Fixed-effects TSSEM with several clusters
#### Create a variable for different samples
#### Younger participants: Children and Adolescents
#### Older participants: others
cluster <- ifelse(Digman97$cluster %in% c("Children","Adolescents"),
                  yes="Younger participants", no="Older participants")

#### Show the cluster
cluster

## Example of Fixed-effects TSSEM with several clusters
fixed1.cluster <- tssem1(Digman97$data, Digman97$n, method="FEM",
                         cluster=cluster)
summary(fixed1.cluster)

fixed2.cluster <- tssem2(fixed1.cluster, Amatrix=A1, Smatrix=S1, Fmatrix=F1)
#### Please note that the estimates for the younger participants are problematic.
summary(fixed2.cluster)

## Setup two plots
layout(t(1:2))

## Plot the first group
plot(fixed2.cluster[[1]])
title("Younger participants")

## Plot the second group
plot(fixed2.cluster[[2]])
title("Older participants")

#### Random-effects TSSEM with random effects on the diagonals
random1 <- tssem1(Digman97$data, Digman97$n, method="REM",
                  RE.type="Diag")
summary(random1)

random2 <- tssem2(random1, Amatrix=A1, Smatrix=S1, Fmatrix=F1)
summary(random2)

## Display the model with the parameter estimates
plot(random2, color="green")

Description

It includes two datasets in multiple-treatment studies and multiple-endpoint studies reported by Gleser and Olkin (1994).

Usage

data("Gleser94")

Format

A list of two data frames.

MTS

A data frame of multiple-treatment studies.

MES

A data frame of multiple-endpoint studies.

Source

Gleser, L. J., & Olkin, I. (1994). Stochastically dependent effect sizes. In H. Cooper & L. V. Hedges (Eds.), The handbook of research synthesis. (pp. 339-355). New York: Russell Sage Foundation.

See Also

smdMTS, smdMES

Examples

data(Gleser94)

#### Multiple-treatment studies
Gleser94$MTS

## Assuming homogeneity of variances
my.MTS <- t(apply(Gleser94$MTS, MARGIN=1,
            function(x)
            smdMTS(m=x[c("Mean.C", "Mean.E1", "Mean.E2", "Mean.E3", "Mean.E4", "Mean.E5")],
                   v=x[c("SD.C", "SD.E1", "SD.E2", "SD.E3", "SD.E4", "SD.E5")]^2,
                   n=x[c("N.C", "N.E1", "N.E2", "N.E3", "N.E4", "N.E5")],
                   homogeneity="variance", list.output=FALSE)))

## Fixed-effects multivariate meta-analysis
fit.MTS <- meta(y=my.MTS[, 1:5], 
                v=my.MTS[, 6:20], 
                RE.constraints = diag(0, ncol=5, nrow=5),
                model.name="MTS")
summary(fit.MTS)

#### Multiple-endpoint studies
Gleser94$MES

## Calculate the sampling variances and covariance and amend into the data set
Gleser94$MES$Uncoached.V11 <- with(Gleser94$MES, SD.Uncoached.Math^2)
Gleser94$MES$Uncoached.V21 <- with(Gleser94$MES,
                                   SD.Uncoached.Math*Cor.Math.Verbal*SD.Uncoached.Verbal)
Gleser94$MES$Uncoached.V22 <- with(Gleser94$MES, SD.Uncoached.Verbal^2)

Gleser94$MES$Coached.V11 <- with(Gleser94$MES, SD.Coached.Math^2)
Gleser94$MES$Coached.V21 <- with(Gleser94$MES,
                                 SD.Coached.Math*Cor.Math.Verbal*SD.Coached.Verbal)
Gleser94$MES$Coached.V22 <- with(Gleser94$MES, SD.Coached.Verbal^2)

## Assuming homogeneity of covariance matrices
my.MES <- t(apply(Gleser94$MES, MARGIN=1,
            function(x)
            smdMES(m1=x[c("Mean.Uncoached.Math", "Mean.Uncoached.Verbal")],
                   m2=x[c("Mean.Coached.Math", "Mean.Coached.Verbal")],
                   V1=vec2symMat(x[c("Uncoached.V11", "Uncoached.V21", "Uncoached.V22")]),
                   V2=vec2symMat(x[c("Coached.V11", "Coached.V21", "Coached.V22")]),
                   n1=x["N.Uncoached"],
                   n2=x["N.Coached"],
                   homogeneity="covariance", list.output=FALSE)))

## Fixed-effects multivariate meta-analysis
fit.MES <- meta(y=my.MES[, 1:2], 
                v=my.MES[, 3:5], 
                RE.constraints = diag(0, ncol=2, nrow=2),
                model.name="MES")
summary(fit.MES)

Description

The data set includes 113 correlation matrices on the Rosenberg Self-Esteem Scale reported by Gnambs, Scharl, and Schroeders (2018). Thirty-six studies were based on the reported correlation matrices (CorMat=1) whereas the correlation matrices of the other 77 studies were calculated from the reported factor loadings.

Usage

data(Gnambs18)

Details

A list of data with the following structure:

data

A list of 113 correlation matrices. The variable names are from I1 to I10.

n

A vector of sample sizes.

Year

The year of publications.

Country

The country of studies conducted.

Language

The language used in the studies.

Publication

Whether the studies were published (1) or unpublished (0).

MeanAge

Mean age of the participants.

FemaleProp

Proportion of the female participants.

Individualism

Individualism score of the country.

CorMat

Whether the correlation matrices are obtained from the original studies (1) or reproduced from the factor loadings (0).

Source

Gnambs, T., Scharl, A., & Schroeders, U. (2018). The structure of the Rosenberg Self-Esteem Scale. Zeitschrift Fur Psychologie, 226(1), 14-29. https://doi.org/10.1027/2151-2604/a000317

Description

Effects of open education on attitude toward school and on reading achievement reported by Hedges and Olkin (1985).

Usage

data(HedgesOlkin85)

Details

The variables are:

study

Study number

d_att

Standardized mean difference on attitude

d_ach

Standardized mean difference on achievement

var_att

Sampling variance of the effect size of attitude

cov_att_ach

Sampling covariance between the effect sizes

var_ach

Sampling variance of the effect size of achievement

Source

Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando, FL: Academic Press.

References

Cheung, M. W.-L. (2010). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling, 17, 481-509.

Examples

data(HedgesOlkin85)

## Fixed-effects meta-analysis
summary( meta(y=cbind(d_att, d_ach),
              v=cbind(var_att, cov_att_ach, var_ach),
              data=HedgesOlkin85,
              RE.constraints=matrix(0, nrow=2, ncol=2)) )

Description

It tests the homogeneity of univariate and multivariate effect sizes.

Usage

homoStat(y, v)

Arguments

Value

A list of

Author(s)

Mike W.-L. Cheung <[email protected]>

References

Becker, B. J. (1992). Using results from replicated studies to estimate linear models. Journal of Educational Statistics, 17, 341-362.

Cheung, M. W.-L. (2010). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling, 17, 481-509.

Cochran, W. G. (1954). The combination of estimates from different experiments. Biometrics, 10, 101-129.

See Also

meta

Examples

with( Hox02, homoStat(yi, vi) )

with( HedgesOlkin85, homoStat(y=cbind(d_att, d_ach),
      v=cbind(var_att, cov_att_ach, var_ach)) )

Description

Twenty stimulated studies on standardized mean difference and one continuous study characteristic reported by Hox (2002).

Usage

data(Hox02)

Details

The variables are:

study

Study number

yi

Effect size (standardized mean difference)

vi

Sampling variance of the effect size

weeks

Duration of the experimental intervention in terms of weeks

Source

Hox, J. J. (2002). Multilevel analysis: Techniques and applications. Mahwah, N.J.: Lawrence Erlbaum Associates.

References

Cheung, M. W.-L. (2008). A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling. Psychological Methods, 13, 182-202.

Examples

data(Hox02)

#### ML estimation method
## Random-effects meta-analysis
summary( meta(y=yi, v=vi, data=Hox02, I2=c("I2q", "I2hm"), intervals.type="LB") ) 

## Fixed-effects meta-analysis
summary( meta(y=yi, v=vi, data=Hox02, RE.constraints=0,
              model.name="Fixed effects model") )

## Mixed-effects meta-analysis with "weeks" as a predictor
## Request likelihood-based CI
summary( meta(y=yi, v=vi, x=weeks, data=Hox02, intervals.type="LB",
              model.name="Mixed effects meta analysis with LB CI") )

#### REML estimation method
## Random-effects meta-analysis with REML
summary( VarComp <- reml(y=yi, v=vi, data=Hox02) )

## Extract the variance component
VarComp_REML <- matrix( coef(VarComp), ncol=1, nrow=1 )

## Meta-analysis by treating the variance component as fixed
summary( meta(y=yi, v=vi, data=Hox02, RE.constraints=VarComp_REML) )


## Mixed-effects meta-analysis with "weeks" as a predictor
## Request Wald CI
summary( reml(y=yi, v=vi, x=weeks, intervals.type="z",
              data=Hox02, model.name="REML with LB CI") )

Description

This dataset includes fourteen studies of Correlation Matrices reported by Hunter (1983)

Usage

data(Hunter83)

Details

A list of data with the following structure:

data

A list of 14 studies of correlation matrices. The variables are Ability, Job knowledge, Work sample and Supervisor rating

n

A vector of sample sizes

Source

Hunter, J. E. (1983). A causal analysis of cognitive ability, job knowledge, job performance, and supervisor ratings. In F. Landy, S. Zedeck, & J. Cleveland (Eds.), Performance Measurement and Theory (pp. 257-266). Hillsdale, NJ: Erlbaum.

Examples

data(Hunter83)

#### Fixed-effects model
## First stage analysis
fixed1 <- tssem1(Hunter83$data, Hunter83$n, method="FEM",
                 model.name="TSSEM1 fixed effects model")
summary(fixed1)

#### Second stage analysis
## Model without direct effect from Ability to Supervisor
## A1 <- create.mxMatrix(c(0,"0.1*A2J","0.1*A2W",0,0,0,"0.1*J2W","0.1*J2S",
##                         0,0,0,"0.1*W2S",0,0,0,0),
##                         type="Full", ncol=4, nrow=4, as.mxMatrix=FALSE)

## ## This step is not necessary but it is useful for inspecting the model.
## dimnames(A1)[[1]] <- dimnames(A1)[[2]] <- c("Ability","Job","Work","Supervisor") 
## A1

## S1 <- create.mxMatrix(c(1,"0.1*Var_e_J", "0.1*Var_e_W", "0.1*Var_e_S"),
##                       type="Diag", as.mxMatrix=FALSE)
## dimnames(S1)[[1]] <- dimnames(S1)[[2]] <- c("Ability","Job","Work","Supervisor") 
## S1

################################################################################
## Model specification in lavaan model syntax
## The "ind" effect can be defined within the syntax
model1 <- "## Regression paths
           Job_knowledge ~ A2J*Ability
           Work_sample ~ A2W*Ability + J2W*Job_knowledge
           Supervisor ~ J2S*Job_knowledge + W2S*Work_sample

           ## Fix the variance of Ability at 1
           Ability ~~ 1*Ability

           ## Label the error variances of the dependent variables
           Job_knowledge ~~ VarE_J*Job_knowledge
           Work_sample ~~ VarE_W*Work_sample
           Supervisor ~~ VarE_S*Supervisor

           ## Define an indirect effect
           ind := A2J*J2S+A2J*J2W*W2S+A2W*W2S"

## Display the model
plot(model1, layout="spring", sizeMan=10)

RAM1 <- lavaan2RAM(model1, obs.variables=c("Ability","Job_knowledge",
                   "Work_sample","Supervisor"))
RAM1

################################################################################
fixed2 <- tssem2(fixed1, RAM=RAM1, intervals.type="z",
                 diag.constraints=FALSE,
                 model.name="TSSEM2 fixed effects model")
summary(fixed2)

## Display the model with the parameter estimates
plot(fixed2, layout="spring")

## Coefficients
coef(fixed2)

## VCOV based on parametric bootstrap
vcov(fixed2)

#### Random-effects model with diagonal elements only
## First stage analysis
random1 <- tssem1(Hunter83$data, Hunter83$n, method="REM", RE.type="Diag", 
                  acov="weighted", model.name="TSSEM1 random effects model")
summary(random1)

model2 <- "## Regression paths
           Job_knowledge ~ A2J*Ability
           Work_sample ~ A2W*Ability + J2W*Job_knowledge
           Supervisor ~ J2S*Job_knowledge + W2S*Work_sample

           ## Fix the variance of Ability at 1
           Ability ~~ 1*Ability

           ## Label the error variances of the dependent variables
           Job_knowledge ~~ VarE_J*Job_knowledge
           Work_sample ~~ VarE_W*Work_sample
           Supervisor ~~ VarE_S*Supervisor"

RAM2 <- lavaan2RAM(model2, obs.variables=c("Ability","Job_knowledge",
                   "Work_sample","Supervisor"))
RAM2

## Second stage analysis
## Model without direct effect from Ability to Supervisor

## The "ind" effect is defined in tssem2().
random2 <- tssem2(random1, RAM=RAM2, intervals.type="LB",
                  diag.constraints=FALSE,
                  mx.algebras=
                      list(ind=mxAlgebra(A2J*J2S+A2J*J2W*W2S+A2W*W2S, name="ind")),
                  model.name="TSSEM2 random effects model")
                  
summary(random2)

## Display the model with the parameter estimates
plot(random2, layout="spring")

Description

It creates or generates the model implied correlation or covariance matrices based on the RAM model specification.

Usage

impliedR(RAM, Amatrix, Smatrix, Fmatrix, Mmatrix, corr=TRUE, labels, ...)
rimpliedR(RAM, Amatrix, Smatrix, Fmatrix, AmatrixSD, SmatrixSD,
          k=1, corr=TRUE, nonPD.pop=c("replace", "nearPD", "accept"))

Arguments

Details

This function can be used to generate the model implied correlation matrix for the standardized parameters with the corr=TRUE argument. Suppose we want to calculate the population correlation matrix for a mediation model with x, m, and y. We only need to specify the population path coefficients among x, m, and y in the Amatrix. We do not need to specify the population error variances of m and y. We treat the error variances as unknown parameters by giving them starting values in the Smatrix matrix. When the covariance matrix is requested by specifying corr=FALSE, it simply calculates the population model covariance matrix by treating the values in Smatrix as the population values.

Value

A list of RAM matrices, the model implied correlation or covariance matrix of the observed variables (SigmaObs), of both observed and latent variables (SigmaAll), the minimum fit (minFit) which should be zero, and the status code of the optimization (status) which should also be zero when the optimization is fine. The last object is mx.fit which is the output after running the model. It can be used in the diagnosis.