Type:	Package
Title:	Meta-Analysis using Structural Equation Modeling
Version:	1.5.0
Date:	2024-09-26
Depends:	R (≥ 3.4.0), OpenMx
Imports:	Matrix, MASS, ellipse, graphics, stats, utils, mvtnorm, numDeriv, lavaan
Suggests:	metafor, semPlot, R.rsp, testthat, matrixcalc
VignetteBuilder:	R.rsp
Maintainer:	Mike Cheung <mikewlcheung@nus.edu.sg>
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, see Cheung (2015) <doi:10.3389/fpsyg.2014.01521>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
LazyLoad:	yes
LazyData:	yes
ByteCompile:	yes
URL:	https://github.com/mikewlcheung/metasem
BugReports:	https://github.com/mikewlcheung/metasem/issues
NeedsCompilation:	no
Packaged:	2024-09-25 23:56:01 UTC; mikewlcheung
Author:	Mike Cheung [aut, cre]
Repository:	CRAN
Date/Publication:	2024-09-26 02:30:02 UTC

Meta-Analysis using Structural Equation Modeling

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 <mikewlcheung@nus.edu.sg>

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

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

Multivariate effect sizes between classroom management self-efficacy (CMSE) and other variables reported by Aloe et al. (2014)

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)

Dataset on the Effectiveness of the BCG Vaccine for Preventing Tuberculosis

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)

Ten Studies of Correlation Matrices used by Becker (2009)

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)

Studies on Sex Differences in Conformity Reported by Becker (1983)

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) )

Six Studies of Correlation Matrices reported by Becker (1992; 1995)

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)

Five Studies of Ten Correlation Matrices reported by Becker and Schram (1994)

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)

Five Published Trails from Berkey et al. (1998)

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) )

Correlation Matrices from Boer et al. (2016)

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")    

A Dataset from Bornmann et al. (2007)

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))

Dataset from Chan, Jones, Jamieson, and Albarracin (2017)

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.

Fifty Studies of Correlation Matrices used in Cheung and Chan (2000)

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")

A Dataset from TSSEM User's Guide Version 1.11 by Cheung (2009)

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")

Correlation Matrices from Cooke et al. (2016)

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")

Selected effect sizes from Cooper et al. (2003)

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") )

Convert correlation or covariance matrices into a dataframe of correlations or covariances with their sampling covariance matrices

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 <mikewlcheung@nus.edu.sg>

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)

Matrix Diagonals

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 <mikewlcheung@nus.edu.sg>

See Also

diag

Examples

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

Factor Correlation Matrices of Big Five Model from Digman (1997)

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")

Two Datasets from Gleser and Olkin (1994)

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)

Correlation Matrices from Gnambs, Scharl, and Schroeders (2018)

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

Effects of Open Education Reported by Hedges and Olkin (1985)

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)) )

Simulated Effect Sizes Reported by Hox (2002)

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") )

Fourteen Studies of Correlation Matrices reported by Hunter (1983)

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")

Dataset from Jaramillo, Mulki and Marshall (2005)

Description

A dataset of the relationship between organizational commitment (OC) and salesperson job performance (JP) from Jaramillo, Mulki & Marshall (2005).

Usage

data(Jaramillo05)

Format

A data frame with 61 observations on the following 10 variables.

Author

a character vector of study

Sample_size

sample size of the study

Sales

sample type; either "mixed", "nonsales" or "sales"

Country

a character vector of country of study

IDV

Hofstede's (1997) individualism index

OC_scale

scale of OC; either "Porter or Mowday", "Meyer" or "other"

OC_alpha

Coefficient alpha of organizational commitment

JP_alpha

Coefficient alpha of job performance

r

correlation between organizational commitment and job performance

r_v

sampling variance of r

Citations

Citations from Google Scholar as of 27 August 2024

Source

Jaramillo, F., Mulki, J. P., & Marshall, G. W. (2005). A meta-analysis of the relationship between organizational commitment and salesperson job performance: 25 years of research. Journal of Business Research, 58(6), 705-714. doi:10.1016/j.jbusres.2003.10.004

Examples

## Research question 4.4.1
summary(meta(r, r_v, data=Jaramillo05))

## Research question 4.4.2
## Select cases with either "sales" or "nonsales"
Sales.df <- subset(Jaramillo05, Sales %in% c("sales", "nonsales"))

## Create a predictor with 1 and 0 when they are "sales" or "nonsales", respectively
predictor <- ifelse(Jaramillo05$Sales=="sales", yes=1, no=0)

## Mixed-effects meta-analysis
summary( meta(y = r, v = r_v, x = predictor, data = Jaramillo05) )

## Research question 4.4.3
summary(meta(r, r_v, x=IDV, data=Jaramillo05))

Multivariate effect sizes reported by Kalaian and Raudenbush (1996)

Description

This data set includes 47 multivariate effect sizes reported by Kalaian and Raudenbush (1996, Table 1).

Usage

data(Kalaian96)

Details

A list of data with the following structure:

Study

Study name

Year

Year of publication

n_e

Sample size of the experimental group

n_c

Sample size of the control group

dSAT_V

Standardized mean difference of the Scholastic Aptitude Test (SAT) on verbal

dSAT_M

Standardized mean difference of SAT on math

var_V

Sampling variance of dSAT_V

cov_VM

Sampling covariance of dSAT_V and dSAT_M with a common correlation of 0.66

var_M

Sampling variance of dSAT_M

Hr

Hours of training

ETS

Educational Testing Service

Study_type

Either Randomized, Matched or Nonequivalent comparison

Home_work

Home work

Source

Kalaian, H. A., & Raudenbush, S. W. (1996). A multivariate mixed linear model for meta-analysis. Psychological Methods, 1(3), 227-235. https://doi.org/10.1037/1082-989X.1.3.227

Examples

data(Kalaian96)

Eight studies from Mak et al. (2009)

Description

Eight studies from Mak et al. (2009) were reported by Cheung et al. (2012).

Usage

data(Mak09)

Format

A data frame with 8 observations on the following 10 variables.

Study

a character vector of study

type

a character vector

AF.BP

a numeric vector

Tot.BP

a numeric vector

AF.non.BP

a numeric vector

Tot.non.BP

a numeric vector

yi

a numeric vector

vi

a numeric vector

age.mean

a numeric vector

study.duration

a numeric vector

Source

Mak, A., Cheung, M. W.-L., Ho, R. C. M., Cheak, A. A. C., & Lau, C. S. (2009). Bisphosphonate and atrial fibrillation: Bayesian meta-analyses of randomized controlled trials and observational studies. BMC Musculoskeletal Disorders, 10(113). doi:10.1186/1471-2474-10-113 Available at https://bmcmusculoskeletdisord.biomedcentral.com/articles/10.1186/1471-2474-10-113.

References

Cheung, M. W.-L., Ho, R. C. M., Lim, Y., & Mak, A. (2012). Conducting a meta-analysis: Basics and good practices. International Journal of Rheumatic Diseases, 15(2), 129-135. doi: 10.1111/j.1756-185X.2012.01712.x

Examples

## Random-effects meta-analysis
( meta1 <- summary(meta(y=yi, v=vi, data=Mak09, I2=c("I2q", "I2hm"))) )

## Convert the estimates back into odds ratio 
OR <- with(coef(meta1), exp(c(Estimate[1], lbound[1], ubound[1])))
names(OR) <- c("Estimate in OR", "lbound in OR", "ubound in OR")
OR
  
## Mixed-effects meta-analysis with mean age as a predictor
summary( meta(y=yi, v=vi, x=age.mean, data=Mak09) )

Correlation Matrices from Mathieu et al. (2015)

Description

The data set includes a list of correlation matrices of panel studies between cohesion (C) and performance (P) in Mathieu et al. (2015, Table 1).

Usage

data(Mathieu15)

Details

A list of data with the following structure:

data

A list of studies of correlation matrices. The variables are C1, P1, C2, and P2.

n

A vector of sample sizes.

Year

Year of publication.

Sample

Sample characteristics.

Student

Whether the samples are student or non-student based on Sample.

Source

Mathieu, J. E., Kukenberger, M. R., D'Innocenzo, L., & Reilly, G. (2015). Modeling reciprocal team cohesion-performance relationships, as impacted by shared leadership and members' competence. Journal of Applied Psychology, 100(3), 713-734. https://doi.org/10.1037/a0038898

Examples

# TSSEM
## Model 1: no constraint
## Stage 1 analysis
tssem1.fit <- tssem1(Mathieu15$data, Mathieu15$n)
summary(tssem1.fit)

## Proposed model in lavaan syntax
model1 <- 'C2 ~ c2c*C1 + p2c*P1
           P2 ~ c2p*C1 + p2p*P1
           C1 ~~ c1withp1*P1
           C1 ~~ 1*C1
           P1 ~~ 1*P1
           C2 ~~ c2withp2*P2'

## Convert the lavaan model to RAM specification
RAM1 <- lavaan2RAM(model1, obs.variables=c("C1", "P1", "C2", "P2"))
RAM1

## Stage 2 analysis
tssem1b.fit <- tssem2(tssem1.fit, RAM=RAM1)
summary(tssem1b.fit)

plot(tssem1b.fit, col="yellow", edge.label.position=0.58)

## Model 2: Equality constraints on the path coefficient
## Proposed model with equal effects time 1 to time 2
model2 <- 'C2 ~ same*C1 + diff*P1
           P2 ~ diff*C1 + same*P1
           C1 ~~ c1withp1*P1
           C1 ~~ 1*C1
           P1 ~~ 1*P1
           C2 ~~ c2withp2*P2'

## Convert the lavaan model to RAM specification
RAM2 <- lavaan2RAM(model2, obs.variables=c("C1", "P1", "C2", "P2"))
RAM2
    
## Stage 2 analysis
tssem2b.fit <- tssem2(tssem1.fit, RAM=RAM2)
summary(tssem2b.fit)

## Compare the models with and without the constraints. 
anova(tssem1b.fit, tssem2b.fit)

## Plot the model
plot(tssem2b.fit, col="yellow", edge.label.position=0.60)


## OSMASEM
my.df <- Cor2DataFrame(Mathieu15)
    
head(my.df$data)

## Model without any moderator
osmasem.fit1 <- osmasem(model.name="No moderator", RAM=RAM1, data=my.df)
summary(osmasem.fit1)

## Extract the heterogeneity variance-covariance matrix
diag(VarCorr(osmasem.fit1))

plot(osmasem.fit1, col="yellow", edge.label.position=0.6)

## Model with student sample as a moderator on the regression coefficients
A1 <- create.modMatrix(RAM1, output="A", "Student")
A1

## Model with a moderator    
osmasem.fit2 <- osmasem(model.name="Student sample as a moderator", RAM=RAM1, 
                        Ax=A1, data=my.df)
summary(osmasem.fit2)

## Compare the models with and without the moderator
anova(osmasem.fit2, osmasem.fit1)

## Get the R2 of the moderator
osmasemR2(osmasem.fit2, osmasem.fit1)

Dataset on the Environmental Tobacco Smoke (ETS) on children's health

Description

This dataset includes 59 studies reported by Nam, Mengersen, and Garthwaite (2003) on the potential health effects among children exposed to environmental tobacco smoke (ETS), or passive smoking. The effect sizes are the log odds ratios of asthma and lower respiratory disease (LRD).

Usage

data(Nam03)

Details

A list of data with the following structure:

ID

Study identification number.

Size

Total number of valid subjects in the study.

Age

Mean age of participants.

Year

Year of publication.

Country

Country code.

Smoke

Source of ETS.

Adj

Whether the reported odds ratio is adjusted for covariates.

Asthma_logOR

Log odds ratio of asthma.

LRD_logOR

Log odds ratio of lower respiratory disease.

Asthma_v

Sampling variance of Asthma_logOR.

AsthmaLRD_cov_05

Sampling covariance between Asthma_logOR and LRD_logOR by assuming a correlation of 0.5

LRD_v

Sampling variance of LRD_logOR.

Source

Nam, I.-S., Mengersen, K., & Garthwaite, P. (2003). Multivariate meta-analysis. Statistics in Medicine, 22(14), 2309-2333. https://doi.org/10.1002/sim.1410

Examples

data(Nam03)

Correlation Matrices from Nohe et al. (2015)

Description

The data sets include two lists of correlation matrices of panel studies between work-family conflict and strain reported in Table A1 (Nohe15A1) and Table A2 (Nohe15A2) by Nohe et al. (2015).

Usage

data(Nohe15A1)
data(Nohe15A2)

Details

A list of data with the following structure:

data

A list of studies of correlation matrices. The variables are W1, S1, W2, and S2 in Nohe15A1 and F1, S1, F2, and S2 in Nohe15A2

n

A vector of sample sizes

RelXX

The reliabilities of W1, S1, W2 and S2 in Nohe15A1 and the reliabilities of F1 S1, F2 , and S2 in Nohe15A2

FemalePer

Percentage of female participants

Publication

Whether the studies were published (P) or unpublished (U)

Lag

Time lag between the coded measurement waves in months

Source

Nohe, C., Meier, L. L., Sonntag, K., & Michel, A. (2015). The chicken or the egg? A meta-analysis of panel studies of the relationship between work-family conflict and strain. Journal of Applied Psychology, 100(2), 522-536.

Examples

#### TSSEM
    
## Set seed for replicability    
set.seed(23891)
    
## Table A1
randA1a <- tssem1(Nohe15A1$data, Nohe15A1$n, method="REM", RE.type="Diag")
summary(randA1a)

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'

## Display the model
plot(model1, layout="spring")    
    
RAM1 <- lavaan2RAM(model1, obs.variables=c("W1", "S1", "W2", "S2"))
RAM1

randA1b <- tssem2(randA1a, Amatrix=RAM1$A, Smatrix=RAM1$S)
summary(randA1b)

## Display the model with the parameter estimates
plot(randA1b, layout="spring")    

## Table A2
randA2a <- tssem1(Nohe15A2$data, Nohe15A2$n, method="REM", RE.type="Diag")
## Rerun to remove error code
randA2a <- rerun(randA2a)
summary(randA2a)

model2 <- 'F2 ~ f2f*F1 + s2F*S1
           S2 ~ f2s*F1 + s2s*S1
           F1 ~~ f1WITHs1*S1
           F2 ~~ f2WITHs2*S2
           F1 ~~ 1*F1
           S1 ~~ 1*S1
           F2 ~~ Errf2*F2
           S2 ~~ Errs2*S2'

## Display the model
plot(model2, layout="spring")
    
RAM2 <- lavaan2RAM(model2, obs.variables=c("F1", "S1", "F2", "S2"))
RAM2

randA2b <- tssem2(randA2a, Amatrix=RAM2$A, Smatrix=RAM2$S)
summary(randA2b)

## Display the model with the parameter estimates
plot(randA2b, layout="spring")  
    
## Estimate the heterogeneity of the parameter estimates
tssemParaVar(randA1a, randA2b)    

## Parametric bootstrap based on Yu et al. (2016)
## I assume that you know what you are doing!

## Set seed for reproducibility
set.seed(39128482)

## Average the correlation coefficients with the univariate-r approach
uni1 <- uniR1(Nohe15A1$data, Nohe15A1$n)
uni1

## Generate random correlation matrices
boot.cor <- bootuniR1(uni1, Rep=50)

## Display the quality of the generated correlation matrices
summary(boot.cor)

## Proposed saturated model
model1 <- 'W2 + S2 ~ W1 + S1'

## Use the harmonic mean of the sample sizes as n in SEM
n <- uni1$n.harmonic    
    
boot.fit1 <- bootuniR2(model=model1, data=boot.cor, n=n)
summary(boot.fit1)

## Proposed model with equal regression coefficients
model2 <- 'W2 ~ Same*W1 + Cross*S1
           S2 ~ Cross*W1 + Same*S1'

boot.fit2 <- bootuniR2(model=model2, data=boot.cor, n=n)
summary(boot.fit2)

#### OSMASEM    

## Calculate the sampling variance-covariance matrix of the correlation matrices.    
my.df <- Cor2DataFrame(Nohe15A1)

## Standardize the moderator "Lag"
my.df$data$Lag <- scale(my.df$data$Lag)
    
head(my.df$data)

## 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'
plot(model1)     

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

## Create vechs of the model implied correlation matrix
## with implicit diagonal constraints
## M0 <- create.vechsR(A0=RAM1$A, S0=RAM1$S)

## Create heterogeneity variances
## RE.type= either "Diag" or "Symm"
##
## Transform= either "expLog" or "sqSD" for better estimation on variances
## T0 <- create.Tau2(RAM=RAM1, RE.type="Diag")
##
## Fit the model    
## fit0 <- osmasem(model.name="No moderator", Mmatrix=M0, Tmatrix=T0, data=my.df)

## Fit the model
fit0 <- osmasem(model.name="No moderator", RAM=RAM1, data=my.df)
summary(fit0)

## Get the SRMR
osmasemSRMR(fit0)

## Get the transformed variance component of the random effects    
VarCorr(fit0)
    
## "lag" as a moderator on A matrix
A1 <- 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, Ax=A1)
##
## Fit the nodel
## fit1 <- osmasem(model.name="Lag as a moderator for Amatrix", Mmatrix=M1,
##                 Tmatrix=T0, data= my.df)
    
fit1 <- osmasem(model.name="Lag as a moderator for Amatrix",
                RAM=RAM1, Ax=A1, data= my.df)
summary(fit1)
VarCorr(fit1)

## Compare the models with and without the moderator "lag"
anova(fit1, fit0)

## Calculate the R2    
osmasemR2(fit0, fit1)

Studies on the Hospital Anxiety and Depression Scale Reported by Norton et al. (2013)

Description

The data set includes 28 studies on 14 items measuring the Hospital Anxiety and Depression Scale (HADS) Reported by Norton et al. (2013).

Usage

data(Norton13)

Details

The variables are:

data

A list of 28 studies of correlation matrices. The variables are 14 items (x1 to x14) measuring HADS.

n

A vector of sample sizes

population

A vector of the population of the data

group

A vector of classification into patients vs. non-patients based on population

Source

Norton, S., Cosco, T., Doyle, F., Done, J., & Sacker, A. (2013). The Hospital Anxiety and Depression Scale: A meta confirmatory factor analysis. Journal of Psychosomatic Research, 74(1), 74-81.

References

Jak, S., & Cheung, M. W.-L. (2018). Addressing heterogeneity in meta-analytic structural equation modeling using subgroup analysis. Behavior Research Methods, 50, 1359-1373.

Examples

data(Norton13)

Studies on Students' School Engagement and Achievement Reported by Roorda et al. (2011)

Description

The data set includes 45 studies on the influence of affective teacher-student relationships on students' school engagement and achievement reported by Roorda et al. (2011).

Usage

data(Roorda11)

Details

The variables are:

data

A list of 45 studies of correlation matrices. The variables are pos (positive teacher-student relations), neg (negative teacher-student relations), enga (student engagement), and achiev (student achievement).

n

A vector of sample sizes

SES

A vector of average socio-economic status (SES) of the samples

Source

Roorda, D. L., Koomen, H. M. Y., Spilt, J. L., & Oort, F. J. (2011). The influence of affective teacher-student relationships on students' school engagement and achievement a meta-analytic approach. Review of Educational Research, 81(4), 493-529.

References

Jak, S., & Cheung, M. W.-L. (2018). Addressing heterogeneity in meta-analytic structural equation modeling using subgroup analysis. Behavior Research Methods, 50, 1359-1373.

Examples

## Random-effects model: First stage analysis
random1 <- tssem1(Cov = Roorda11$data, n = Roorda11$n, method = "REM",
                  RE.type = "Diag")
summary(random1)

varnames <- c("pos", "neg", "enga", "achiev")

## Prepare a regression model using create.mxMatrix()
A <- create.mxMatrix(c(0,0,0,0,
                       0,0,0,0,
                       "0.1*b31","0.1*b32",0,0,
                       0,0,"0.1*b43",0),
                     type = "Full", nrow = 4, ncol = 4, byrow = TRUE,
                     name = "A", as.mxMatrix = FALSE)

## This step is not necessary but it is useful for inspecting the model.
dimnames(A) <- list(varnames, varnames)
A

S <- create.mxMatrix(c(1,
                       ".5*p21",1,
                       0,0,"0.6*p33",
                       0,0,0,"0.6*p44"), 
                     type="Symm", byrow = TRUE,
                     name="S", as.mxMatrix = FALSE)

## This step is not necessary but it is useful for inspecting the model.
dimnames(S) <- list(varnames, varnames)
S

## Random-effects model: Second stage analysis
random2 <- tssem2(random1, Amatrix=A, Smatrix=S, diag.constraints=TRUE, 
                  intervals="LB")
summary(random2)

## Display the model with the parameter estimates    
plot(random2)

Correlation Matrices from Scalco et al. (2017)

Description

The data set includes correlation matrices using the theory of planned behavior to predict organic food consumption reported by Scalco17 et al. (2017).

Usage

data(Scalco17)

Details

A list of data with the following structure:

data

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

n

A vector of sample sizes

Age

A vector of the mean age of the samples

Female

A vector of the percentage of the female samples

Source

Scalco, A., Noventa, S., Sartori, R., & Ceschi, A. (2017). Predicting organic food consumption: A meta-analytic structural equation model based on the theory of planned behavior. Appetite, 112, 235-248.

Examples

data(Scalco17)

Correlations from Stadler et al. (2015)

Description

The data set includes correlations between complex problem solving and intelligence reported by Stadler et al. (2015).

Usage

data(Stadler15)

Details

A list of data with the following structure:

ID

ID of the effect sizes

Authors

Authors of the studies

Year

Year of the studies

N

Sample size

CPSMeasure

Complex problem solving (CPS) measure

IntelligenceMeasure

Intelligence measure

r

Correlation between CPS and intelligence

v

Sampling variance of r

Source

Stadler, M., Becker, N., Godker, M., Leutner, D., & Greiff, S. (2015). Complex problem solving and intelligence: A meta-analysis. Intelligence, 53, 92-101.

Correlation coefficients reported by Tenenbaum and Leaper (2002)

Description

Forty-eight studies reported by Tenenbaum and Leaper (2002, Table 1).

Usage

data(Tenenbaum02)

Details

The variables are:

Authors

Authors of the study

Year

Year of publication

N

Sample size

r

Correlation between parents' gender schemas and their offspring's gender-related cognitions.

v

Sampling variance of r

Publication_source

Publication source: 1="top-tier journal", 2="second-tier journal or book chapter", 3="dissertation", 4="other unpublished study"

Author_gender

Gender of the first author: "W"="woman", "M"="man"

Parent_type

Parent type: "M"="mother", "F"="father", "MF"="mother and father"

Parent_predictor

Parent predictor: "S"="self gender schema", "A"="gender attitudes about others"

Offspring_age

Offspring age (months)

Offspring_type

Offspring type: "D"="daughter", "S"="son", "DS"="daughter and son"

Offspring_outcome

Offspring outcome: "S"="gender schema for self", "A"="gender attitudes toward others", "I"="gender-related interests and preferences", "W"="work-related attitudes"

Source

Tenenbaum, H. R., & Leaper, C. (2002). Are parents' gender schemas related to their children's gender-related cognitions? A meta-analysis. Developmental Psychology, 38(4), 615-630. https://doi.org/10.1037/0012-1649.38.4.615

Examples

data(Tenenbaum02)

Extract Variance-Covariance Matrix of the Random Effects

Description

It extracts the variance-covariance matrix of the random effects (variance component) from either the meta or osmasem objects.

Usage

VarCorr(x, ...)

Arguments

Value

A variance-covariance matrix of the random effects.

Note

It is similar to coef(object, select="random") in tssem. The main difference is that coef() returns a vector while VarCorr() returns its correspondent matrix.

Author(s)

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

See Also

coef, vcov

Examples

## 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))
VarCorr(bcg)

Compare Nested Models with Likelihood Ratio Statistic

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 <mikewlcheung@nus.edu.sg>

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) 

Convert a Character Matrix into MxAlgebra-class

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 <mikewlcheung@nus.edu.sg>

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

Convert a Matrix into MxMatrix-class

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 <mikewlcheung@nus.edu.sg>

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)

Convert a Character Matrix with Starting Values to a Character Matrix without Starting Values

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 <mikewlcheung@nus.edu.sg>

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"

Compute Asymptotic Covariance Matrix of a Correlation/Covariance Matrix

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 <mikewlcheung@nus.edu.sg>

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)

Create a Block Diagonal Matrix

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 <mikewlcheung@nus.edu.sg>

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" 

Create a Block Diagonal Matrix by Repeating the Input

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 <mikewlcheung@nus.edu.sg>

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" 

Parametric bootstrap on the univariate R (uniR) object

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 <mikewlcheung@nus.edu.sg>

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

Fit Models on the bootstrapped correlation matrices

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 <mikewlcheung@nus.edu.sg>

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

Calculate Effect Sizes using lavaan Models

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 <mikewlcheung@nus.edu.sg>

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)  

Check the correctness of the RAM formulation

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 <mikewlcheung@nus.edu.sg>

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)   

Extract Parameter Estimates from various classes.

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 <mikewlcheung@nus.edu.sg>

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")

Create an F matrix to select observed variables

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 <mikewlcheung@nus.edu.sg>

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

Create a variance component of the heterogeneity of the random effects

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 <mikewlcheung@nus.edu.sg>

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

Create a V-known matrix

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 <mikewlcheung@nus.edu.sg>

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

Create a moderator matrix used in OSMASEM

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 <mikewlcheung@nus.edu.sg>

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

Create a Vector into MxMatrix-class

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 <mikewlcheung@nus.edu.sg>

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"))

Create a model implied correlation matrix with implicit diagonal constraints

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 <mikewlcheung@nus.edu.sg>

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)

Test the Homogeneity of Effect Sizes

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 <mikewlcheung@nus.edu.sg>

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)) )

Create or Generate the Model Implied Correlation or Covariance Matrices

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.

Note

It is important to ensure that all the population values in Amatrix must be set as fixed parameters; otherwise, these values may be altered with the corr=TRUE argument. When there is an error or warning message about the status code, there is a high chance that some of the values in Amatrix are incorrectly set as free parameters.

Author(s)

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

Examples

set.seed(100)

## A one-factor CFA model
model <- "f =~ 0.3*x1 + 0.4*x2 + 0.5*x3
          f ~~ 1*f"

RAM <- lavaan2RAM(model)

impliedR(RAM, corr=TRUE)

## A simple mediation model
## All are population parameters in the A matrix
A1 <- matrix(c(0, 0, 0,
               0.3, 0, 0,
               0.2, 0.4, 0), nrow=3, ncol=3, byrow=TRUE,
             dimnames=list(c("x", "m", "y"), c("x", "m", "y")))
A1             

## Variance of x is fixed at 1 while the other variances are free.
S1 <- matrix(c(1, 0, 0,
               0, "0.1*ErrVarM",0,
               0, 0, "0.1*ErrVarY"), nrow=3, ncol=3,
             dimnames=list(c("x", "m", "y"), c("x", "m", "y")))
S1

impliedR(Amatrix=A1, Smatrix=S1)

## SD of A1
A1SD <- matrix(c(0, 0, 0,
                 0.1, 0, 0,
                 0.1, 0.1, 0), nrow=3, ncol=3, byrow=TRUE,
               dimnames=list(c("x", "m", "y"), c("x", "m", "y")))
A1SD

rimpliedR(Amatrix=A1, Smatrix=S1, AmatrixSD=A1SD, k=2)

## A CFA model
A2 <- matrix(c(0, 0, 0, 0.3,
               0, 0, 0, 0.4,
               0, 0, 0, 0.5,
               0, 0, 0, 0), nrow=4, ncol=4, byrow=TRUE,
             dimnames=list(c("x1", "x2", "x3", "f"),
                           c("x1", "x2", "x3", "f")))
A2

## Variance of f is fixed at 1 while the other variances are free.
S2 <- matrix(c("0.7*Err1", 0, 0, 0,
                0, "0.7*Err2", 0, 0,
                0, 0, "0.7*Err3", 0,
                0, 0, 0, 1), nrow=4, ncol=4,
            dimnames=list(c("x1", "x2", "x3", "f"), c("x1", "x2", "x3", "f")))
S2

F2 <- create.Fmatrix(c(1,1,1,0), as.mxMatrix=FALSE)
F2

## Model implied correlation matrix
impliedR(Amatrix=A2, Smatrix=S2, Fmatrix=F2, corr=TRUE)

## Model implied covariance matrix
impliedR(Amatrix=A2, Smatrix=S2, Fmatrix=F2, corr=FALSE)

## SD of A2
A2SD <- matrix(c(0, 0, 0, 0.1,
                 0, 0, 0, 0.1,
                 0, 0, 0, 0.1,
                 0, 0, 0, 0), nrow=4, ncol=4, byrow=TRUE,
               dimnames=list(c("x1", "x2", "x3", "f"),
                             c("x1", "x2", "x3", "f")))               
A2SD

## SD of S2: correlated between x1 and x2
S2SD <- matrix(c(0, 0.1, 0, 0,
                 0.1, 0, 0, 0,
                 0, 0, 0, 0.1,
                 0, 0, 0, 0), nrow=4, ncol=4, byrow=TRUE,
               dimnames=list(c("x1", "x2", "x3", "f"),
                             c("x1", "x2", "x3", "f")))               
S2SD

rimpliedR(Amatrix=A2, Smatrix=S2, Fmatrix=F2, AmatrixSD=A2SD,
          SmatrixSD=S2SD, k=2)

Estimate the asymptotic covariance matrix of standardized or unstandardized indirect and direct effects

Description

It estimates the standardized or unstandardized indirect and direct effects and their asymptotic sampling covariance matrix.

Usage

indirectEffect(x, n, standardized = TRUE, direct.effect = TRUE, run = TRUE)

Arguments

Details

Cheung (2009) estimated the standardized indirect effect and its standard error with non-linear constraints. Since OpenMx does not generate standard errors when there are non-linear constraints, Kwan and Chan's (2011) approach is used in this function. Delta method is used to calculate the asymptotic covariance matrix.

Value

A vector (or a matrix if the input is a list of matrices) of (standardized) indirect effect, standardized direct effect, and their asymptotic sampling covariance matrices

Author(s)

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

References

Cheung, M. W.-L. (2009). Comparison of methods for constructing confidence intervals of standardized indirect effects. Behavior Research Methods, 41, 425-438.

Kwan, J., & Chan, W. (2011). Comparing standardized coefficients in structural equation modeling: a model reparameterization approach. Behavior Research Methods, 43, 730-745.

Examples

## A correlation matrix as input
x <- matrix(c(1, 0.4, 0.2, 0.4, 1, 0.3, 0.2, 0.3, 1), ncol=3)
dimnames(x) <- list( c("y", "m", "x"), c("y", "m", "x") )
indirectEffect(x, n=300)

## A list of correlation matrices
indirectEffect( list(x, x), n=c(300,500), standardized=FALSE )

Test Positive Definiteness of a List of Square Matrices

Description

It tests the positive definiteness of a square matrix or a list of square matrices. It returns TRUE if the matrix is positive definite. It returns FALSE if the matrix is either non-positive definite or not symmetric. Variables with NA in the diagonals will be removed before testing. It returns NA when there are missing correlations even after deleting the missing variables.

Usage

is.pd(x, check.aCov=FALSE, cor.analysis=TRUE, tol=1e-06)

Arguments

Value

If the input is a matrix, it returns TRUE, FALSE or NA. If the input is a list of matrices, it returns a list of TRUE, FALSE or NA.

Author(s)

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

Examples

A <- diag(1,3)
is.pd(A)
# TRUE

B <- matrix(c(1,2,2,1), ncol=2)
is.pd(B)
# FALSE

is.pd(list(A, B))
# TRUE FALSE

C <- A
C[2,1] <- C[1,2] <- NA
is.pd(C)
# NA

A Dataset from ISSP (2005)

Description

Thirty-two covariance matrices on work-related attitudes were extracted from the International Social Survey Programme 2005: Work Orientation III (ISSP, 2005). Seven variables were selected for demonstration purposes. They were grouped into three constructs: Importance of Job Prospects measured by job security (JP1), high income (JP2), and opportunity for advancement (JP3); Importance of Job Autonomy measured by work independently (JA1) and decide time of work (JA2); and Importance of Contributions to Society measured by help other people (CS1) and a job useful to society (CS2).

Usage

data(issp05)

Details

A list of data with the following structure:

data

A list of 32 covariance matrices

n

A vector of sample sizes

means

A matrix of means

pdi

Hofstede's Power Distance Index

idv

Hofstede's Individualism

mas

Hofstede's Masculinity

uai

Hofstede's Uncertainty Avoidance Index

ltowvs

Hofstede's Long- Versus Short-Term Orientation

ivr

Hofstede's Indulgence Versus Restraint

Author(s)

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

Source

ISSP Research Group (2007): International Social Survey Programme 2005: Work Orientation III (ISSP 2005). GESIS Data Archive, Cologne. ZA4350 Data file Version 1.0.0, doi:10.4232/1.4350

https://geerthofstede.com/research-and-vsm/dimension-data-matrix/

See Also

issp89

Examples

data(issp05)

#### TSSEM random-effects model with covariance matrices

## Stage 1 analysis
rand1 <- tssem1(issp05$data, issp05$n, method="REM", cor.analysis=FALSE)
summary(rand1)

## Proposed model
model1 <- "JP =~ JP1 + JP2 + JP3
           JA =~ JA1 + JA2
           CS =~ CS1 + CS2
           JP ~~ JA + CS
           JA ~~ CS"

ram1 <- lavaan2RAM(model1, obs.variables=c("JP1", "JP2", "JP3", "JA1", "JA2",
                                           "CS1", "CS2"))

## Stage 2 analysis
rand2 <- tssem2(rand1, RAM=ram1)
summary(rand2)

plot(rand2)

#### OSMASEM with covariance matrices
## Create a data frame for the OSMASEM
df <- Cor2DataFrame(issp05$data, n=issp05$n, Means=issp05$means,
                    cor.analysis=FALSE)

## Standardize idv
idv <- scale(issp05$idv)

## Replace missing values with mean
idv[is.na(idv)] <- mean(idv, na.rm=TRUE)
df$data$idv <- idv

## No moderator 
fit1 <- osmasem2(model.name="No_moderator", RAM=ram1, data=df,
                 cor.analysis=FALSE, mean.analysis=FALSE)
summary(fit1, fitIndices = TRUE)

## Proposed model with idv as a moderator
model2 <- "JP =~ a*JP1 + b*JP2 + c*JP3
           JA =~ d*JA1 + e*JA2
           CS =~ f*CS1 + g*CS2
           JP ~~ JA + CS
           JA ~~ CS
           a == a0 + a1*data.idv
           b == b0 + b1*data.idv
           c == c0 + c1*data.idv
           d == d0 + d1*data.idv
           e == e0 + e1*data.idv
           f == f0 + f1*data.idv
           g == g0 + g1*data.idv"

ram2 <- lavaan2RAM(model2, obs.variables=c("JP1", "JP2", "JP3", "JA1", "JA2",
                                           "CS1", "CS2"))

fit2 <- osmasem2(RAM=ram2, data=df, cor.analysis=FALSE, mean.analysis=FALSE,
                 replace.constraints = TRUE)
summary(fit2)

## Compare fit1 and fit2
anova(fit2, fit1)

A Dataset from Cheung and Chan (2005; 2009)

Description

Eleven covariance matrices on work-related attitudes were extracted from the Inter-University Consortium for Political and Social Research (1989). Nine variables were selected by Cheung and Chan (2005; 2009) for demonstration purposes. They were grouped into three constructs: Job Prospects measured by job security (JP1), income (JP2), and advancement opportunity (JP3); Job Nature measured by interesting job (JN1), independent work (JN2), help other people (JN3), and useful to society (JN4); and Time Demand measured by flexible working hours (TD1) and lots of leisure time (TD2).

Usage

data(issp89)

Details

A list of data with the following structure:

data

A list of 11 studies of covariance matrices

n

A vector of sample sizes

Author(s)

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

Source

Inter-University Consortium for Political and Social Research. (1989). International Social Survey Program: Work orientation. Ann Arbor, MI: Author.

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.

See Also

issp05

Examples

data(issp89)

#### Analysis of correlation structure in Cheung and Chan (2005)
#### Fixed-effects model: Stage 1 analysis
cor1 <- tssem1(issp89$data, issp89$n, method="FEM", cor.analysis=TRUE)
summary(cor1)
  
## 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
cor2 <- tssem2(cor1, Amatrix=A1, Smatrix=S1, Fmatrix=F1, intervals.type="LB")
summary(cor2)

## Display the model with the parameter estimates
plot(cor2, nDigits=1)

#### Analysis of covariance structure in Cheung and Chan (2009)
#### Fixed-effects model: Stage 1 analysis
cov1 <- tssem1(issp89$data, issp89$n, method="FEM", cor.analysis=FALSE)
summary(cov1)
  
#### Fixed-effects model: Stage 2 analysis
cov2 <- tssem2(cov1, Amatrix=A1, Smatrix=S1, Fmatrix=F1)              
summary(cov2)

## Display the model with the parameter estimates
plot(cov2, nDigits=1)

Convert lavaan models to RAM models

Description

It converts models specified in lavaan model syntax to RAM models.

Usage

lavaan2RAM(model, obs.variables = NULL, A.notation = "ON",
           S.notation = "WITH", M.notation = "mean",
           A.start=0.1, S.start=0.5, M.start=0,
           auto.var = TRUE, std.lv = TRUE, ngroups = 1, ...)

Arguments

. Please note that fixed.x is set at FALSE. Thus it cannot be passed to ....

Details

It uses the model.syntax to do the conversion. Experimental: functions of parameters (':=' in lavaan) and constraints ('==', '>', and '<' in lavaan) will be converted to mxAlgebra and mxConstraint in OpenMx. As there are differences between lavaan and OpenMx, they may not work properly.

Value

A list of RAM specification with A, S, F, and M matrices.

Author(s)

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

See Also

ramModel, Becker92, Becker09, Digman97, Hunter83, as.mxMatrix, checkRAM

Examples

## Regression model on correlation matrix
model1 <- "## y is modelled by x1, x2, and x3
           y ~ b1*x1 + b2*x2 + b3*x3
           ## Fix the independent variables at 1
           x1 ~~ 1*x1
           x2 ~~ 1*x2
           x3 ~~ 1*x3
           ## Declare the correlations among the independent variables
           x1 ~~ x2
           x1 ~~ x3
           x2 ~~ x3
           ## Constraint
           b3 == b1 + b2
           ## Function of parameters
           fn1 := b1*b2^b3"

## Compare the arrangements of variables with and without
## specifying the obs.variables arguments. 
lavaan2RAM(model1, obs.variables=c("y", "x1", "x2", "x3"))

## Two-factor CFA model
model2 <- "f1 =~ x1 + x2 + x3
           f2 =~ x4 + x5 + x6
           ## Declare the correlation between f1 and f2
           ## and label it with cor_f1f2
           f1 ~~ cor_f1f2*f2"

lavaan2RAM(model2)

## Regression model with the mean structure
model3 <- "y ~ x
           ## Intercept of y
           y ~ 1
           ## Mean of x
           x ~ 1"

lavaan2RAM(model3)

## Multiple group regression model
## Different intercepts with a common slope
model4 <- "y ~ c(a1, a2)*1 + c(b, b)*x"

lavaan2RAM(model4, ngroups=2)

Convert a List of Symmetric Matrices into a Stacked Matrix

Description

It converts a list of symmetric matrices into a stacked matrix. Dimensions of the symmetric matrices have to be the same. It tries to preserve the dimension names if possible. Dimension names will be created if there are no dimension names in the first symmetric matrix.

Usage

list2matrix(x, diag = FALSE)

Arguments

Value

A k x p* stacked matrix where p* = p(p-1)/2 for diag=FALSE or p* = p(p+1)/2 for diag=TRUE.

Author(s)

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

Examples

C1 <- matrix(c(1,0.5,0.4,0.5,1,0.2,0.4,0.2,1), ncol=3)  
C2 <- matrix(c(1,0.4,NA,0.4,1,NA,NA,NA,NA), ncol=3)  

## A list without dimension names 
list2matrix(list(C1, C2))
#      x2_x1 x3_x1 x3_x2
# [1,]   0.5   0.4   0.2
# [2,]   0.4    NA    NA

dimnames(C1) <- list( c("x","y","z"), c("x","y","z") )
dimnames(C2) <- list( c("x","y","z"), c("x","y","z") )

## A list with dimension names
list2matrix(list(C1, C2))
#      y_x z_x z_y
# [1,] 0.5 0.4 0.2
# [2,] 0.4  NA  NA

Convert a Matrix into a Block Diagonal Matrix

Description

It converts a matrix into a block diagonal matrix.

Usage

matrix2bdiag(x, ...)

Arguments

Details

Each row of x is converted into a symmetric matrix via vec2symMat. Then the list of the symmetric matrices is converted into a block diagonal matrix via a function written by Scott Chasalow posted at http://www.math.yorku.ca/Who/Faculty/Monette/pub/stmp/0827.html.

Author(s)

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

See Also

vec2symMat

Examples

(m1 <- matrix(1:12, ncol=6, byrow=TRUE))
#      [,1] [,2] [,3] [,4] [,5] [,6]
# [1,]    1    2    3    4    5    6
# [2,]    7    8    9   10   11   12

matrix2bdiag(m1)
#      [,1] [,2] [,3] [,4] [,5] [,6]
# [1,]    1    2    3    0    0    0
# [2,]    2    4    5    0    0    0
# [3,]    3    5    6    0    0    0
# [4,]    0    0    0    7    8    9
# [5,]    0    0    0    8   10   11
# [6,]    0    0    0    9   11   12

Univariate and Multivariate Meta-Analysis with Maximum Likelihood Estimation

Description

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

Usage

meta(y, v, x, data, intercept.constraints = NULL, coef.constraints = NULL,
     RE.constraints = NULL, RE.startvalues=0.1, RE.lbound = 1e-10,
     intervals.type = c("z", "LB"), I2="I2q", R2=TRUE,
     model.name="Meta analysis with ML", suppressWarnings = TRUE,
     silent = TRUE, run = TRUE, ...)
metaFIML(y, v, x, av, data, intercept.constraints=NULL,
         coef.constraints=NULL, RE.constraints=NULL,
         RE.startvalues=0.1, RE.lbound=1e-10,
         intervals.type=c("z", "LB"), R2=TRUE,
         model.name="Meta analysis with FIML",
         suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)

Arguments

Value

An object of class meta with a list of

Note

Missing values (NA) in y and their related elements in v will be removed automatically. When there are missing values in v but not in y, missing values will be replaced by 1e5. Effectively, these effect sizes will have little impact on the analysis. metaFIML() uses FIML to handle missing covariates in X. It is experimental. It may not be stable.

Author(s)

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

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.

Cheung, M. W.-L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267-294.

Cheung, M. W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20, 429-454.

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

Hardy, R. J., & Thompson, S. G. (1996). A likelihood approach to meta-analysis with random effects. Statistics in Medicine, 15, 619-629.

Neale, M. C., & Miller, M. B. (1997). The use of likelihood-based confidence intervals in genetic models. Behavior Genetics, 27, 113-120.

Raudenbush, S. W. (2009). Analyzing effect sizes: random effects models. In H. M. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 295-315). New York: Russell Sage Foundation.

Xiong, C., Miller, J. P., & Morris, J. C. (2010). Measuring study-specific heterogeneity in meta-analysis: application to an antecedent biomarker study of Alzheimer's disease. Statistics in Biopharmaceutical Research, 2(3), 300-309. doi:10.1198/sbr.2009.0067

See Also

reml, Hox02, Berkey98, wvs94a

Convert metaSEM objects into semPlotModel objects for plotting

Description

It converts objects in class wls into objects of class semPlotModel.

Usage

meta2semPlot(object, manNames = NULL, latNames = NULL, labels = c("labels", "RAM"), ...)

Arguments

Details

It uses the ramModel() to do the conversion.

Value

A "semPlotModel" object.

Author(s)

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

See Also

ramModel, Becker92, Becker09, Digman97, Hunter83

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

## Depreciated in the future
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, ...)
## Depreciated in the future
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, ...)
meta3L(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, ...)
meta3LFIML(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

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.

meta3L() 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. meta3LFIML() 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 meta3LFIML(). Moreover, auxiliary variables av2 at level-2 and av3 at level-3 may be included to improve the estimation. Although meta3LFIML() is more flexible in handling missing covariates, it is more likely to encounter estimation problems.

Author(s)

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

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

reml3L, Cooper03, Bornmann07

One-stage meta-analytic structural equation modeling

Description

It fits MASEM with the one-stage MASEM (OSMASEM) approach.

Usage

osmasem(model.name="osmasem", RAM=NULL, Mmatrix=NULL,
        Tmatrix=NULL, Jmatrix=NULL, Ax=NULL, Sx=NULL,
        A.lbound=NULL, A.ubound=NULL,
        RE.type=c("Diag", "Symm", "Zero"), data,
        subset.variables=NULL, subset.rows=NULL, 
        intervals.type = c("z", "LB"),
        mxModel.Args=NULL, mxRun.Args=NULL,
        suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)
osmasem2(model.name="osmasem2", RAM, data, cor.analysis=TRUE,
         RE.type.Sigma=c("Diag", "Symm", "Zero"),
         RE.type.Mu=c("Symm", "Diag", "Zero"),
         RE.type.SigmaMu=c("Zero", "Full"),
         mean.analysis=FALSE, intervals.type=c("z", "LB"),
         startvalues=NULL, replace.constraints=FALSE,
         mxModel.Args=NULL, run=TRUE, ...)

Arguments

Details

osmasem was implemented based on Jak and Cheung (2020) for meta-analyzing correlation matrices. osmasem2 was a rewrite designed to handle correlation or covariance matrices, including the means. There are several major differences between them: 1. osmasem allows the use of RAM or (Mmatrix and Tmatrix), while osmasem2 calculates the Mmatrix and Tmatrix based on the RAM input. 2. RE.type is used to specify the type of random effects on the correlations in osmasem. On the contrary, osmasem2 has three types of random effects: correlations/covariances, means, and covariance between correlations/covariance and means. 3. osmasem reports the transformed random effects in the parameter table. Users have to use VarCorr to obtain the heterogeneity matrix of the random effects. In contrast, osmasem2 reports the heterogeneity matrix in the parameter table. 4. osmasem2 allows the imposition of linear and nonlinear constraints and the creation of parameter functions in RAM, which osmasem does not.

Value

An object of class osmasem

Author(s)

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

References

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

See Also

Cor2DataFrame, create.vechsR, create.Tau2, create.V, osmasem, Nohe15, issp05

Calculate the R2 in OSMASEM and OSMASEM3L

Description

It calculates the R2 of the moderators in explaining the variances in the heterogeneity variances.

Usage

osmasemR2(model1, model0, R2.truncate=TRUE)

Arguments

Value

model1 and model0 are the models with and without the moderators, respectively. The function does not check whether the models are nested. It is the users' responsibility to make sure that the models with and without the moderators are nested. It returns a list of the diagonals of the heterogeneity variances of the models without and with the moderators, and the R2.

Author(s)

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

See Also

osmasem

Calculate the SRMR in OSMASEM and OSMASEM3L

Description

It calculates the standardized root mean squared residuals (SRMR) in OSMASEM and OSMASEM3L.

Usage

osmasemSRMR(x)

Arguments

Value

It calculates the model implied correlation matrix and its saturated counterpart to calculate the SRMR. It should be noted that the heterogeneity variances are ignored in the calculations.

Author(s)

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

See Also

osmasem, Nohe15

Display the Accumulative Sample Sizes for the Covariance Matrix

Description

It displays the accumulative sample sizes for the covariance matrix.

Usage

pattern.n(x, n)

Arguments

Value

A square matrix of the accumulative sample sizes of the input matrices.

Author(s)

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

Examples

## Show the pattern of missing data
pattern.n(Hunter83$data, Hunter83$n)

#             Ability Knowledge Work sample Supervisor
# Ability        3815      3372        3281       3605
# Knowledge      3372      3532        2998       3322
# Work sample    3281      2998        3441       3231
# Supervisor     3605      3322        3231       3765

Display the Pattern of Missing Data of a List of Square Matrices

Description

It displays the pattern of missing data (or pattern of data that are present) of a list of square matrices with the same dimensions.

Usage

pattern.na(x, show.na = TRUE, type=c("tssem", "osmasem"))

Arguments

Value

A square matrix of numerical values with the same dimensions of the input matrices.

Author(s)

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

Examples

## Show the pattern of missing data
pattern.na(Hunter83$data, show.na=TRUE)

#             Ability Knowledge Work sample Supervisor
# Ability           1         3           3          2
# Knowledge         3         2           4          3
# Work sample       3         4           2          3
# Supervisor        2         3           3          1

## Show the pattern of data that are present
pattern.na(Hunter83$data, show.na=FALSE)

#             Ability Knowledge Work sample Supervisor
# Ability          13        11          11         12
# Knowledge        11        12          10         11
# Work sample      11        10          12         11
# Supervisor       12        11          11         13

Plot methods for various objects

Description

It plots the models from either the lavaan model or meta, wls, and osmasem objects.

Usage

## S3 method for class 'meta'
plot(x, effect.sizes, add.margin = 0.1, interval = 0.95,
     main= "Effect Sizes and their Confidence Ellipses",
     axis.labels= paste("Effect size ", effect.sizes, sep = ""),
     study.col = "black", study.pch = 19, study.min.cex = 0.8,
     study.weight.plot = FALSE, study.ellipse.plot = TRUE,
     study.ellipse.col = "black", study.ellipse.lty = 2,
     study.ellipse.lwd = 0.5, estimate.col = "blue",
     estimate.pch = 18, estimate.cex = 2,
     estimate.ellipse.plot = TRUE, estimate.ellipse.col = "red",
     estimate.ellipse.lty = 1, estimate.ellipse.lwd = 2,
     randeff.ellipse.plot = TRUE, randeff.ellipse.col = "green",
     randeff.ellipse.lty = 1, randeff.ellipse.lwd = 2,
     univariate.plot = TRUE, univariate.lines.col = "gray",
     univariate.lines.lty = 3, univariate.lines.lwd = 1,
     univariate.polygon.width = 0.02,
     univariate.polygon.col = "red",
     univariate.arrows.col = "green", univariate.arrows.lwd = 2,
     diag.panel = FALSE, xlim=NULL, ylim=NULL, ...)
## S3 method for class 'character'
plot(x, fixed.x=FALSE, nCharNodes=0, nCharEdges=0,
     layout=c("tree", "circle", "spring", "tree2", "circle2"),
     sizeMan=8, sizeLat=8, edge.label.cex=1.3, color="white", ...)
## S3 method for class 'wls'
plot(x, manNames=NULL, latNames=NULL, labels=c("labels", "RAM"),
     what="est", nCharNodes=0, nCharEdges=0,
     layout=c("tree", "circle", "spring", "tree2", "circle2"),
     sizeMan=8, sizeLat=8, edge.label.cex=1.3, color="white",
     weighted=FALSE, ...)
## S3 method for class 'osmasem'
plot(x, manNames=NULL, latNames=NULL, labels=c("labels", "RAM"),
     what="est", nCharNodes=0, nCharEdges=0,
     layout=c("tree", "circle", "spring", "tree2", "circle2"),
     sizeMan=8, sizeLat=8, edge.label.cex=1.3, color="white",
  weighted=FALSE, ...)
## S3 method for class 'osmasem2'
plot(x, manNames=NULL, latNames=NULL, labels=c("labels", "RAM"),
     what="est", nCharNodes=0, nCharEdges=0,
     layout=c("tree", "circle", "spring", "tree2", "circle2"),
     sizeMan=8, sizeLat=8, edge.label.cex=1.3, color="white",
  weighted=FALSE, ...)
## S3 method for class 'mxsem'
plot(x, manNames=NULL, latNames=NULL, labels=c("labels", "RAM"),
     what="est", nCharNodes=0, nCharEdges=0,
     layout=c("tree", "circle", "spring", "tree2", "circle2"),
     sizeMan=8, sizeLat=8, edge.label.cex=1.3, color="white",
    weighted=FALSE, ...)

Arguments

Note

The estimated effect sizes and random effects are based on the labels Intercept1, Intercept2, ... and Tau2_1_1, Tau2_2_1, Tau2_2_2, etc. At least two effect sizes are required for this function.

Author(s)

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

References

Cheung, M. W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20, 429-454.

See Also

Berkey98, wvs94a meta2semPlot semPaths

Examples

## lavaan model
model <- "y ~ m + x
          m ~ x"
plot(model)

Print Methods for various Objects

Description

Print methods for the tssem1FEM, tssem1FEM.cluster, tssem1REM, wls, meta, meta3LFIML, reml, uniR1 and impliedR objects.

Usage

## S3 method for class 'tssem1FEM'
print(x, ...)
## S3 method for class 'tssem1FEM.cluster'
print(x, ...)
## S3 method for class 'tssem1REM'
print(x, ...)
## S3 method for class 'wls'
print(x, ...)
## S3 method for class 'meta'
print(x, ...)
## S3 method for class 'meta3LFIML'
print(x, ...)
## S3 method for class 'reml'
print(x, ...)
## S3 method for class 'uniR1'
print(x, ...)
## S3 method for class 'impliedR'
print(x, ...)

Arguments

Author(s)

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

See Also

tssem1, wls, meta, reml

Generate (Nested) Sample/Population Correlation/Covariance Matrices

Description

It generates (nested) random sample or population correlation or covariance matrices. rCor() is a wrapper to call rCorPop() and then rCorSam().

Usage

rCor(Sigma, V, n, corr=TRUE, raw.data=FALSE,
     nonPD.pop=c("replace", "nearPD", "accept"),
     nonPD.sam=c("stop", "nearPD"))
rCorPop(Sigma, V, k, corr=TRUE, 
        nonPD.pop=c("replace", "nearPD", "accept"))
rCorSam(Sigma, n, corr=TRUE, raw.data=FALSE, 
        nonPD.sam=c("stop", "nearPD"))
rCor3L(Sigma, V.B, V.W, n, cluster, corr=TRUE, raw.data=FALSE,
       nonPD.pop=c("replace", "nearPD", "accept"),
       nonPD.sam=c("stop", "nearPD"))

Arguments

Value

An object of the generated population/sample correlation/covariance matrices.

Author(s)

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

Examples

Sigma <- matrix(c(1, .2, .3,
                  .2, 1, .4,
                  .3, .4, 1), ncol=3, nrow=3)
V <- diag(c(.1, .1, .1))

## Generate two population correlation matrices
Pop.corr <- rCorPop(Sigma, V, k=2)
Pop.corr
                  
summary(Pop.corr)
                  
## Generate two sample correlation matrices
rCorSam(Sigma=Pop.corr, n=c(10, 10))
                 
## The above code is the same as the following one
rCor(Sigma, V, n=c(10, 10))                  

Read External Correlation/Covariance Matrices

Description

It reads full/lower triangle/stacked vectors of correlation/covariance data into a list of correlation/covariance matrices.

Usage

readFullMat(file, ...)
readStackVec(file, ...)
readLowTriMat(file, no.var, ...)

Arguments

Value

A list of correlation/covariance matrices.

Author(s)

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

Examples

## Write two full correlation matrices into a file named "fullmat.dat".
## x2 is missing in the second matrix.
## The content of "fullmat.dat" is
# 1.0 0.3 0.4
# 0.3 1.0 0.5
# 0.4 0.5 1.0
# 1.0 NA 0.4
# NA NA NA
# 0.4 NA 1.0

## cat("1.0 0.3 0.4\n0.3 1.0 0.5\n0.4 0.5 1.0
## 1.0 NA 0.4\nNA NA NA\n0.4 NA 1.0",
## file="fullmat.dat", sep="")

## Read the correlation matrices from a file
## my.full <- readFullMat("fullmat.dat")

## Read the correlation matrices from a string
x <-
"1.0 0.3 0.4
0.3 1.0 0.5
0.4 0.5 1.0
1.0 NA 0.4
NA NA NA
0.4 NA 1.0"

my.full <- readFullMat(textConnection(x))

## my.full
# $`1`
#     x1  x2  x3
# x1 1.0 0.3 0.4
# x2 0.3 1.0 0.5
# x3 0.4 0.5 1.0
#
# $`2`
#     x1 x2  x3
# x1 1.0 NA 0.4
# x2  NA NA  NA
# x3 0.4 NA 1.0

## Write two lower triangle correlation matrices into a file named "lowertriangle.dat".
## x2 is missing in the second matrix.
## The content of "lowertriangle.dat" is
# 1.0 
# 0.3 1.0 
# 0.4 0.5 1.0
# 1.0
# NA NA 
# 0.4 NA 1.0
## cat("1.0\n0.3 1.0\n0.4 0.5 1.0\n1.0\nNA NA\n0.4 NA 1.0",
##     file="lowertriangle.dat", sep="")

## Read the lower triangle correlation matrices from a file
## my.lowertri <- readLowTriMat(file = "lowertriangle.dat", no.var = 3)

## Read the correlation matrices from a string
x <-
"1.0 
0.3 1.0 
0.4 0.5 1.0
1.0
NA NA 
0.4 NA 1.0"

my.lowertri <- readLowTriMat(textConnection(x), no.var = 3)

## my.lowertri
# $`1`
#     x1  x2  x3
# x1 1.0 0.3 0.4
# x2 0.3 1.0 0.5
# x3 0.4 0.5 1.0
#
# $`2`
#     x1 x2  x3
# x1 1.0 NA 0.4
# x2  NA NA  NA
# x3 0.4 NA 1.0

## Write two vectors of correlation coefficients based on
##  column major into a file named "stackvec.dat".
## x2 is missing in the second matrix.
## The content of "stackvec.dat" is
# 1.0 0.3 0.4 1.0 0.5 1.0
# 1.0 NA 0.4 NA NA 1.0
## cat("1.0 0.3 0.4 1.0 0.5 1.0\n1.0 NA 0.4 NA NA 1.0\n",
##     file="stackvec.dat", sep="")

## Read the stack vectors from a file
## my.vec <- readStackVec("stackvec.dat")

## Read the stack vectors from a string
x <- "
1.0 0.3 0.4 1.0 0.5 1.0
1.0 NA 0.4 NA NA 1.0"

my.vec <- readStackVec(textConnection(x))

## my.vec
# $`1`
#     x1  x2  x3
# x1 1.0 0.3 0.4
# x2 0.3 1.0 0.5
# x3 0.4 0.5 1.0
#
# $`2`
#    x1 x2  x3
# x1 1.0 NA 0.4
# x2  NA NA  NA
# x3 0.4 NA 1.0

Estimate Variance Components with Restricted (Residual) Maximum Likelihood Estimation

Description

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

Usage

reml(y, v, x, data, RE.constraints = NULL, RE.startvalues = 0.1,
     RE.lbound = 1e-10, intervals.type = c("z", "LB"),
     model.name="Variance component with REML",
     suppressWarnings = TRUE, silent = TRUE, run = TRUE, ...)

Arguments

Details

Restricted (residual) maximum likelihood obtains the parameter estimates on the transformed data that do not include the fixed-effects parameters. A transformation matrix M=I-X(X'X)^{-1}X is created based on the design matrix X which is just a column vector when there is no predictor in x. The last N redundant rows of M is removed where N is the rank of X. After pre-multiplying by M on y, the parameters of fixed-effects are removed from the model. Thus, only the parameters of random-effects are estimated.

An alternative but equivalent approach is to minimize the -2*log-likelihood function:

\log(\det|V+T^2|)+\log(\det|X'(V+T^2)^{-1}X|)+(y-X\hat{\alpha})'(V+T^2)^{-1}(y-X\hat{\alpha})

where V is the known conditional sampling covariance matrix of y, T^2 is the variance component of the random effects, and \hat{\alpha}=(X'(V+T^2)^{-1}X)^{-1} X'(V+T^2)^{-1}y. reml() minimizes the above likelihood function to obtain the parameter estimates.

Value

An object of class reml with a list of

Note

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

Author(s)

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

References

Cheung, M. W.-L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling, 20(1), 157-167.

Mehta, P. D., & Neale, M. C. (2005). People Are Variables Too: Multilevel Structural Equations Modeling. Psychological Methods, 10(3), 259-284.

Searle, S. R., Casella, G., & McCulloch, C. E. (1992). Variance components. New York: Wiley.

Viechtbauer, W. (2005). Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30(3), 261-293.

See Also

meta, reml3, Hox02, Berkey98

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

Description

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

Usage

## Depreciated in the future
reml3(y, v, cluster, x, data, RE2.startvalue=0.1, RE2.lbound=1e-10,
      RE3.startvalue=RE2.startvalue, RE3.lbound=RE2.lbound, RE.equal=FALSE,
      intervals.type=c("z", "LB"), model.name="Variance component with REML",
      suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)
reml3L(y, v, cluster, x, data, RE2.startvalue=0.1, RE2.lbound=1e-10,
      RE3.startvalue=RE2.startvalue, RE3.lbound=RE2.lbound, RE.equal=FALSE,
      intervals.type=c("z", "LB"), model.name="Variance component with REML",
      suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...) 

Arguments

Details

Restricted (residual) maximum likelihood obtains the parameter estimates on the transformed data that do not include the fixed-effects parameters. A transformation matrix M=I-X(X'X)^{-1}X is created based on the design matrix X which is just a column vector when there is no predictor in x. The last N redundant rows of M is removed where N is the rank of X. After pre-multiplying by M on y, the parameters of fixed-effects are removed from the model. Thus, only the parameters of random-effects are estimated.

An alternative but the equivalent approach is to minimize the -2*log-likelihood function:

\log(\det|V+T^2|)+\log(\det|X'(V+T^2)^{-1}X|)+(y-X\hat{\alpha})'(V+T^2)^{-1}(y-X\hat{\alpha})

where V is the known conditional sampling covariance matrix of y, T^2 is the variance component combining level-2 and level-3 random effects, and \hat{\alpha}=(X'(V+T^2)^{-1}X)^{-1} X'(V+T^2)^{-1}y. reml() minimizes the above likelihood function to obtain the parameter estimates.

Value

An object of class reml with a list of

Note

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

Author(s)

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

References

Cheung, M. W.-L. (2013). Implementing restricted maximum likelihood estimation in structural equation models. Structural Equation Modeling, 20(1), 157-167.

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

Mehta, P. D., & Neale, M. C. (2005). People Are Variables Too: Multilevel Structural Equations Modeling. Psychological Methods, 10(3), 259-284.

Searle, S. R., Casella, G., & McCulloch, C. E. (1992). Variance components. New York: Wiley.

See Also

meta3L, reml, Cooper03, Bornmann07

Rerun models via mxTryHard()

Description

It reruns models via mxTryHard().

Usage

rerun(object, autofixtau2=FALSE, extraTries=10, ...)

Arguments

Author(s)

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

Examples

random1 <- tssem1(Digman97$data, Digman97$n, method="REM", RE.type="Diag")
random1_rerun <- rerun(random1)
summary(random1_rerun)

Fit a structural equation model using OpenMx

Description

It fits a structural equation model by creating a mxModel from a RAM object.

Usage

## Depreciated in the future
create.mxModel(model.name="sem", RAM=NULL, data=NULL,
               Cov=NULL, means=NULL, numObs,
               intervals.type=c("z", "LB"), startvalues=NULL,
               replace.constraints=FALSE, mxModel.Args=NULL,
               run=TRUE, silent=TRUE, ...)
sem(model.name="sem", RAM=NULL, data=NULL,
    Cov=NULL, means=NULL, numObs,
    intervals.type=c("z", "LB"), startvalues=NULL,
    lbound=NULL, ubound=NULL, replace.constraints=FALSE,
    mxModel.Args=NULL, run=TRUE, silent=TRUE, ...)

Arguments

Value

An object of class mxsem

Note

when there are constraints with replace.constraints=TRUE and intervals.type="LB", it returns an error because some parameters in the model are replaced with the new parameters in the constraints. However, the names of these new parameters are not captured in the CI object. Users are advised to use intervals.type="z" before it is fixed.

Author(s)

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

Examples

## Generate data
set.seed(100)
n <- 100
x <- rnorm(n)
y <- 0.5*x + rnorm(n, mean=0, sd=sqrt(1-0.5^2))
my.df <- data.frame(y=y, x=x)

## A regression model
model <- "y ~ x   # Regress y on x
          y ~ 1   # Intercept of y
          x ~ 1   # Mean of x"
plot(model)

RAM <- lavaan2RAM(model, obs.variables=c("y", "x"))

my.fit <- sem(RAM=RAM, data=my.df)
summary(my.fit)

## A meta-analysis
model <- "f =~ 1*yi
          f ~ mu*1          ## Average effect
          f ~~ tau2*f       ## Heterogeneity variance
          yi ~~ data.vi*yi  ## Known sampling variance"
plot(model)

## Do not standardize the latent variable (f): std.lv=FALSE 
RAM <- lavaan2RAM(model, obs.variables="yi", std.lv=FALSE)

## Use likelihood-based CI 
my.fit <- sem(RAM=RAM, data=Hox02, intervals="LB")
summary(my.fit)

Compute Effect Sizes for Multiple End-point Studies

Description

It computes the standardized mean differences and their asymptotic sampling covariance matrix for two multiple end-point studies with p effect sizes.

Usage

smdMES(m1, m2, V1, V2, n1, n2,
       homogeneity=c("covariance", "correlation", "none"), 
       bias.adjust=TRUE, list.output=TRUE, lavaan.output=FALSE) 

Arguments

Details

Gleser and Olkin (2009) introduce formulas to calculate the standardized mean differences and their sampling covariance matrix for multiple end-point studies under the assumption of homogeneity of the covariance matrix. This function uses a structural equation modeling (SEM) approach introduced in Chapter 3 of Cheung (2015) to calculate the same estimates. The SEM approach is more flexible in two ways: (1) it allows homogeneity of covariance or correlation matrices or not; and (2) it allows users to test this assumption by checking the fitted lavaan-class object.

Author(s)

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

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

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). New York: Russell Sage Foundation.

See Also

Gleser94, smdMTS, calEffSizes

Examples

    
## Sample means for the two constructs in Group 1  
m1 <- c(2.5, 4.5)

## Sample means for the two constructs in Group 2     
m2 <- c(3, 5)

## Sample covariance matrix in Group 1    
V1 <- matrix(c(3,2,2,3), ncol=2)

## Sample covariance matrix in Group 2
V2 <- matrix(c(3.5,2.1,2.1,3.5), ncol=2)

## Sample size in Group 1
n1 <- 20

## Sample size in Group 2    
n2 <- 25

## SMD with the assumption of homogeneity of covariance matrix    
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cov", bias.adjust=TRUE,
       lavaan.output=FALSE)

## SMD with the assumption of homogeneity of correlation matrix 
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cor", bias.adjust=TRUE,
       lavaan.output=FALSE)

## SMD without any assumption of homogeneity
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="none", bias.adjust=TRUE,
       lavaan.output=FALSE)

## Output the fitted lavaan model
## It provides a likelihood ratio test to test the null hypothesis of
## homogeneity of variances.     
fit <- smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cov", bias.adjust=TRUE,
              lavaan.output=TRUE)

lavaan::summary(fit)

lavaan::parameterestimates(fit)

Compute Effect Sizes for Multiple Treatment Studies

Description

It computes the standardized mean differences and their asymptotic sampling covariance matrix for k multiple treatment studies. The first group is assumed as the control group.

Usage

smdMTS(m, v, n, homogeneity=c("variance", "none"), bias.adjust=TRUE, 
       all.comparisons=FALSE, list.output=TRUE, lavaan.output=FALSE) 

Arguments

Details

Gleser and Olkin (2009) introduce formulas to calculate the standardized mean differences and their sampling covariance matrix for multiple treatment studies under the assumption of homogeneity of the covariance matrix. This function uses a structural equation modeling (SEM) approach introduced in Chapter 3 of Cheung (2015) to calculate the same estimates. The SEM approach is more flexible in three ways: (1) it allows homogeneity of variances or not; (2) it allows users to test the assumption of homogeneity of variances by checking the fitted lavaan-class object; and (3) it may calculate all pairwise comparisons.

Author(s)

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

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

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). New York: Russell Sage Foundation.

See Also

Gleser94, smdMES, calEffSizes

Examples

  
## Sample means for groups 1 to 3
m <- c(5,7,9)

## Sample variances
v <- c(10,11,12)

## Sample sizes
n <- c(50,52,53)

## Assuming homogeneity of variances
smdMTS(m, v, n, homogeneity = "var", bias.adjust=TRUE, all.comparisons=FALSE,
       lavaan.output=FALSE)

## Not assuming homogeneity of variances and comparing all pairwise groups
## Please note that the SD of the first group is used as the standardizer    
smdMTS(m, v, n, homogeneity = "none", bias.adjust=TRUE, all.comparisons=TRUE,
       lavaan.output=FALSE)

## Output the fitted lavaan model
## It provides a likelihood ratio test to test the null hypothesis of
## homogeneity of variances.    
fit <- smdMTS(m, v, n, homogeneity = "var", bias.adjust=FALSE, all.comparisons=FALSE,
              lavaan.output=TRUE)

lavaan::summary(fit)
    
lavaan::parameterestimates(fit)

Summary Method for tssem1, wls, meta, and meta3LFIML Objects

Description

It summaries results for various class.

Usage

## S3 method for class 'tssem1FEM'
summary(object, ...)
## S3 method for class 'tssem1FEM.cluster'
summary(object, ...)
## S3 method for class 'tssem1REM'
summary(object, robust=FALSE, ...)
## S3 method for class 'wls'
summary(object, df.adjustment=0, ...)
## S3 method for class 'wls.cluster'
summary(object, df.adjustment=0, ...)
## S3 method for class 'meta'
summary(object, homoStat=TRUE, robust=FALSE, ...)
## S3 method for class 'meta3LFIML'
summary(object, allX=FALSE, robust=FALSE, ...)
## S3 method for class 'reml'
summary(object, ...)
## S3 method for class 'mxsem'
summary(object, robust=FALSE, ...)
## S3 method for class 'CorPop'
summary(object, ...)
## S3 method for class 'Cor3L'
summary(object, ...)
## S3 method for class 'bootuniR2'
summary(object, probs=c(0, 0.1, 0.5, 0.9, 1),
        cutoff.chisq.pvalue=0.05, cutoff.CFI=0.9, cutoff.SRMR=0.1,
        cutoff.RMSEA=0.05, ...)
## S3 method for class 'osmasem'
summary(object, fitIndices=FALSE, numObs, robust=FALSE, ...)
## S3 method for class 'osmasem'
summary(object, fitIndices=FALSE, numObs, robust=FALSE, ...)
## S3 method for class 'tssem1FEM'
print.summary(x, ...)
## S3 method for class 'wls'
print.summary(x, ...)
## S3 method for class 'meta'
print.summary(x, ...)
## S3 method for class 'meta3LFIML'
print.summary(x, ...)
## S3 method for class 'reml'
print.summary(x, ...)
## S3 method for class 'mxsem'
print.summary(x, ...)
## S3 method for class 'CorPop'
print.summary(x, ...)
## S3 method for class 'Cor3L'
print.summary(x, ...)
## S3 method for class 'bootuniR2'
print.summary(x, ...)

Arguments

Note

If the OpenMx status1 is either 0 or 1, the estimation is considered fine. If the OpenMx status1 is other values, it indicates estimation problems. Users should refer to 'OpenMx' website for more details.

Author(s)

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

See Also

tssem1, wls, meta, reml, rCor, bootuniR2, osmasem

First Stage of the Two-Stage Structural Equation Modeling (TSSEM)

Description

It conducts the first stage analysis of TSSEM by pooling correlation/covariance matrices. tssem1FEM() and tssem1REM() use fixed- and random-effects models, respectively. tssem1() is a wrapper of these functions.

Usage

tssem1(Cov, n, method=c("REM","FEM"), cor.analysis = TRUE, cluster=NULL,
       RE.type=c("Diag", "Symm", "Zero", "User"), RE.startvalues=0.1,
       RE.lbound=1e-10, RE.constraints=NULL, I2="I2q",
       acov=c("weighted", "individual", "unweighted"), asyCovOld=FALSE,
       model.name=NULL, suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)
tssem1FEM(Cov, n, cor.analysis=TRUE, model.name=NULL,
          cluster=NULL, suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)
tssem1REM(Cov, n, cor.analysis=TRUE, RE.type=c("Diag", "Symm", "Zero","User"),
          RE.startvalues=0.1, RE.lbound=1e-10, RE.constraints=NULL,
          I2="I2q", acov=c("weighted", "individual", "unweighted"),
          asyCovOld=FALSE, model.name=NULL, suppressWarnings=TRUE,
          silent=TRUE, run=TRUE, ...)

Arguments

Value

Either an object of class tssem1FEM for fixed-effects TSSEM, an object of class tssem1FEM.cluster for fixed-effects TSSEM with cluster argument, or an object of class tssem1REM for random-effects TSSEM.

Author(s)

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

References

Cheung, M. W.-L. (2014). Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods, 46, 29-40.

Cheung, M. W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20, 429-454.

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.

See Also

wls, Cheung09, Becker92, Digman97, issp89, issp05

Estimate the heterogeneity (SD) of the parameter estimates of the TSSEM object

Description

It estimates the heterogeneity of the parameter estimates of the TSSEM objects using either the bootstrap or the delta methods.

Usage

tssemParaVar(tssem1.obj, tssem2.obj, method=c("bootstrap", "delta"),
             interval=0.8, Rep=50, output=c("data.frame", "matrices"),
             nonPD.pop=c("replace", "nearPD", "accept"))

Arguments

Details

The bootstrap method is based on the discussion in Cheung (2018) and Yu et al. (2016). The delta method is an alternative method to obtain the heterogeneity.

Value

Either a data.frame or matrices of the output.

Author(s)

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

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

bootuniR1, bootuniR2, Nohe15

First Stage analysis of the univariate R (uniR) approach

Description

It conducts the first stage analysis of the uniR analysis by pooling elements of the correlation coefficients individually.

Usage

uniR1(Cor, n, ...)

Arguments

Details

This function implements the univariate r approach proposed by Viswesvaran and Ones (1995) to conduct meta-analytic structural equation modeling (MASEM). It uses Schmidt and Hunter's approach to combine correlation coefficients. It is included in this package for research interests. The two-stage structural equation modeling (TSSEM) approach is preferred (e.g., Cheung, 2015; Cheung & Chan, 2005).

Value

An object of class uniR1 of the original data, the sample sizes, the harmonic mean of sample sizes, the average correlation matrix, the standard errors of the correlation matrix, and the standard deviations (heterogeneity) of the correlation matrix.

Author(s)

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

References

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

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

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: Sage.

Viswesvaran, C., & Ones, D. S. (1995). Theory testing: Combining psychometric meta-analysis and structural equations modeling. Personnel Psychology, 48, 865-885.

See Also

uniR2, Becker09

Second Stage analysis of the univariate R (uniR) approach

Description

It conducts the second stage analysis of the uniR analysis by fitting structural equation models on the average correlation matrix.

Usage

uniR2mx(x, RAM = NULL, Amatrix = NULL, Smatrix = NULL, Fmatrix = NULL,
        model.name=NULL, suppressWarnings=TRUE, silent=TRUE,
        run=TRUE, ...)
uniR2lavaan(x, model, ...)

Arguments

Details

This function implements the univariate r approach proposed by Viswesvaran and Ones (1995) to conduct meta-analytic structural equation modeling (MASEM). It treats the average correlation matrix as if it was a covariance matrix in fitting structural equation models. The harmonic mean of the sample sizes in combining correlation coefficients is used as the sample size in fitting structural equation models. It is included in this package for research interests. The two-stage structural equation modeling (TSSEM) approach is preferred (e.g., Cheung, 2015; Cheung & Chan, 2005).

Value

A fitted object returned from mxRun or sem.

Author(s)

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

References

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

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

Viswesvaran, C., & Ones, D. S. (1995). Theory testing: Combining psychometric meta-analysis and structural equations modeling. Personnel Psychology, 48, 865-885.

See Also

uniR1, lavaan2RAM, Becker09

Dataset on the effectiveness of multidimensional family therapy in treating adolescents with multiple behavior problems

Description

This dataset includes 61 effect sizes from 19 manuscripts nested from 8 studies reported by van der Pol et al. (2017). It studies the effectiveness of multidimensional family therapy in treating adolescents with multiple behavior problems.

Usage

data(vanderPol17)

Details

A list of data with the following structure:

Number

Number of the effect size.

Study

Authors of the studies.

N

Total sample size.

N_target

Sample size in the target group.

N_control

Sample size in the control group.

Comparison_condition

Either cognitive behavioral therapy (CBT), combined treatment (CT) or group therapy (Group).

Study_ID

Level-3 cluster.

Age_mean

Mean age of the participants.

Fllow_up

Follow-up duration (in months).

Per_Males

Percentage of males.

Per_Minorities

Percentage of minorities.

Per_Conduct_disorder

Percentage of participants with conduct disorder

Per_Severe_cannabis_users

Percentage of participants of severe cannabis use.

Outcome_measure

Either substance abuse, delinquency, externalizing and internalizing psychopathology, and family functioning

d

Effect size in Cohen's d.

v

Sampling variance of d

Source

van der Pol, T. M., Hoeve, M., Noom, M. J., Stams, G. J. J. M., Doreleijers, T. A. H., van Domburgh, L., & Vermeiren, R. R. J. M. (2017). Research Review: The effectiveness of multidimensional family therapy in treating adolescents with multiple behavior problems - a meta-analysis. Journal of Child Psychology and Psychiatry, 58(5), 532-545. https://doi.org/10.1111/jcpp.12685

Examples

data(vanderPol17)

Extract Covariance Matrix Parameter Estimates from Objects of Various Classes

Description

It extracts the variance-covariance matrix of the parameter estimates from objects of various classes.

Usage

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

Arguments

Value

A variance-covariance matrix of the parameter estimates.

Note

vcov returns NA when the diag.constraints=TRUE argument is used in wls objects.

Author(s)

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

See Also

tssem1, wls, meta, reml

Examples

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

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

## Random-effects only
vcov(model1, select="random")

Convert a Vector into a Symmetric Matrix

Description

It converts a vector into a symmetric matrix by filling up the elements into the lower triangle of the matrix.

Usage

vec2symMat(x, diag = TRUE, byrow = FALSE)

Arguments

Value

A symmetric square matrix based on column major

Author(s)

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

See Also

matrix2bdiag

Examples

vec2symMat(1:6)
#      [,1] [,2] [,3]
# [1,]    1    2    3
# [2,]    2    4    5
# [3,]    3    5    6

vec2symMat(1:6, diag=FALSE)
#      [,1] [,2] [,3] [,4]
# [1,]    1    1    2    3
# [2,]    1    1    4    5
# [3,]    2    4    1    6
# [4,]    3    5    6    1

vec2symMat(letters[1:6])
#      [,1] [,2] [,3]
# [1,] "a"  "b"  "c" 
# [2,] "b"  "d"  "e" 
# [3,] "c"  "e"  "f" 

Conduct a Correlation/Covariance Structure Analysis with WLS

Description

It fits a correlation or covariance structure with weighted least squares (WLS) estimation method where the inverse of the asymptotic covariance matrix is used as the weight matrix. tssem2 conducts the second stage analysis of the two-stage structural equation modeling (TSSEM). tssem2 is a wrapper of wls.

Usage

wls(Cov, aCov, n, RAM=NULL, Amatrix=NULL, Smatrix=NULL, Fmatrix=NULL,
    diag.constraints=FALSE, cor.analysis=TRUE, intervals.type=c("z","LB"),
    mx.algebras=NULL, mxModel.Args=NULL, subset.variables=NULL, model.name=NULL,
    suppressWarnings=TRUE, silent=TRUE, run=TRUE, ...)
tssem2(tssem1.obj, RAM=NULL, Amatrix=NULL, Smatrix=NULL, Fmatrix=NULL,
       diag.constraints=FALSE, intervals.type=c("z", "LB"), mx.algebras=NULL,
       mxModel.Args=NULL, subset.variables=NULL, model.name=NULL, suppressWarnings=TRUE,
       silent=TRUE, run=TRUE, ...)

Arguments

Value

An object of class wls with a list of

Note

If the input is a list of tssem1.obj, it returns a list of results for each cluster.

Author(s)

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

References

Bentler, P.M., & Savalei, V. (2010). Analysis of correlation structures: current status and open problems. In Kolenikov, S., Thombs, L., & Steinley, D. (Eds.). Recent Methodological Developments in Social Science Statistics (pp. 1-36). Hoboken, NJ: Wiley.

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

Cheung, M. W.-L. (2014). Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods, 46, 29-40.

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.

Joreskog, K. G., Sorbom, D., Du Toit, S., & Du Toit, M. (1999). LISREL 8: New Statistical Features. Chicago: Scientific Software International.

McArdle, J. J., & MacDonald, R. P. (1984). Some algebraic properties of the Reticular Action Model for moment structures. British Journal of Mathematical and Statistical Psychology, 37, 234-251.

See Also

tssem1, Becker92, Digman97, Hunter83, issp89, issp05

Examples

#### Analysis of correlation structure
R1.labels <- c("a1", "a2", "a3", "a4")

R1 <- matrix(c(1.00, 0.22, 0.24, 0.18,
               0.22, 1.00, 0.30, 0.22,
               0.24, 0.30, 1.00, 0.24,
               0.18, 0.22, 0.24, 1.00), ncol=4, nrow=4,
               dimnames=list(R1.labels, R1.labels))
n <- 1000
acovR1 <- asyCov(R1, n)

#### One-factor CFA model using lavaan specification
model1 <- "f =~ a1 + a2 + a3 + a4"

RAM1 <- lavaan2RAM(model1, obs.variables=R1.labels)

wls.fit1a <- wls(Cov=R1, aCov=acovR1, n=n, RAM=RAM1,
                 cor.analysis=TRUE, intervals="LB")
summary(wls.fit1a)

## One-factor CFA model using RAM specification
(A1 <- cbind(matrix(0, nrow=5, ncol=4),
             matrix(c("0.2*a1","0.2*a2","0.2*a3","0.2*a4",0),
             ncol=1)))

(S1 <- Diag(c("0.2*e1","0.2*e2","0.2*e3","0.2*e4",1)))

## The first 4 variables are observed while the last one is latent.
(F1 <- create.Fmatrix(c(1,1,1,1,0), name="F1"))

wls.fit1b <- wls(Cov=R1, aCov=acovR1, n=n, Fmatrix=F1, Smatrix=S1, Amatrix=A1,
                 cor.analysis=TRUE, intervals="LB")
summary(wls.fit1b)

## Select 3 variables to analyze
model2 <- "f =~ a1 + a2 + a3"

RAM2 <- lavaan2RAM(model2, obs.variables=R1.labels[-4])

wls.fit1c <- wls(Cov=R1, aCov=acovR1, n=n, RAM=RAM2,
                 cor.analysis=TRUE, subset.variables=c("a1", "a2", "a3"))
summary(wls.fit1c)

#### Multiple regression analysis using lavaan specification
R2.labels <- c("y", "x1", "x2")

R2 <- matrix(c(1.00, 0.22, 0.24, 
               0.22, 1.00, 0.30, 
               0.24, 0.30, 1.00), ncol=3, nrow=3,
               dimnames=list(R2.labels, R2.labels))
acovR2 <- asyCov(R2, n)

model3 <- "y ~ x1 + x2
           ## Variances of x1 and x2 are 1
           x1 ~~ 1*x1
           x2 ~~ 1*x2
           ## x1 and x2 are correlated
           x1 ~~ x2"

RAM3 <- lavaan2RAM(model3, obs.variables=R2.labels)

wls.fit2a <- wls(Cov=R2, aCov=acovR2, n=n, RAM=RAM3,
                 cor.analysis=TRUE, intervals="z")
summary(wls.fit2a)


#### Multiple regression analysis using RAM specification

## A2: Regression coefficents
#    y x1 x2
# y  F T  T 
# x1 F F  F 
# x2 F F  F 
(A2 <- mxMatrix("Full", ncol=3, nrow=3, byrow=TRUE,
               free=c(FALSE, rep(TRUE, 2), rep(FALSE, 6)), name="A2"))

## S2: Covariance matrix of free parameters
#    y x1 x2
# y  T F  F 
# x1 F F  F 
# x2 F T  F
(S2 <- mxMatrix("Symm", ncol=3, nrow=3, values=c(0.2,0,0,1,0.2,1),
                labels=c("Var_y", NA, NA, NA, "Cov_x1_x2", NA),
                free=c(TRUE,FALSE,FALSE,FALSE,TRUE,FALSE), name="S2"))

## F may be ignored as there is no latent variable.
wls.fit2b <- wls(Cov=R2, aCov=acovR2, n=n, Amatrix=A2, Smatrix=S2,
                 cor.analysis=TRUE, intervals="LB")
summary(wls.fit2b)


#### Analysis of covariance structure using lavaan specification
R3.labels=c("a1", "a2", "a3", "a4")

R3 <- matrix(c(1.50, 0.22, 0.24, 0.18,
               0.22, 1.60, 0.30, 0.22,
               0.24, 0.30, 1.80, 0.24,
               0.18, 0.22, 0.24, 1.30), ncol=4, nrow=4,
               dimnames=list(R3.labels, R3.labels))
n <- 1000
acovS3 <- asyCov(R3, n, cor.analysis=FALSE)

model3 <- "f =~ a1 + a2 + a3 + a4"

RAM3 <- lavaan2RAM(model3, obs.variables=R3.labels)

wls.fit3a <- wls(Cov=R3, aCov=acovS3, n=n, RAM=RAM3,
                 cor.analysis=FALSE)
summary(wls.fit3a)

#### Analysis of covariance structure using RAM specification
(A3 <- cbind(matrix(0, nrow=5, ncol=4),
             matrix(c("0.2*a1","0.2*a2","0.2*a3","0.2*a4",0),ncol=1)))

(S3 <- Diag(c("0.2*e1","0.2*e2","0.2*e3","0.2*e4",1)))

F3 <- c(TRUE,TRUE,TRUE,TRUE,FALSE)
(F3 <- create.Fmatrix(F3, name="F3", as.mxMatrix=FALSE))

wls.fit3b <- wls(Cov=R3, aCov=acovS3, n=n, Amatrix=A3, Smatrix=S3,
                Fmatrix=F3, cor.analysis=FALSE)
summary(wls.fit3b)

Forty-four Studies from Cheung (2013)

Description

Between 1990 and 1993, 57,561 adults aged 18 and above from 42 nations were interviewed by local academic institutes in Eastern European nations and by professional survey organizations in other nations. The standardized mean difference (SMD) between males and females on life satisfaction and life control in each country were calculated as the effect sizes. Positive values indicate that males have higher scores than females do.

Usage

data(wvs94a)

Details

The variables are:

country

Country

lifesat

SMD on life satisfaction

lifecon

SMD on life control

lifesat_var

Sampling variance of lifesat

inter_cov

Sampling covariance between lifesat and lifecon

lifecon_var

Sampling variance of lifecon

gnp

Gross National Product

Source

World Values Study Group. (1994). World Values Survey, 1981-1984 and 1990-1993 [Computer file]. Ann Arbor, MI: Inter-university Consortium for Political and Social Research.

References

Au, K., & Cheung, M. W.-L. (2004). Intra-cultural variation and job autonomy in 42 countries. Organization Studies, 25, 1339-1362.

Cheung, M. W.-L. (2013). Multivariate meta-analysis as structural equation models. Structural Equation Modeling, 20, 429-454.

Examples

data(wvs94a)

## Random-effects model
random.ma1 <- meta(y=cbind(lifesat, lifecon),
                   v=cbind(lifesat_var, inter_cov, lifecon_var), data=wvs94a,
                   model.name="Random effects model")
summary(random.ma1)

## Random-effects model with both population effect sizes fixed at 0
random.ma2 <- meta(y=cbind(lifesat, lifecon),
                   v=cbind(lifesat_var, inter_cov, lifecon_var), data=wvs94a,
                   intercept.constraints=matrix(0, nrow=1, ncol=2),
                   model.name="Effect sizes are fixed at 0")
summary(random.ma2)

## Compare the nested models
anova(random.ma1, random.ma2)

## Fixed-effects model by fixing the variance component at 0 
fixed.ma <- meta(y=cbind(lifesat, lifecon),
                 v=cbind(lifesat_var, inter_cov, lifecon_var), data=wvs94a,
                 RE.constraints=matrix(0, ncol=2, nrow=2),
                 model.name="Fixed effects model")
summary(fixed.ma)

## Mixed-effects model
## gnp is divided by 10000 and centered by using 
## scale(gnp/10000, scale=FALSE)
mixed.ma1 <- meta(y=cbind(lifesat, lifecon),
                  v=cbind(lifesat_var, inter_cov, lifecon_var),
                  x=scale(gnp/10000, scale=FALSE), data=wvs94a,
                  model.name="GNP as a predictor")
summary(mixed.ma1)

## Mixed-effects model with equal regression coefficients
mixed.ma2 <- meta(y=cbind(lifesat, lifecon),
                  v=cbind(lifesat_var, inter_cov, lifecon_var),
                  x=scale(gnp/10000, scale=FALSE), data=wvs94a,
                  coef.constraints=matrix(c("0.0*Eq_slope",
                                            "0.0*Eq_slope"), nrow=2),
                  model.name="GNP as a predictor with equal slope")
summary(mixed.ma2)

## Compare the nested models
anova(mixed.ma1, mixed.ma2)

## Plot the multivariate effect sizes
plot(random.ma1, main="Estimated effect sizes and their 95% confidence ellipses",
     axis.label=c("Gender difference on life satisfaction",
                  "Gender difference on life control"))

Forty-four Covariance Matrices on Life Satisfaction, Job Satisfaction, and Job Autonomy

Description

Between 1990 and 1993, 57,561 adults aged 18 and above from 42 nations were interviewed by local academic institutes in Eastern European nations and by professional survey organizations in other nations. The covariance matrices among Life Satisfaction, Job Satisfaction, and Job Autonomy were calculated.

Usage

data(wvs94b)

Details

The variables are:

data

Covariance matrix among Life Satisfaction (LS), Job Satisfaction (JS), and Job Autonomy (JA)

n

Sample size in the country

Source

World Values Study Group. (1994). World Values Survey, 1981-1984 and 1990-1993 [Computer file]. Ann Arbor, MI: Inter-university Consortium for Political and Social Research.

References

Au, K., & Cheung, M. W.-L. (2004). Intra-cultural variation and job autonomy in 42 countries. Organization Studies, 25, 1339-1362.

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(wvs94b)

## Get the indirect and the direct effects and
## their sampling covariance matrices for each study
indirect1 <- indirectEffect(wvs94b$data, wvs94b$n)
indirect1

## Multivariate meta-analysis on the indirect and direct effects
indirect2 <- meta(indirect1[, c("ind_eff", "dir_eff")],
                  indirect1[, c("ind_var", "ind_dir_cov", "dir_var")])

summary(indirect2)