wls: 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, ...)

Value

An object of class wls with a list of

call: The matched call
Cov: Input data of either a covariance or correlation matrix
asyCov: The asymptotic covariance matrix of the input data
noObservedStat: Number of observed statistics
n: Sample size
cor.analysis: logical
noConstraints: Number of constraints imposed on S
indepModelChisq: Chi-square statistic of the independent model returned by .indepwlsChisq
indepModelDf: Degrees of freedom of the independent model returned by .indepwlsChisq
mx.fit: A fitted object returned from mxRun

Arguments

tssem1.obj: An object of either class tssem1FEM, class tssem1FEM.cluster or class tssem1REM returned from tssem1()
Cov: A \(p\) x \(p\) sample correlation/covariance matrix where \(p\) is the number of variables.
aCov: A \(p*\) x \(p*\) asymptotic sampling covariance matrix of either vechs (Cov) or vech (Cov) where \(p* = p(p-1)/2 \) for correlation matrix and \(p* = p(p+1)/2 \) for covariance matrix.
n: Sample size.
RAM: A RAM object including a list of matrices of the model returned from lavaan2RAM.
Amatrix: If RAM is not specified, an Amatrix is required. An asymmetric matrix in the RAM specification with MxMatrix-class. If it is NULL, a matrix of zero will be created. If it is a matrix, it will be converted into MxMatrix-class by the as.mxMatrix function.
Smatrix: If RAM is not specified, an Smatrix is required. A symmetric matrix in the RAM specification with MxMatrix-class. If it is a matrix, it will be converted into MxMatrix-class by the as.mxMatrix function.
Fmatrix: A filter matrix in the RAM specification with MxMatrix-class. If it is NULL (the default), an identity matrix with the same dimensions of Cov will be created. If it is a matrix, it will be converted into MxMatrix-class by the as.mxMatrix function. It is not required when there is no latent variable.
diag.constraints: Logical. This argument is ignored when cor.analysis=FALSE. If diag.constraints=TRUE, the diagonals of the model implied matrix would be constrained at 1 by nonlinear constraints. The drawback is that standard error will not be generated. Parametric bootstrap is used to estimate the standard error by drawing samples from \(\mathcal{N}(vech(Cov), asyCov)\) for covariance analysis and \(\mathcal{N}(vechs(Cov), asyCov)\) for correlation analysis while asyCov is treated as fixed. This process is computationally intensive. A better approach is to request likelihood-based confidence intervals (CIs) by specifying intervals.type="LB". If diag.constraints=FALSE and cor.analysis=TRUE, the diagonals are automatically constrained as ones by treating the error variances as computed values rather than as parameters. Since the error variances are not parameters, they are not reported.
cor.analysis: Logical. Analysis of correlation or covariance structure. If cor.analysis=TRUE, vechs is used to vectorize S; otherwise, vech is used to vectorize S.
intervals.type: Either z (default if missing) or LB. If it is z, it calculates the 95% Wald CIs based on the z statistic. If it is LB, it calculates the 95% likelihood-based CIs on the parameter estimates. Please note that the z values and their associated p values are based on the z statistic. They are not related to the likelihood-based CIs.
mx.algebras: A list of mxMatrix or mxAlgebra objects on the Amatrix, Smatrix, and Fmatrx. It can be used to define new functions of parameters and their LBCIs. For example, if the regression coefficients to calculate an indirect effect are stored in A[1,2] and A[1,3], we may define list(ind=mxAlgebra(Amatrix[1,2]*Amatrix[1,3], name="ind")) See the examples in Becker92 and Hunter83. It should be noted that Fmatrix, Amatrix, Smatrix, Iden (a \(p\) x \(p\) identity matrix), sampleS (sample correlation or covariance matrix), impliedS1, impliedS (model implied correlation or covariance matrix), vecS, invAcov, obj, One, select and constraint and Ematrix (computed error variances when diag.constraints=FALSE) have been defined internally. You should not create new matrices using these names.
mxModel.Args: A list of arguments passed to mxModel. These include, for example, additional mxMatrix and mxConstraint.
model.name: A string for the model name in mxModel. If it is missing, the default is "TSSEM2 (or WLS) Analysis of Correlation Structure" for cor.analysis=TRUE and "TSSEM2 (or WLS) Analysis of Covariance Structure" for cor.analysis=FALSE.
subset.variables: An optional character vector of variable names to select variables in the analysis. For example, there are 10 variables in Cov, say, x1 to x10. We may use c("x1", "x2", "x3") to select three variables in the analysis. Please note that this argument does not reorder the data. That is, c("x3", "x2", "x1") is the same as c("x1", "x2", "x3").
suppressWarnings: Logical. If TRUE, warnings are suppressed. The argument to be passed to mxRun.
silent: Logical. An argument to be passed to mxRun
run: Logical. If FALSE, only return the mx model without running the analysis.
...: Further arguments to be passed to mxRun.

Author

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.

Examples

Run this code

# \donttest{
#### 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)
# }

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