esem: Perform and Exploratory Structural Equation Model (ESEM) by...
In psych: Procedures for Psychological, Psychometric, and Personality Research

esem	R Documentation

Perform and Exploratory Structural Equation Model (ESEM) by using factor extension techniques

Description

Structural Equation Modeling (SEM) is a powerful tool for confirming multivariate structures and is well done by the lavaan, sem, or OpenMx packages. Because they are confirmatory, SEM models test specific models. Exploratory Structural Equation Modeling (ESEM), on the other hand, takes a more exploratory approach. By using factor extension, it is possible to extend the factors of one set of variables (X) into the variable space of another set (Y). Using this technique, it is then possible to estimate the correlations between the two sets of latent variables, much the way normal SEM would do. Based upon exploratory factor analysis (EFA) this approach provides a quick and easy approach to do exploratory structural equation modeling.

Usage

esem(r, varsX, varsY, nfX = 1, nfY = 1, n.obs = NULL, fm = "minres", 
      rotate = "oblimin", rotateY="oblimin", plot = TRUE, cor = "cor",
       use = "pairwise",weight=NULL, ...)
esemDiagram(esem=NULL,labels=NULL,cut=.3,errors=FALSE,simple=TRUE,
	regression=FALSE,lr=TRUE, digits=1,e.size=.1,adj=2,
     main="Exploratory Structural Model", ...)
esem.diagram(esem=NULL,labels=NULL,cut=.3,errors=FALSE,simple=TRUE,
	regression=FALSE,lr=TRUE, digits=1,e.size=.1,adj=2,
     main="Exploratory Structural Model", ...) #deprecated
cancorDiagram(fit, cut=.1,digits=2, nfX = NULL, nfY = NULL,simple=FALSE,e.size=.1,
     main="Canonical Correlation",...)    
interbattery(r, varsX, varsY, nfX = 1, nfY = 1, n.obs = NULL,cor = "cor", 
       use = "pairwise",weight=NULL)

Arguments

`r`	A correlation matrix or a raw data matrix suitable for factor analysis
`varsX`	The variables defining set X
`varsY`	The variables defining set Y
`nfX`	The number of factors to extract for the X variables
`nfY`	The number of factors to extract for the Y variables
`n.obs`	Number of observations (needed for eBIC and chi square), can be ignored.
`fm`	The factor method to use, e.g., "minres", "mle" etc. (see fa for details)
`rotate`	Which rotation to use. (see fa for details)
`rotateY`	Which rotatation to use for the Y variables.
`plot`	If TRUE, draw the esemDiagram
`cor`	What options for to use for correlations (see fa for details)
`use`	"pairwise" for pairwise complete data, for other options see cor
`weight`	Weights to apply to cases when finding wt.cov
`...`	other parameters to pass to fa or to esemDiagram functions.
`esem`	The object returned from esem and passed to esemDiagram
`fit`	The object returned by lmCor with canonical variates
`labels`	Variable labels
`cut`	Loadings with abs(loading) > cut will be shown
`simple`	Only the biggest loading per item is shown
`errors`	include error estimates (as arrows)
`e.size`	size of ellipses (adjusted by the number of variables)
`digits`	Round coefficient to digits
`adj`	loadings are adjusted by factor number mod adj to decrease likelihood of overlap
`main`	Graphic title, defaults to "Exploratory Structural Model"
`lr`	draw the graphic left to right (TRUE) or top to bottom (FALSE)
`regression`	Not yet implemented

Details

Factor analysis as implemented in fa attempts to summarize the covariance (correlational) structure of a set of variables with a small set of latent variables or “factors". This solution may be ‘extended’ into a larger space with more variables without changing the original solution (see fa.extension. Similarly, the factors of a second set of variables (the Y set) may be extended into the original (X ) set. Doing so allows two independent measurement models, a measurement model for X and a measurement model for Y. These two sets of latent variables may then be correlated for an Exploratory Structural Equation Model. (This is exploratory because it is based upon exploratory factor analysis (EFA) rather than a confirmatory factor model (CFA) using more traditional Structural Equation Modeling packages such as sem, lavaan, or Mx.)

Although the output seems very similar to that of a normal EFA using fa, it is actually two independent factor analyses (of the X and the Y sets) that are then mutually extended into each other. That is, the loadings and structure matrices from sets X and Y are merely combined, and the correlations between the two sets of factors are found.

Interbattery factor analysis was developed by Tucker (1958) as a way of comparing the factors in common to two batteries of tests. (Currently under development and not yet complete). Using some straight forward linear algebra It is easy to find the factors of the intercorrelations between the two sets of variables. This does not require estimating communalities and is highly related to the procedures of canonical correlation.

The difference between the esem and the interbattery approach is that the first factors the X set and then relates those factors to factors of the Y set. Interbattery factor analysis, on the other hand, tries to find one set of factors that links both sets but is still distinct from factoring both sets together.

Value

`communality`	The amount of variance in each of the X and Y variables accounted for by the total model.
`sumsq`	The amount of variance accounted for by each factor – independent of the other factors.
`dof`	Degrees of freedom of the model.
`esem.dof`	Alternatively consider the df1 for the X model and df2 for the Y model. esem.dof = df1 + df2
`null.dof`	Degrees of freedom of the null model (the correlation matrix)
`ENull`	chi square of the null model
`chi`	chi square of the model. This is found by examining the size of the residuals compared to their standard error.
`rms`	The root mean square of the residuals.
`nh`	Harmonic sample size if using min.chi for factor extraction.
`EPVAL`	Probability of the Emprical Chi Square given the hypothesis of an identity matrix.
`crms`	Adjusted root mean square residual
`EBIC`	When normal theory fails (e.g., in the case of non-positive definite matrices), it useful to examine the empirically derived EBIC based upon the empirical `\chi^2` - 2 df.
`ESABIC`	Sample size adjusted empirical BIC
`fit`	sum of squared residuals versus sum of squared original values
`fit.off`	fit applied to the off diagonal elements
`sd`	standard deviation of the residuals
`factors`	Number of factors extracted
`complexity`	Item complexity
`n.obs`	Number of total observations
`loadings`	The factor pattern matrix for the combined X and Y factors
`Structure`	The factor structure matrix for the combined X and Y factors
`loadsX`	Just the X set of loadings (pattern) without the extension variables.
`loadsY`	Just the Y set of loadings (pattern) without the extension variables.
`PhiX`	The correlations of the X factors
`PhiY`	the correlations of the Y factors
`Phi`	the correlations of the X and Y factors within the selves and across sets.
`fm`	The factor method used
`fx`	The complete factor analysis output for the X set
`fy`	The complete factor analysis output for the Y set
`residual`	The residual correlation matrix (R - model). May be examined by a call to residual().
`Call`	Echo back the original call to the function.
`model`	model code for SEM and for lavaan to do subsequent confirmatory modeling

Note

Developed September, 2016, revised December, 2018 to produce code for lavaan and sem from the esem and esemDiagram functions. Suggestions or comments are most welcome.

This is clearly an exploratory approach and the degrees of freedom for the model are unclear.

In December 2023, I added the canCorDiagram function to help understand canonical correlations.

Author(s)

William Revelle

References

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at https://personality-project.org/r/book/

Tucker, Ledyard (1958) An inter-battery method of factor analysis, Psychometrika, 23, 111-136.

Examples

#make up a sem like problem using sim.structure
fx <-matrix(c( .9,.8,.6,rep(0,4),.6,.8,-.7),ncol=2)  
fy <- matrix(c(.6,.5,.4),ncol=1)
rownames(fx) <- c("V","Q","A","nach","Anx")
rownames(fy)<- c("gpa","Pre","MA")
Phi <-matrix( c(1,0,.7,.0,1,.7,.7,.7,1),ncol=3)
gre.gpa <- sim.structural(fx,Phi,fy)
print(gre.gpa)

#now esem it:
example <- esem(gre.gpa$model,varsX=1:5,varsY=6:8,nfX=2,nfY=1,n.obs=1000,plot=FALSE)
example
esemDiagram(example,simple=FALSE)
#compare this to the canonical solution
mod <- lmCor(y=6:8, x =1:5, data=gre.gpa$model, plot=FALSE)
cancorDiagram(mod)  #does not work because of imaginary roots
#compare two alternative solutions to the first 2 factors of the neo.
#solution 1 is the normal 2 factor solution.
#solution 2 is an esem with 1 factor for the first 6 variables, and 1 for the second 6.

if(require(psychTools)){
f2 <- fa(psychTools::neo[1:12,1:12],2)
es2 <- esem(psychTools::neo,1:6,7:12,1,1)
summary(f2)
summary(es2)
fa.congruence(f2,es2)
}
interbattery(Thurstone.9,1:4,5:9,2,2)  #compare to the solution of Tucker.  We are not there yet.

psych documentation built on June 27, 2024, 5:07 p.m.