Hierarchical factor structures represent the correlations between variables in terms of a smaller set of correlated factors which themselves can be represented by a higher order factor.
Two alternative solutions to such structures are found by the omega function. The correlated factors solutions represents the effect of the higher level, general factor, through its effect on the correlated factors. The other representation makes use of the Schmid Leiman transformation to find the direct effect of the general factor upon the original variables as well as the effect of orthogonal residual group factors upon the items.
Graphic presentations of these two alternatives are helpful in understanding the structure. omega.graph and omega.diagram draw both such structures. Graphs are drawn directly onto the graphics window or expressed in ``dot" commands for conversion to graphics using implementations of Graphviz (if using omega.graph).
Using Graphviz allows the user to clean up the Rgraphviz output. However, if Graphviz and Rgraphviz are not available, use omega.diagram.
See the other structural diagramming functions, fa.diagram and structure.diagram.
In addition
omega.diagram(om.results,sl=TRUE,sort=TRUE,labels=NULL,flabels=NULL,cut=.2,
gcut=.2,simple=TRUE, errors=FALSE, digits=1,e.size=.1,rsize=.15,side=3,
main=NULL,cex=NULL,color.lines=TRUE,marg=c(.5,.5,1.5,.5),adj=2, ...)
omega.graph(om.results, out.file = NULL, sl = TRUE, labels = NULL, size = c(8, 6),
node.font = c("Helvetica", 14), edge.font = c("Helvetica", 10),
rank.direction=c("RL","TB","LR","BT"), digits = 1, title = "Omega", ...)The output from the omega function
Optional output file for off line analysis using Graphviz
Orthogonal clusters using the Schmid-Leiman transform (sl=TRUE) or oblique clusters
variable labels
Labels for the factors (not counting g)
size of graphics window
What font to use for the items
What font to use for the edge labels
Defaults to left to right
Precision of labels
control font size
Use black for positive, red for negative
The margins for the figure are set to be wider than normal by default
Adjust the location of the factor loadings to vary as factor mod 4 + 1
Figure title
main figure caption
Other options to pass into the graphics packages
the size to draw the ellipses for the factors. This is scaled by the number of variables.
Minimum path coefficient to draw
Minimum general factor path to draw
draw just one path per item
sort the solution before making the diagram
on which side should errors be drawn?
show the error estimates
size of the rectangles
A graph object
A matrix suitable to be run throughe the sem function in the sem package.
While omega.graph requires the Rgraphviz package, omega.diagram does not. codeomega requires the GPArotation package.
https://personality-project.org/r/r.omega.html
Revelle, W. (in preparation) An Introduction to Psychometric Theory with applications in R. https://personality-project.org/r/book
Revelle, W. (1979). Hierarchical cluster analysis and the internal structure of tests. Multivariate Behavioral Research, 14, 57-74. (https://personality-project.org/revelle/publications/iclust.pdf)
Zinbarg, R.E., Revelle, W., Yovel, I., & Li. W. (2005). Cronbach's Alpha, Revelle's Beta, McDonald's Omega: Their relations with each and two alternative conceptualizations of reliability. Psychometrika. 70, 123-133. https://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf
Zinbarg, R., Yovel, I., Revelle, W. & McDonald, R. (2006). Estimating generalizability to a universe of indicators that all have one attribute in common: A comparison of estimators for omega. Applied Psychological Measurement, 30, 121-144. DOI: 10.1177/0146621605278814 https://journals.sagepub.com/doi/10.1177/0146621605278814
# NOT RUN {
#24 mental tests from Holzinger-Swineford-Harman
if(require(GPArotation) ) {om24 <- omega(Harman74.cor$cov,4) } #run omega
#
#example hierarchical structure from Jensen and Weng
if(require(GPArotation) ) {jen.omega <- omega(make.hierarchical())}
# }
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