omega {psych}R Documentation

Calculate the omega estimate of factor saturation

Description

McDonald has proposed coefficient omega as an estimate of the general factor saturation of a test. One way to find omega is to do a factor analysis of the original data set, rotate the factors obliquely, do a Schmid Leiman transformation, and then find omega. This function estimates omega as suggested by McDonald by using hierarchical factor analysis (following Jensen).

Usage

omega(m, nfactors, pc = "mle",key = NULL, flip=TRUE, digits=NULL, ...)

Arguments

m A correlation matrix or a data.frame/matrix of data
nfactors Number of factors believed to be group factors
pc pc="pa" for principal axes, pc="pc" for principal components, pc="mle" for maximum likelihood.
key a vector of +/- 1s to specify the direction of scoring of items. The default is to assume all items are positively keyed, but if some items are reversed scored, then key should be specified.
flip If flip is TRUE, then items are automatically flipped to have positive correlations on the general factor. Items that have been reversed are shown with a - sign.
digits if specified, round the output to digits
... Allows additional parameters to be passed through to the factor routines

Details

``Many scales are assumed by their developers and users to be primarily a measure of one latent variable. When it is also assumed that the scale conforms to the effect indicator model of measurement (as is almost always the case in psychological assessment), it is important to support such an interpretation with evidence regarding the internal structure of that scale. In particular, it is important to examine two related properties pertaining to the internal structure of such a scale. The first property relates to whether all the indicators forming the scale measure a latent variable in common.

The second internal structural property pertains to the proportion of variance in the scale scores (derived from summing or averaging the indicators) accounted for by this latent variable that is common to all the indicators (Cronbach, 1951; McDonald, 1999; Revelle, 1979). That is, if an effect indicator scale is primarily a measure of one latent variable common to all the indicators forming the scale, then that latent variable should account for the majority of the variance in the scale scores. Put differently, this variance ratio provides important information about the sampling fluctuations when estimating individuals' standing on a latent variable common to all the indicators arising from the sampling of indicators (i.e., when dealing with either Type 2 or Type 12 sampling, to use the terminology of Lord, 1956). That is, this variance proportion can be interpreted as the square of the correlation between the scale score and the latent variable common to all the indicators in the infinite universe of indicators of which the scale indicators are a subset. Put yet another way, this variance ratio is important both as reliability and a validity coefficient. This is a reliability issue as the larger this variance ratio is, the more accurately one can predict an individual's relative standing on the latent variable common to all the scale's indicators based on his or her observed scale score. At the same time, this variance ratio also bears on the construct validity of the scale given that construct validity encompasses the internal structure of a scale." (Zinbarg, Yovel, Revelle, and McDonald, 2006).

McDonald has proposed coefficient omega as an estimate of the general factor saturation of a test. Zinbarg, Revelle, Yovel and Li (2005) http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf compare McDonald's Omega to Cronbach's alpha and Revelle's beta. They conclude that omega is the best estimate. (See also Zinbarg et al., 2006)

One way to find omega is to do a factor analysis of the original data set, rotate the factors obliquely, do a Schmid-Leiman (schmid) transformation, and then find omega. Here we present code to do that.

Omega differs as a function of how the factors are estimated. Three options are available, pc="pa" does a principle axes factor analysis (factor.pa), pc="mle" uses the factanal function, and pc="pc" does a principal components analysis (principal).

For ability items, it is typically the case that all items will have positive loadings on the general factor. However, for non-cognitive items it is frequently the case that some items are to be scored positively, and some negatively. Although probably better to specify which directions the items are to be scored by specifying a key vector, if flip =TRUE (the default), items will be reversed so that they have positive loadings on the general factor. The keys are reported so that scores can be found using the score.items function.

Output from omega can be shown using the omega.graph function.

Beta, an alternative to omega, is defined as the worst split half reliability. It can be estimated by using ICLUST (a hierarchical clustering algorithm originally developed for main frames and written in Fortran and that is now available in R. (For a very complimentary review of why the ICLUST algorithm is useful in scale construction, see Cooksey and Soutar, 2005).

The omega function uses exploratory factor analysis to estimate the omega_h coefficient. It is important to remember that ``A recommendation that should be heeded, regardless of the method chosen to estimate omega_h, is to always examine the pattern of the estimated general factor loadings prior to estimating omega_h. Such an examination constitutes an informal test of the assumption that there is a latent variable common to all of the scale's indicators that can be conducted even in the context of EFA. If the loadings were salient for only a relatively small subset of the indicators, this would suggest that there is no true general factor underlying the covariance matrix. Just such an informal assumption test would have afforded a great deal of protection against the possibility of misinterpreting the misleading omega_h estimates occasionally produced in the simulations reported here." (Zinbarg et al., 2006, p 137).

A simple demonstration of the problem of an omega estimate reflecting just one of two group factors can be found in the last example.

Value

alpha Cronbach's alpha
schmid The Schmid Leiman transformed factor matrix and associated matrices
schmid$sl The g factor loadings as well as the residualized factors
schmid$orthog Varimax rotated solution of the original factors
schmid$oblique The oblimin transformed factors
schmid$fcor the correlation matrix of the oblique factors
schid$gloading The loadings on the higher order, g, factor of the oblimin factors
key A vector of -1 or 1 showing which direction the items were scored.

Note

Requires the GPArotation package

Author(s)

http://personality-project.org/revelle.html
Maintainer: William Revelle revelle@northwestern.edu

References

http://personality-project.org/r/r.omega.html

Revelle, W. (1979). Hierarchical cluster analysis and the internal structure of tests. Multivariate Behavioral Research, 14, 57-74. (http://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. http://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 http://apm.sagepub.com/cgi/reprint/30/2/121

See Also

omega.graph ICLUST, ICLUST.graph, VSS, schmid , make.hierarchical

Examples

## Not run: 
test.data <- Harman74.cor$cov
my.omega <- omega(test.data,3)       
print(my.omega,digits=2)
#

## End(Not run)
#create 9 variables with a hierarchical structure
jen.data <- make.hierarchical()
#with correlations of
jen.data
#find omega 
jen.omega <- omega(jen.data,digits=2)
jen.omega

#create 8 items with a two factor solution, showing the use of the flip option
sim2 <- item.sim(8)
omega(sim2)   #an example of misidentification-- remember to look at the loadings matrices.


[Package psych version 1.0-33 Index]