omegaStats: Omega Statistics from a Factor Loading Matrix

In psych: Procedures for Psychological, Psychometric, and Personality Research

Omega Statistics from a Factor Loading Matrix

Description

Among the many ways of estimating internal consistency, the family of model based statistics known as omega (omega_hierarchical, omega_total, omega_subtest, omega_bifactor) provide a number of solutions. omegaStats provides common output for several different estimation procedures. The omega function will find three versions of omega, as will the CFA function. Because of their different assumptions, the omega_hierarchical and omega_bifactor will provide different estimates of the general factor saturation of a test. Users of other packages (e.g., lavaan, may find the basic statistics by providing a correlation matrix and factor loadings matrix. See the last example.

Usage

omegaStats(r, loads = NULL, Phi = NULL, gload = NULL, group = NULL, h2 = NULL, 
    covar = FALSE, lavaan=NULL)

Arguments

r	A correlation matrix
loads	A factor loadings matrix with the first column representing a general factor. Or alternatively users may specify the general factor and group factor loadings separately.
Phi	Factor intercorrelations
gload	A matrix of loadings on a general factor (if loads is not specified)
group	A matrix of group factor loadings (if loads is not specified)
h2	Communalities of the variables (r and loads are not specified)
covar	If TRUE, treat the r matrix as a covariance matrix to find u2
lavaan	To find omega from lavaan output, specify the lavaan object (see example)

Details

Model based estimates of internal consistency include the family of estimates known as omega. Omega_hierarchical (found by the omega function) estimates the general factor saturation of a test, while CFA finds the amount of variance accounted by the first factor in a bifactor solution. Although both procedures agree in terms of their estimates for the total amount of reliable variance, they disagree with respect to the amount that is common to all items.

lavaan based estimates differ yet again (see the final example which is not run).

Value

omega hierarchical	The \omega_h coefficient
omega.lim	The limit of \omega_h as the test becomes infinitly large
omega total	The \omega_t coefficient
stats	as found by fa.stats
...	Many other statistics as reported by fa
Call	echoes the call to the function

Note

What is interesting is finding omega from the bifactor solution of non-correlated group factors. Direct CFA.bifactor will find a general factor (even though not there), although conventional fa procedures will fail to find a general factor.

Author(s)

William Revelle

References

Jensen, Arthur R. and Li-Jen Weng (1994) What is a good g? Intelligence, 18, 3, 231-258

Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at 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)

Revelle, W. and Condon, D.M. (2019) Reliability from alpha to omega: A tutorial. Psychological Assessment, 31, 12, 1395-1411. https://doi.org/10.1037/pas0000754. https://osf.io/preprints/psyarxiv/2y3w9 Preprint available from PsyArxiv

Revelle, W. and Zinbarg, R. E. (2009) Coefficients alpha, beta, omega and the glb: comments on Sijtsma. Psychometrika, 74, 1, 145-154. (https://personality-project.org/revelle/publications/rz09.pdf

Waller, N. G. (2017) Direct Schmid-Leiman Transformations and Rank-Deficient Loadings Matrices. Psychometrika. DOI: 10.1007/s11336-017-9599-0

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. (2007). Estimating omega for structures containing two group factors: Perils and prospects. Applied Psychological Measurement. 31 (2), 135-157.

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

See Also

fa, CFA, omega

Examples

#Create a 3 correlated factor model using default values and 10000 observations
v9 <- sim.hierarchical(n=10000)

model <- 'F1=~ V1 + V2 + V3
          F2=~ V4 + V5 + V6
          F3 =~ V7 +V8 + V9'
test <-  CFA.bifactor(model,v9$observed)
test   #show the results
omegaStats(test$r,test$loadings)


om <- omega(v9$observed)
omegaStats(r=om$R,loads=om$schmid$sl[,1:4])

##Now creat a three factor model with orthogonal factors
model <- 'F1=~ V1 + V2 + V3
          F2=~ V4 + V5 + V6
          F3 =~ V7 +V8 + V9'



model9 <- 'F1 =~ .9*V1 + .8*V2 + .7*V3
           F2 =~ .8*V4 + .7*V5 +.6*V6
           F3 =~ .7*V7 + .6*V8 +.5*V9'

set.seed(42)          
v9s <- sim(model9,n=500)
 test <- CFA(model9,v9s$observed )  #do a cfa using Lavaan syntax
 test.bi <- CFA.bifactor(model9,v9s$observed)
 test.hi <- CFA.bifactor(model9,v9s$observed,g=TRUE)
 om <- omega(v9s$observed)
 omegaStats(test.bi$r,test.bi$loadings)

###do not run the following test of lavaan
if(FALSE) {
lav.model <- 'g=~ V1 + V2 + V3+ V4 + V5 + V6+V7 +V8 + V9
          F1=~ V1 + V2 + V3
          F2=~ V4 + V5 + V6
          F3 =~ V7 +V8 + V9'
library(lavaan)
fit <- cfa(lav.model,data=v9$observed,orthogonal=TRUE,std.lv=TRUE, estimator="ULS")
omegaStats(r=v9$r,lavaan=fit)   #from lavaan
fa.congruence(fit,test) 
}  #end of lavaan test

psych documentation built on Feb. 3, 2026, 9:08 a.m.