omega(m,nfactors=3,fm="minres",n.iter=1,p=.05,poly=FALSE,key=NULL,flip=TRUE,digits=2,title="Omega",sl=TRUE,labels=NULL, plot=TRUE,n.obs=NA,rotate="oblimin",Phi=NULL,option="equal", ...)
omegaSem(m,nfactors=3,fm="minres",key=NULL,flip=TRUE, digits=2,title="Omega",sl=TRUE,labels=NULL, plot=TRUE,n.obs=NA,rotate="oblimin",Phi = NULL,option="equal",...)
omegaFromSem(m,s,flip=TRUE)
omegah(m,nfactors=3,fm="minres",key=NULL,flip=TRUE, digits=2,title="Omega",sl=TRUE,labels=NULL, plot=TRUE,n.obs=NA,rotate="oblimin",Phi = NULL,option="equal",...)omega.graph may be used separately.target.rot)schmid for treating the case of 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 (hierarchical ($\omega_h$) as an estimate of the general factor saturation of a test. Zinbarg, Revelle, Yovel and Li (2005)
One way to find $\omega_h$ is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (schmid) transformation to find general factor loadings, and then find $\omega_h$. Here we present code to do that.
$\omega_h$ differs as a function of how the factors are estimated. Four options are available, the default does a minres factor solution, fm="pa" does a principle axes factor analysis (factor.pa), fm="mle" uses the factanal function, and fm="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. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex).
$\beta$, 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 part of the psych package. (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.
Diagnostic statistics that reflect the quality of the omega solution include a comparison of the relative size of the g factor eigen value to the other eigen values, the percent of the common variance for each item that is general factor variance (p2), the mean of p2, and the standard deviation of p2. Further diagnostics can be done by describing (describe) the $schmid$sl results.
Although omega_h is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the schmid extraction to treat the two subfactors as having equal loadings.
There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be "equal" which will be the sqrt(oblique correlations between the factors) or to "first" or "second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option="first" or option="second" to the call.
Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints.
In addition to $\omega_h$, another of McDonald's coefficients is $\omega_t$. This is an estimate of the total reliability of a test.
McDonald's $\omega_t$, which is similar to Guttman's $\lambda_6$, guttman but uses the estimates of uniqueness ($u^2$ from factor analysis to find $e_j^2$. This is based on a decomposition of the variance of a test score, $V_x$ into four parts: that due to a general factor, $\vec{g}$, that due to a set of group factors, $\vec{f}$, (factors common to some but not all of the items), specific factors, $\vec{s}$ unique to each item, and $\vec{e}$, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error).
Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e}$ then the communality of item$_j$, based upon general as well as group factors, $h_j^2 = c_j^2 + \sum{f_{ij}^2}$ and the unique variance for the item $u_j^2 = \sigma_j^2 (1-h_j^2)$ may be used to estimate the test reliability. That is, if $h_j^2$ is the communality of item$_j$, based upon general as well as group factors, then for standardized items, $e_j^2 = 1 - h_j^2$ and $$\omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x}$$ Because $h_j^2 \geq r_{smc}^2$, $\omega_t \geq \lambda_6$.
It is important to distinguish here between the two $\omega$ coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, $\omega_t$ and $\omega_h$. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor. $$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$$
Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by $$\omega_limit = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$$
The input to omega may be a correlation matrix or a raw data matrix, or a factor pattern matrix with the factor intercorrelations (Phi) matrix.
omega is an exploratory factor analysis function that uses a Schmid-Leiman transformation. omegaSem first calls omega and then takes the Schmid-Leiman solution, converts this to a confirmatory sem model and then calls the sem package to conduct a confirmatory model. $\omega_h$ is then calculated from the CFA output. Although for well behaved problems, the efa and cfa solutions will be practically identical, the CFA solution will not always agree with the EFA solution. In particular, the estimated $R^2$ will sometimes exceed 1. (An example of this is the Harman 24 cognitive abilities problem.)
In addition, not all EFA solutions will produce workable CFA solutions. Model misspecifications will lead to very strange CFA estimates.
omegaFromSem takes the output from a sem model and uses it to find
$\omega_h$. The estimate of factor indeterminacy, found by the multiple $R^2$ of the variables with the factors, will not match that found by the EFA model. In particular, the estimated $R^2$ will sometimes exceed 1. (An example of this is the Harman 24 cognitive abilities problem.)
Revelle, W. (1979). Hierarchical cluster analysis and the internal structure of tests. Multivariate Behavioral Research, 14, 57-74. (
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.
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
omega.graph ICLUST, ICLUST.graph, VSS, schmid, make.hierarchicaltest.data <- Harman74.cor$cov
if(!require(GPArotation)) {message("Omega requires GPA rotation" )} else {my.omega <- omega(test.data)
print(my.omega,digits=2)}
#create 9 variables with a hierarchical structure
v9 <- sim.hierarchical()
#with correlations of
round(v9,2)
#find omega
v9.omega <- omega(v9,digits=2)
v9.omega
#create 8 items with a two factor solution, showing the use of the flip option
sim2 <- item.sim(8)
omega(sim2) #an example of misidentification-- remember to look at the loadings matrices.
omega(sim2,2) #this shows that in fact there is no general factor
omega(sim2,2,option="first") #but, if we define one of the two group factors as a general factor, we get a falsely high omega
#apply omega to analyze 6 mental ability tests
data(ability.cov) #has a covariance matrix
omega(ability.cov$cov)Run the code above in your browser using DataLab