Omega

Reliability is concerned with the consistency of scores on a measure when the measure is administered to the same group of individuals under comparable circumstances. Scores on a measure are often computed as the sum of scores across items in [Page 1177]the measure and are referred to as scale scores. Reliability of scale scores is defined in classical test theory as the ratio of scale true score variance to observed scale score variance. It can also be defined as the correlation between scale scores and the scores from its parallel form, or the square of correlation between scale true scores and observed scores. That is, ${\rho }_{X{X}^{\prime }}={\sigma }_T^2/{\sigma }_X^2=1-{\sigma }_E^2/{\sigma }_X^2$. The population ρXX′ is unknown in practice and needs to be estimated from data.

Omega ω is ground on factor analysis. It is one of the popular methods for estimating reliability for scale scores. It was initially termed by David R. Heise and George W. Bohrnstedt in 1970 and has been discussed extensively by Roderick P. McDonald since 1978. The classic notion of ω has four assumptions: (1) a set of items measures a single-latent construct (or factor) of interest; (2) the latent factor is the only common cause of the inter-item correlation, and consequently, residual scores are independent across items; (3) the relationship between the latent factor and item scores is linear; and (4) a one-factor model adequately represents the data. After further defining and describing how to calculate ω, this entry details the difference between ω and the coefficient α, which is another common way to estimate reliability of scale scores. Next, the entry describes how to obtain an ω estimate from sample data. Finally, consideration is given to some situations that do not conform to the common assumptions of the classic notion of ω.

Within the factor analysis framework, an individual’s scores on an item xij is expressed as

$$x_{ij}={\tau }_j+{\lambda }_j{F}_i+e_{ij},$$

where τj is the intercept for item j, Fi is the factor score for individual i, λj is the factor loading of the jth item on the factor, and eij is the individual’s residual score on item j. The terms τj+λjFi and eij can be conceptualized as true and error scores, respectively, as defined in classical test theory. But note that eij includes both specific and random error components. The metric of latent factor is arbitrary. By fixing the variance of factor at one and following the assumption of independency between factor scores and residuals, the variance of observed scores on the jth item and the variance of scale scores on the measure are, respectively,

$${\sigma }_{x_j}^2={\lambda }_j^2+\psi {\text{}}_j^{\text{}\text{}2}\text{and}$$

$${\sigma }_X^2={\left({\displaystyle \sum _{j=1}^J{\lambda }_j}\right)}^2+{\displaystyle \sum _{j=1}^J{\psi }_j^2,}$$

where ${\psi }_j^2$ denotes the residual score variance for item j. ω is computed as

$$\omega =\frac{{\left({\displaystyle \sum _{j=1}^J{\lambda }_j}\right)}^2}{{\sigma }_X^2},$$

or

$$\omega =1-\frac{{\displaystyle \sum _{j=1}^J{\psi }_j^2}}{{\sigma }_X^2}.$$

It is interpreted as the proportion of scale score variance that is attributable to the common latent factor or that is not attributable to the uniqueness of items. Under the four assumptions previously described, ω is an accurate estimate of reliability of scale scores, that is, ρXX′ = ω.