Description

Calculate reliability values of factors by coefficient omega

Usage

Arguments

Details

The coefficient alpha (Cronbach, 1951) can be calculated by

α = \frac{k}{k - 1}≤ft[ 1 - \frac{∑^{k}_{i = 1} σ_{ii}}{∑^{k}_{i = 1} σ_{ii} + 2∑_{i < j} σ_{ij}} \right],

where k is the number of items in a factor, σ_{ii} is the item i observed variances, σ_{ij} is the observed covariance of items i and j.

The coefficient omega (Bollen, 1980; see also Raykov, 2001) can be calculated by

ω_1 =\frac{≤ft( ∑^{k}_{i = 1} λ_i \right)^{2} Var≤ft( ψ \right)}{≤ft( ∑^{k}_{i = 1} λ_i \right)^{2} Var≤ft( ψ \right) + ∑^{k}_{i = 1} θ_{ii} + 2∑_{i < j} θ_{ij} },

where λ_i is the factor loading of item i, ψ is the factor variance, θ_{ii} is the variance of measurement errors of item i, and θ_{ij} is the covariance of measurement errors from item i and j.

The second coefficient omega (Bentler, 1972, 2009) can be calculated by

ω_2 = \frac{≤ft( ∑^{k}_{i = 1} λ_i \right)^{2} Var≤ft( ψ \right)}{\bold{1}^\prime \hat{Σ} \bold{1}},

where \hat{Σ} is the model-implied covariance matrix, and \bold{1} is the k-dimensional vector of 1. The first and the second coefficients omega will have the same value when the model has simple structure, but different values when there are (for example) cross-loadings or method factors. The first coefficient omega can be viewed as the reliability controlling for the other factors (like η^2_{partial} in ANOVA). The second coefficient omega can be viewed as the unconditional reliability (like η^2 in ANOVA).

The third coefficient omega (McDonald, 1999), which is sometimes referred to hierarchical omega, can be calculated by

ω_3 =\frac{≤ft( ∑^{k}_{i = 1} λ_i \right)^{2} Var≤ft( ψ \right)}{\bold{1}^\prime Σ \bold{1}},

where Σ is the observed covariance matrix. If the model fits the data well, the third coefficient omega will be similar to the ω_2. Note that if there is a directional effect in the model, all coefficients omega will use the total factor variances, which is calculated by lavInspect(object, "cov.lv").

In conclusion, ω_1, ω_2, and ω_3 are different in the denominator. The denominator of the first formula assumes that a model is congeneric factor model where measurement errors are not correlated. The second formula accounts for correlated measurement errors. However, these two formulas assume that the model-implied covariance matrix explains item relationships perfectly. The residuals are subject to sampling error. The third formula use observed covariance matrix instead of model-implied covariance matrix to calculate the observed total variance. This formula is the most conservative method in calculating coefficient omega.

The average variance extracted (AVE) can be calculated by

AVE = \frac{\bold{1}^\prime \textrm{diag}≤ft(ΛΨΛ^\prime\right)\bold{1}}{\bold{1}^\prime \textrm{diag}≤ft(\hat{Σ}\right) \bold{1}},

Note that this formula is modified from Fornell & Larcker (1981) in the case that factor variances are not 1. The proposed formula from Fornell & Larcker (1981) assumes that the factor variances are 1. Note that AVE will not be provided for factors consisting of items with dual loadings. AVE is the property of items but not the property of factors.

Regarding categorical indicators, coefficient alpha and AVE are calculated based on polychoric correlations. The coefficient alpha from this function differs from the standard alpha calculation, which does not assume items are continuous, so numerically weighted categories can be treated as numeric. Researchers may check the alpha function in the psych package for the standard coefficient alpha calculation.

Item thresholds are not accounted for. Coefficient omega for categorical items, however, is calculated by accounting for both item covariances and item thresholds using Green and Yang's (2009, formula 21) approach. Three types of coefficient omega indicate different methods to calculate item total variances. The original formula from Green and Yang is equivalent to ω_3 in this function. Green and Yang did not propose a method for calculating reliability with a mixture of categorical and continuous indicators, and we are currently unaware of an appropriate method. Therefore, when reliability detects both categorical and continuous indicators in the model, an error is returned. If the categorical indicators load on a different factor(s) than continuous indicators, then reliability can be calculated separately for those scales by fitting separate models and submitting each to the reliability function.

Value

Reliability values (coefficient alpha, coefficients omega, average variance extracted) of each factor in each group. If there are multiple factors, a total column can optionally be included.

Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

Yves Rosseel (Ghent University; Yves.Rosseel@UGent.be)

Terrence D. Jorgensen (University of Amsterdam; TJorgensen314@gmail.com)

References

Bollen, K. A. (1980). Issues in the comparative measurement of political democracy. American Sociological Review, 45(3), 370–390. doi: 10.2307/2095172

Bentler, P. M. (1972). A lower-bound method for the dimension-free measurement of internal consistency. Social Science Research, 1(4), 343–357. doi: 10.1016/0049-089X(72)90082-8

Bentler, P. M. (2009). Alpha, dimension-free, and model-based internal consistency reliability. Psychometrika, 74(1), 137–143. doi: 10.1007/s11336-008-9100-1

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334. doi: 10.1007/BF02310555

Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement errors. Journal of Marketing Research, 18(1), 39–50. doi: 10.2307/3151312

Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74(1), 155–167. doi: 10.1007/s11336-008-9099-3

McDonald, R. P. (1999). Test theory: A unified treatment. Mahwah, NJ: Erlbaum.

Raykov, T. (2001). Estimation of congeneric scale reliability using covariance structure analysis with nonlinear constraints British Journal of Mathematical and Statistical Psychology, 54(2), 315–323. doi: 10.1348/000711001159582

See Also

reliabilityL2 for reliability value of a desired second-order factor, maximalRelia for the maximal reliability of weighted composite

Examples