semTools (version 0.5-4)

reliability: Calculate reliability values of factors

Description

Calculate reliability values of factors by coefficient omega

Usage

reliability(object, return.total = FALSE, dropSingle = TRUE,
  omit.imps = c("no.conv", "no.se"))

Arguments

object

A '>lavaan or '>lavaan.mi object, expected to contain only exogenous common factors (i.e., a CFA model).

return.total

logical indicating whether to return a final column containing the reliability of a composite of all items. Ignored in 1-factor models, and should only be set TRUE if all factors represent scale dimensions that could nonetheless be collapsed to a single scale composite (scale sum or scale mean).

dropSingle

logical indicating whether to exclude factors defined by a single indicator from the returned results. If TRUE (default), single indicators will still be included in the total column when return.total = TRUE.

omit.imps

character vector specifying criteria for omitting imputations from pooled results. Can include any of c("no.conv", "no.se", "no.npd"), the first 2 of which are the default setting, which excludes any imputations that did not converge or for which standard errors could not be computed. The last option ("no.npd") would exclude any imputations which yielded a nonpositive definite covariance matrix for observed or latent variables, which would include any "improper solutions" such as Heywood cases. NPD solutions are not excluded by default because they are likely to occur due to sampling error, especially in small samples. However, gross model misspecification could also cause NPD solutions, users can compare pooled results with and without this setting as a sensitivity analysis to see whether some imputations warrant further investigation.

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.

Details

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

$$\alpha =\frac{k}{k-1}[1-\frac{\sum _{i=1}^k{\sigma }_{ii}}{\sum _{i=1}^k{\sigma }_{ii}+2\sum _{i<j}{\sigma }_{ij}}],$$

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

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

$${\omega }_1=\frac{{\left(\sum _{i=1}^k{\lambda }_i\right)}^{2}Var\left(\psi \right)}{{\left(\sum _{i=1}^k{\lambda }_i\right)}^{2}Var\left(\psi \right)+\sum _{i=1}^k{\theta }_{ii}+2\sum _{i<j}{\theta }_{ij}},$$

where ${\lambda }_i$ is the factor loading of item i, $\psi$ is the factor variance, ${\theta }_{ii}$ is the variance of measurement errors of item i, and ${\theta }_{ij}$ is the covariance of measurement errors from item i and j.

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

$${\omega }_2=\frac{{\left(\sum _{i=1}^k{\lambda }_i\right)}^{2}Var\left(\psi \right)}{\text{bold}{1}^{\prime }\hat{\mathrm{\Sigma }}\text{bold}1},$$

where $\hat{\mathrm{\Sigma }}$ is the model-implied covariance matrix, and $\text{bold}1$ 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 ${\eta }_{partial}^2$ in ANOVA). The second coefficient omega can be viewed as the unconditional reliability (like ${\eta }^2$ in ANOVA).

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

$${\omega }_3=\frac{{\left(\sum _{i=1}^k{\lambda }_i\right)}^{2}Var\left(\psi \right)}{\text{bold}{1}^{\prime }\mathrm{\Sigma }\text{bold}1},$$

where $\mathrm{\Sigma }$ is the observed covariance matrix. If the model fits the data well, the third coefficient omega will be similar to the ${\omega }_2$. 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, ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ 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

$$AVE=\frac{\text{bold}{1}^{\prime }\text{diag}\left(\mathrm{\Lambda }\mathrm{\Psi }{\mathrm{\Lambda }}^{\prime }\right)\text{bold}1}{\text{bold}{1}^{\prime }\text{diag}\left(\hat{\mathrm{\Sigma }}\right)\text{bold}1},$$

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 ${\omega }_3$ 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.

References

Bollen, K. A. (1980). Issues in the comparative measurement of political democracy. American Sociological Review, 45(3), 370--390. 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. 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. 10.1007/s11336-008-9100-1

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297--334. 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. 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. 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. 10.1348/000711001159582

See Also

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

Examples

# NOT RUN {
HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

fit <- cfa(HS.model, data = HolzingerSwineford1939)
reliability(fit)
reliability(fit, return.total = TRUE)

# }