alpha(x, keys=NULL,title=NULL,na.rm = TRUE)Surprisingly, 105 years after Spearman (1904) introduced the concept of reliability to psychologists, there are still multiple approaches for measuring it. Although very popular, Cronbach's $\alpha$ (1951) underestimates the reliability of a test and over estimates the first factor saturation.
$\alpha$ (Cronbach, 1951) is the same as Guttman's $\lambda$3 (Guttman, 1945) and may be found by $$\lambda_3 = \frac{n}{n-1}\Bigl(1 - \frac{tr(\vec{V})_x}{V_x}\Bigr) = \frac{n}{n-1} \frac{V_x - tr(\vec{V}_x)}{V_x} = \alpha$$
Perhaps because it is so easy to calculate and is available in most commercial programs, alpha is without doubt the most frequently reported measure of internal consistency reliability. Alpha is the mean of all possible spit half reliabilities (corrected for test length). For a unifactorial test, it is a reasonable estimate of the first factor saturation, although if the test has any microstructure (i.e., if it is ``lumpy") coefficients $\beta$ (Revelle, 1979; see ICLUST) and $\omega_h$ (see omega) are more appropriate estimates of the general factor saturation. $\omega_t$ (see omega) is a better estimate of the reliability of the total test.
Guttman's Lambda 6 (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items (the squared multiple correlation or smc), or more precisely, the variance of the errors, $e_j^2$, and is $$\lambda_6 = 1 - \frac{\sum e_j^2}{V_x} = 1 - \frac{\sum(1-r_{smc}^2)}{V_x} .$$
The squared multiple correlation is a lower bound for the item communality and as the number of items increases, becomes a better estimate.
G6 is also sensitive to lumpyness in the test and should not be taken as a measure of unifactorial structure. For lumpy tests, it will be greater than alpha. For tests with equal item loadings, alpha > G6, but if the loadings are unequal or if there is a general factor, G6 > alpha.
Alpha and G6 are both positive functions of the number of items in a test as well as the average intercorrelation of the items in the test. When calculated from the item variances and total test variance, as is done here, raw alpha is sensitive to differences in the item variances. Standardized alpha is based upon the correlations rather than the covariances.
More complete reliability analyses of a single scale can be done using the omega function which finds $\omega_h$ and $\omega_t$ based upon a hierarchical factor analysis.
Alternative functions score.items and cluster.cor will also score multiple scales and report more useful statistics. ``Standardized" alpha is calculated from the inter-item correlations and will differ from raw alpha.
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10 (4), 255-282.
Revelle, W. Hierarchical Cluster Analysis and the Internal Structure of Tests. Multivariate Behavioral Research, 1979, 14, 57-74.
Revelle, W. and Zinbarg, R. E. (2009) Coefficients alpha, beta, omega and the glb: comments on Sijtsma. Psychometrika, 2009.
omega, ICLUST,r4 <- sim.congeneric()
alpha(r4)
r9 <- sim.hierarchical()
alpha(r9)
#an example of two independent factors that produce reasonable alphas
#this is a case where alpha is a poor indicator of unidimensionality
two.f <- sim.item(8)
alpha(two.f,keys=c(rep(1,4),rep(-1,4)))Run the code above in your browser using DataLab