alpha(x, keys=NULL,cumulative=FALSE, title=NULL, max=10,na.rm = TRUE)link{response.frequencies}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 is a generalization of an earlier estimate of reliability for tests with dichotomous items developed by Kuder and Richardson, known as KR20, and a shortcut approximation, KR21. (See Revelle, in prep).
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.
Three alternative item-whole correlations are reported, two are conventional, one unique. r is the correlation of the item with the entire scale, not correcting for item overlap. r.drop is the correlation of the item with the scale composed of the remaining items. Although both of these are conventional statistics, they have the disadvantage that a) item overlap inflates the first and b) the scale is different for each item when an item is dropped. Thus, the third alternative, r.cor, corrects for the item overlap by subtracting the item variance but then replaces this with the best estimate of common variance, the smc.
Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10 (4), 255-282.
Revelle, W. (in preparation) An introduction to psychometric theory with applications in {R}. Springer. (Available online at
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, guttman, score.items, cluster.corr4 <- 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)))
#an example with discrete item responses -- show the frequencies
items <- sim.congeneric(N=500,short=FALSE,low=-2,high=2,categorical=TRUE) #500 responses to 4 discrete items
a4 <- alpha(items$observed) #item response analysis of congeneric measures
a4
#summary just gives Alpha
summary(a4)Run the code above in your browser using DataLab