alpha.cronbach {CMC}R Documentation

Cronbach reliability coefficient alpha


The function computes the Cronbach reliability alpha coefficient denoted with α.





an object of class data.frame or matrix with n subjects in the rows and k items in the columns.


Let X_1,...,X_k be a set of items composing a test and measuring the same underlying unidimensional latent trait. Moreover, let X_{ij} be the observed score (response) of a subject i (i=1,...,n) on an item j (j=1,...,k). Following the classical test theory, X_{ij} can be written as


where τ_{ij}, the true score, and ε_{ij}, the error score, are two unknown random variables generally assumed to be independent (or at least not correlated). In particular, the true score is given by


where μ_j is a fixed effect and a_i is a random effect with zero mean and variance σ^2_a, whereas ε_{ij} is a random effect with zero mean and variance σ^2_ε. Moreover, ε_{ij} and a_{i} are not correlated and for all j=1,...,k and for t &#8800 s, (a_t,ε_{tj}) and (a_s,ε_{sj}) are independent.

The reliability ρ of any item is defined as the ratio of two variances: the variance of the true (unobserved) measure and the variance of the observed measure. Under the parallel model (see Lord and Novick, 1968), it can be shown that

ρ=σ^2_a \ (σ^2_a+σ^2_ε)

where σ^2_a corresponds to the between-subject variability while σ^2_ε is the variance of the measurement error. It is possible to prove that ρ is also the constant correlation between any two items. The reliability of the sum of k items is given by the well-known Spearman-Brown formula:

\tilde ρ=kρ / (kρ+(1-ρ)).

The maximum likelihood estimate of \tildeρ, under the assumption of Normal distribution for the error, is known as the Cronbach alpha coefficient, denoted with α.

The formula for computing α is given by

α= k /(k-1) * [1-∑_{j=1}^n s^2_j / s^2_{TOT}]

where s^2_j=1/(n-1) * ∑_{i=1}^n (X_{ij}-\bar X_j)^2, s^2_{TOT}=1/(nk-1) * ∑_{i=1}^n ∑_{j=1}^k(X_{ij}-\bar X)^2, \bar Xj=1/n * ∑_{i=1}^n X_{ij} and \bar X=1/(nk) * ∑_{i=1}^n ∑_{j=1}^k X_{ij}.


The function returns the value α of the Cronbach reliability coefficient computed as described above. The coefficient takes values in the interval [0,1]. If the actual variation amongst the subjects is very small, then the reliability of the test measured by α tends to be small. On the other hand, if the variance amongst the subject is large, the reliability tends to be large.


No missing values are admitted.


Michela Cameletti and Valeria Caviezel


Cronbach, L.J. (1951) Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.

Lord, F.M. and Novick, M.R. (1968) Statistical Theories of Mental Test Scores. Addison-Wesley Publishing Company, 87–95.

See Also

See Also alpha.curve and cain


out = alpha.cronbach(cain)

[Package CMC version 1.0 Index]