alpha.cronbach {CMC} R Documentation

## Cronbach reliability coefficient alpha

### Description

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

### Usage

alpha.cronbach(x)


### Arguments

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

### Details

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

X_{ij}=τ_{ij}+ε_{ij}

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

τ_{ij}=μ_j+a_i

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}.

### Value

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.

### Author(s)

Michela Cameletti and Valeria Caviezel

### References

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 alpha.curve and cain
data(cain)