CMC (version 1.0)

alpha.cronbach: Cronbach reliability coefficient alpha

Description

The function computes the Cronbach reliability alpha coefficient denoted with $\alpha$.

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.

Value

The function returns the value $\alpha$ 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 $\alpha$ tends to be small. On the other hand, if the variance amongst the subject is large, the reliability tends to be large.

Warning

No missing values are admitted.

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 $$X_{ij}=\tau_{ij}+\epsilon_{ij}$$ where $\tau_{ij}$, the true score, and $\epsilon_{ij}$, 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 $$\tau_{ij}=\mu_j+a_i$$ where $\mu_j$ is a fixed effect and $a_i$ is a random effect with zero mean and variance $\sigma^2_a$, whereas $\epsilon_{ij}$ is a random effect with zero mean and variance $\sigma^2_\epsilon$. Moreover, $\epsilon_{ij}$ and $a_{i}$ are not correlated and for all $j=1,...,k$ and for $t ≠ s$, $(a_t,\epsilon_{tj})$ and $(a_s,\epsilon_{sj})$ are independent.

The reliability $\rho$ 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 $$\rho=\frac{\sigma^2_a}{\sigma^2_a+\sigma^2_\epsilon}$$ where $\sigma^2_a$ corresponds to the between-subject variability while $\sigma^2_\epsilon$ is the variance of the measurement error. It is possible to prove that $\rho$ 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: $$\tilde\rho=\frac{k\rho}{k\rho+(1-\rho)}.$$ The maximum likelihood estimate of $\tilde\rho$, under the assumption of Normal distribution for the error, is known as the Cronbach alpha coefficient, denoted with $\alpha$.

The formula for computing $\alpha$ is given by $$\alpha=\frac{k}{k-1}\left[1-\frac{\sum_{j=1}^n s^2_j}{s^2_{TOT}}\right]$$ where $s^2_j=1/(n-1) * \sum_{i=1}^n (X_{ij}-\bar X_j)^2$, $s^2_{TOT}=1/(nk-1) * \sum_{i=1}^n \sum_{j=1}^k(X_{ij}-\bar X)^2$, $\bar Xj=1/n * \sum_{i=1}^n X_{ij}$ and $\bar X=1/(nk) * \sum_{i=1}^n \sum_{j=1}^k X_{ij}$.

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

See Also alpha.curve and cain

Examples

data(cain)
out = alpha.cronbach(cain)
out