Calculates Cronbach's alpha (Cronbach, 1951),
a coefficient of internal consistency. The coefficient typically serves as an estimate of the reliability of a psychometric test.
Usage
cronbach.alpha(x, standardized = FALSE)
Arguments
x
A numeric data.frame/matrix with rows and columns corresponding to individuals and items,
respectively.
standardized
A logic indicating whether a standardized Cronbach alpha should be calculated (default is FALSE).
Value
Returns Cronbach's alpha as a numeric object.
Details
For a test consisting of $k$ items that measures a quantity $X$,
Cronbach's alpha is defined as
$$\alpha = \frac{k}{k - 1}\left(1 - \frac{\sum_{i=1}^{k}{\sigma_Y}_i^2}{\sigma_X^2}\right)$$
with $X = Y_1 + Y_2 + ... + Y_k$. ${\sigma_Y}_i^2$ is the variance of item $i$,
and $\sigma_X^2$ the variance of the total test score for a sample of individuals that completed the test.
The standardized Cronbach's alpha is defined as
$$\alpha_s = \frac{k\overline{r}}{\left(1 + (k - 1)\overline{r}\right)}$$ where $k$ is the number of items and $\overline{r}$ the mean correlation between the items.
Cases that have missing values on any of the items are excluded.
References
Cronbach,
L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.