Cronbach’s \(\alpha\)(Cronbach 1951) is a widely used measure of internal consistency. If \(K\) scored items \(X_i\) (i.e. questions in a survey) are used to generate the construct \(Y = \sum_{i=1}^{K} X_i\) then Cronbach’s \(\alpha\) is defined as
where \(\alpha \leq 1\) (and can be negative when items are negatively correlated), \(\sigma_Y^2\) is the Variance of the sum of scored items, and \(\sigma_{X_i}^2\) is the variance of item \(i\).
The standardized Cronbach’s \(\alpha\) is obtained by the following expression
where \(\bar{r}\) represents the mean of the off-diagonal coefficients in the upper (or lower) triangular correlation matrix.
Since \(\alpha\) and \(\alpha_s\) are both dependent on the number of items \(K\) it is possible to assess the impact of “leaving one item out” by applying the following derived formula
where \(\alpha_{(i)}\) is Cronbach’s \(\alpha\) without item \(i\), \(Y_{(i)} = \sum_{j=1}^{i-1} X_j + \sum_{j=i+1}^{K} X_j\), and \(\sigma_{Y_{(i)}}^2\) is the variance of \(Y_{(i)}\).
For any index \(i = 1, 2, …, K\) the values of \(\alpha_{(i)}\) can be compared with \(\alpha\). If \(\alpha_{(i)} > \alpha\) then it can be concluded that leaving out the \(i\)-th item improves the internal consistency of the construct.
85.2 R Module
85.2.1 Public website
The Cronbach’s \(\alpha\) R module is available on the public website:
Warning in alpha(x, check.keys = par1): Some items were negatively correlated with the first principal component and were automatically reversed.
This is indicated by a negative sign for the variable name.
To compute the Cronbach’s \(\alpha\), the R code uses the alpha function from the psych library. Note: check.keys = TRUE will automatically reverse-key negatively correlated items; inspect the output to confirm which items were reversed.
85.3 Purpose
Cronbach’s \(\alpha\) is an internal-consistency coefficient for the reliability of a construct based on survey scores; it is a lower bound on reliability only when model assumptions (tau-equivalence and uncorrelated errors) are met. A commonly cited rule of thumb is \(\alpha > 0.7\) for acceptable internal consistency (Nunnally 1978), though appropriate thresholds vary by research context and discipline.
85.4 Pros & Cons
85.4.1 Pros
The Cronbach’s \(\alpha\) has the following advantages:
It is widely known and understood by many readers.
It is relatively easy to interpret and many software packages are able to compute it.
85.4.2 Cons
The Cronbach’s \(\alpha\) has the following disadvantages:
It is a function of \(K\) and does not always truly reflect internal consistency (it should rather be regarded as an indicative degree of internal consistency). As an alternative, McDonald’s \(\omega\) is less sensitive to the number of items and can be computed with the omega function from the psych library.
In the context of confirmatory factor analysis, McDonald’s \(\omega\) is theoretically preferable because \(\alpha\) assumes tau-equivalence of items, an assumption that CFA can directly test and often rejects.
The analysis shows (among others) the values of \(\alpha\), \(\alpha_s\), and \(\alpha_{(i)}\) for a set of four items that belong the construct IM.Know (a form of intrinsic, academic motivation). It can be concluded that leaving out any item does not improve \(\alpha\).
Cronbach, Lee J. 1951. “Coefficient Alpha and the Internal Structure of Tests.”Psychometrika 16 (3): 297–334. https://doi.org/10.1007/BF02310555.
Nunnally, Jum C. 1978. Psychometric Theory. 2nd ed. New York: McGraw-Hill.