calc_alpha() computes Cronbach's coefficient alpha for a set of items from
the unstandardized covariance matrix. Items are coerced to numeric before
calculation. Missing data can be handled via pairwise deletion.
Value
A numeric scalar giving Cronbach's alpha (typically in [0, 1], but can be outside this range for problematic data).
Details
Let \(k\) be the number of items and let \(X_1,\ldots,X_k\) denote the item scores. Define the total score \(T = \sum_{i=1}^k X_i\). Cronbach's alpha can be written without matrix notation as $$\alpha \;=\; \frac{k}{k-1}\left(1 - \frac{\sum_{i=1}^k \mathrm{Var}(X_i)}{\mathrm{Var}(T)}\right).$$ In words: take the sum of the item variances and divide by the variance of the total score; subtract this ratio from 1 and multiply by \(k/(k-1)\).
Both \(\mathrm{Var}(X_i)\) and \(\mathrm{Var}(T)\) are computed using pairwise-complete observations.
Coercion
Each column is coerced via as.numeric(). This means:
factors are converted to integer level codes (not labels);
characters that are not parsable as numbers become
NA;dates/times become their underlying numeric representations. Ensure your columns are truly numeric item responses or pre-process as needed.