(2) use a polychoric correlation matrix input to calculate alpha parallel to Cronbach. However, we can speak of an ordinal reliability alpha. In fact, the sole case where alpha will be essentially the same as reliability is the case of uniformly high factor loadings, no error covariances, and unidimensional instrument (1). The usual assumptions pertaining to a correct interpretation of its value are as follows: (i) no residual correlations, (ii) items have identical loadings, and (iii) the scale is unidimensional. It is a lower bound for reliability, and is essentially used as an indicator of internal consistency of a test or questionnaire. From a practical perspective, I don't see any obvious reason to not use Cronbach's alpha with ordinal items (e.g., Likert-type items), as is commonly done in most of the studies.
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