Suggestion of confidence interval methods for the Cronbach alpha in application to complex survey data

We discuss a relevant inference for the alpha coefficient (Cronbach, 1951) - a popular ratio-type statistic for the covariances and variances in survey sampling including complex survey sampling with unequal selection probabilities. This study can help investigators who wish to evaluate various psychological or social instruments used in large surveys. For the survey data, we investigate workable confidence intervals by using two approaches: (1) the linearization method using the influence function and (2) the coverage-corrected bootstrap method. The linearization method provides adequate coverage rates with correlated ordinal values that many instruments consist of; however, this method may not be as good with some non-normal underlying distributions, e.g., a multi-lognormal distribution. We suggest that the coverage-corrected bootstrap method can be used as a complement to the linearization method, because the coverage-corrected bootstrap method is computer-intensive. Using the developed methods, we provide the confidence intervals for the alpha coefficient to assess various mental health instruments (Kessler 10, Kessler 6 and Sheehan Disability Scale) for different demographics using data from the National Comorbidity Survey Replication (NCS-R).