Keywords: Clustered data, Complex survey, Coverage-correction method, Influence function, Linearization.
We discuss the relevant inference of coefficient alpha (Cronbach, 1951) - a popular ratio-type statistic of the covariances and variances - incorporating complex survey sampling with unequal selection probabilities and clusters. This study will be helpful for investigators who wish to evaluate various psychological or social instruments used in large surveys. For complex survey data, we investigate workable confidence intervals 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 which many instruments consist of, while it may not be as good with some difficult underlying distributions, e.g., multi-lognormal distribution. We suggest that the coverage-corrected bootstrap method can be used as a complementary to the linearization method as the coverage-corrected bootstrap method is compute-intensive. Using the developed methods, we provide the confidence intervals of coefficient alphas to assess mental health instruments (Kessler 10, Kessler 6 and Sheehan disability scale) by different demographics using data from the National Comorbidity Survey Replication (NCS-R).