Suggestion of confidence interval methods for the Cronbach alpha in application to complex survey data
Section 1. Introduction

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In this paper, we propose methods to incorporate the survey designs in confidence intervals for the alpha coefficient (Cronbach, 1951) based on the large sample approximation (linearization) and the “double” bootstrap approach. These methods have not been investigated in the related literature, even though the alpha coefficient is widely used in psychology and other relevant research areas. For a practical application of these methods, we analyze mental health instruments data from the National Comorbidity Survey Replication (NCS-R), a survey conducted between 2001 and 2003 intended to measure the prevalence of mental disorders (Kessler, Berglund, Chiu, Demler, Heeringa, Hiripi, Jin, Pennell, Walters, Zaslavsky and Zheng, 2004). In the analysis, we show the feasibility of the confidence interval method for the alpha coefficient on a survey data set.

A great deal of psychological and sociological research uses assessment instruments (i.e., questionnaires) to obtain quantitative information for a population of interest. Ideally, the different items in one instrument measure the same concepts to achieve a high internal consistency. The alpha coefficient, also known as Cronbach’s alpha (henceforth referred to as $\alpha )$ is a popular statistic (e.g., a quick search of PubMed with the keywords “Cronbach alpha” and “scale” from the years of 2012-2016 brings up more than 700 publications) that is widely used to measure the internal consistency reliability of various instruments.

Let $x$ denote the $p$ -variate column vector of the observations indicating $p$ items from an instrument, and let $\Sigma$ indicate the corresponding covariance matrix. The value $\alpha$ is defined as

$$\alpha =p/\left(p-1\right)\left(1-\text{tr}\Sigma /1^T\Sigma 1\right),$$

where $1$ is the conforming column vector consisting of 1, and $\text{tr}$ indicates the trace of a matrix. The value $\alpha$ shows the ratio between the sum of the covariances and the sum of variances and covariances, thus a high value for $\alpha$ suggests that the items are highly correlated within the instrument. The theoretical values of $\alpha$ range from 0 to 1, where a higher value is considered to be more desirable. The estimator of $\alpha$ (denoted by $\widehat{\alpha })$ is defined as

$$\widehat{\alpha }=p/\left(p-1\right)\left(1-\text{tr}\widehat{\Sigma }/1^T\widehat{\Sigma }1\right),$$

where $\widehat{\Sigma }$ is a consistent estimator of $\Sigma .$ The estimator $\widehat{\alpha }$ can take any value less than or equal to 1, including negative values.

In the literature, many confidence interval strategies for $\alpha$ can be found (e.g., van Zyl, Neudecker and Nel, 2000; Yuan, Guarnaccia and Hayslip, 2003; Kistner and Muller, 2004; Bonett and Wright, 2015), but discussions regarding the applications for complex survey data where observations in the data can have unequal weights due to stratifications and multistage cluster sampling (Lohr, 1999) are largely lacking.

This paper is structured as follows: In Section 2, we propose strategies for obtaining the confidence intervals of $\alpha$ using the linearization method and the coverage-corrected bootstrap method. In Section 3, simulation results are presented based on scenarios of stratified multi-stage cluster sampling and unequal probability sampling scenarios. In Section 4, the developed methods are applied to analyze the NCS-R data sets, and the results comparing different demographics are reported. The Section 5 is devoted to the concluding remarks.

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