Coefficient alpha (Cronbach, 1951) is the primary reliability coefficient in classical test theory for researchers. In practice, it is important to determine whether coefficient alpha is high as well as whether it provides a stable estimation of the true reliability. As with any point estimator, sample coefficient alpha is subject to variability around the true parameter, particularly in small samples. Thus, a better appraisal of the reliability of test scores is obtained by using an interval estimator for coefficient alpha. There is a compelling argument for the use of an interval estimator, instead of a point estimator, for reliability coefficient. The sampling distribution of split-half reliability coefficient is useful to perform inferential statistics to guide the interpretation of the coefficient and testing hypothesis regarding the equality of the coefficients.

Kristof (1963) and Feldt (1965) independently derived the exact distribution of the alpha coefficient under the assumption that the covariance matrix of items has a compound symmetry form. Also, van Zyl, Neudecker, and Nel (2000) derived the asymptotic normal distribution of the maximum likelihood estimator of the alpha coefficient under a more general condition in which compound symmetry of the covariance matrix is not assumed. Charter (2000; 2001) and Feldt & Charter (2003) provided a confidence interval for split-half reliability coefficient as a simple case of the sampling distribution of coefficient alpha for two-part test.

However, the existing methods for estimating the sampling distribution of coefficient alpha were different from the true simulated distribution for small number of parts and small sample size (van Zyl, Neudecker, and Nel, 2000). In addition, procedures based on Kristof (1963) and Feldt (1965) are not general given the assumption of compound symmetry which might not be supported by practice.

The paper aims to propose an asymptotic sampling distribution of the simple case of coefficient alpha for two items or two parts (split-half reliability coefficient) with no assumption regarding the covariance matrix of the two split halves using Delta method. In addition, the paper aims to compare the performance of the proposed sampling distribution and van Zyl, Neudecker, and Nel (2000) under different test conditions.

Sampling distribution of Simple case of Coefficient Alpha

Coefficient alpha gives the internal consistency reliability of test scores composed of n items. I.e.,

 ,

where  is the sum of item score variances and  is the test summed score variance.

Coefficient alpha for two items or two parts is a simple case of coefficient alpha which gives the split-half reliability coefficient (Rulon, 1939).

 ,

where  and  are the covariance and correlation between the summed raw scores on the two parts,  and  are the sample variances of the summed raw scores on the two parts, and , and  is the test summed score variance. The corresponding point estimate is calculated using the sample counterparts from a sample of  examinees, i.e.,

 .

After tedious derivations, the sampling standard error variance for , using delta method under the assumption of a bivariate normal distribution, is

 .

If the assumption of compound symmetry of covariance matrix () is assumed,

,

which equals Kristof’s (1963) and Feldt’s (1965) sampling error variance for coefficient alpha for two-part test.

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