Cumulant-Based Approaches for Testing the Assumption of Independent Errors in Non-Gaussian Parallel and Congeneric Measures

In classical test theory, independence of measurement errors constitutes a central assumption when estimating the reliability of measures. Furthermore, it is well known that this assumption cannot be tested with standard methods that rely on second-order moments (variances, covariances). The present study, therefore, explores properties of non-Gaussian parallel and congeneric measures (i.e., when observed scores deviate from the Gaussian distribution) with respect to their capabilities of identifying violations of the error independence assumption. We show that, under non-Gaussianity and inequality of hidden confounding effects, third and fourth cumulant-based test statistics can be derived which enable researchers to detect non-independent error structures. We describe identifiability conditions under which the proposed test statistics can be expected to have adequate statistical power in parallel and congeneric measures and present results of Monte-Carlo simulation experiments. Results suggest that third-order tests adequately protect the nominal significance level. However, fourth-order tests can produce inflated Type I error rates, in particular, when error variances are unequal. In general, the power to detect non-independent errors increases with the sample size, the magnitude of non-Gaussianity, the degree of inequality of hidden confounding effects, and the degree of error non-independence. A real-world data example is presented for illustrative purposes. The current study presents important insights for developing statistical methods to detect error non-independence. However, these methods also rest on crucial assumptions, emphasizing the severity of the issue of statistically detecting non-independent measurement errors.