Testing Negative Error Variances: Is a Heywood Case a Symptom of Misspecification?

Heywood cases, or negative variance estimates, are a common occurrence in factor analysis and latent variable structural equation models. Though they have several potential causes, structural misspecification is among the most important. This article explains how structural misspecification can lead to a Heywood case in the population, and provides several ways to test whether a negative error variance is a symptom of structural misspecification. The authors consider z-tests based on a variety of standard errors, confidence intervals, bootstrap resampling, and likelihood-ratio-type tests. In their discussion of z-tests, the authors demonstrate which of the standard errors are consistent under different kinds of misspecification. They also introduce new tests based on the scaled chi-square difference: the test on the boundary and the signed root of the scaled chi-square difference. A simulation study assesses the performance of these tests. The authors find that signed root tests and z-tests based on the empirical sandwich and the empirical bootstrap standard errors perform best in detecting negative error variances. They outperform z-tests based on the information matrix or distribution-robust standard errors.