Errata to: PSYCHOMETRIKA, 2012, 77, 561–580 DOI10.1007/s11336-012-9270-8
The Errata includes a corrected figure on the effect of group differences in common factor means and latent slopes on Type I error rates for ordinary least squares tests of intercept differences.
1. Correction to Figure 2
Culpepper (Reference Culpepper2012) examined the performance of ordinary least squares (OLS) as a method for assessing the presence of prediction bias (Millsap, Reference Millsap1997, Reference Millsap1998, Reference Millsap2007; Olivera-Aguilar & Millsap, Reference Olivera-Aguilar and Millsap2013). A seminar student at the University of Minnesota kindly pointed out an error in Figure 2 of Culpepper (Reference Culpepper2012). The error was clerical and the equations in the paper remain unchanged and correct. The purpose of this note is to correct Figure 2 in Culpepper related to the effect of group differences in common factor means (i.e., Δκ) and latent slopes (i.e., ΔΓ) on Type I error rates of OLS tests of group intercept differences.
Recall from Culpepper (Reference Culpepper2012) that Z and Y represented the observed predictor and criterion, respectively, the number of applicants was indicated by n, the proportion in the focal group by p, and P(Z>z ⋆) was the percent of applicants selected in a top-down fashion. Furthermore, the latent measurement model parameters included Δϕ as the difference in group common factor variances, Δξ as the difference in latent prediction error variance, θ z and θ y were the unique factor variances for the predictor and criterion, and the latent measurement intercept and loading was denoted by τ z and λ z for the predictor and τ y and λ y criterion. Figure 1 presents the corrected analytic (and Monte Carlo) Type I errors of tests for intercept differences across values of Δκ and ΔΓ under the assumption of strict invariance with an n= 5,000, p=.8, P(Z>z ⋆)=.5, Δϕ=0, Δξ=0, τ z=τ y=.1, λ z=λ y=.8, and Θ z=Θ y=.2. The dots around each of the Type I error curves in Figure 1 are the Monte Carlo estimates using 5,000 replications.
Figure 1. Impact of Δκ and latent slope differences (\documentclass[12pt]{minimal}
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) on Type I error rates for tests of intercept differences. Note. The dots around each curve are Monte Carlo estimates using 5,000 replications.
Figure 1 demonstrates the results pertaining to Millsap (Reference Millsap1997, Reference Millsap1998, Reference Millsap2007) and Equation (30) of Culpepper (Reference Culpepper2012). Figure 1 shows that Type I error rates for intercept tests are larger than the real rejection level of .05 as Δκ increases. For example, the probability of rejecting a true null hypothesis is approximately 50 % when Δκ=1. In this case, larger latent subgroup slope differences (i.e., \documentclass[12pt]{minimal}
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) slightly reduces intercept Type I error rates. In short, Figure 1 demonstrates that OLS is an inadequate method for testing the presence of subgroup intercept differences whenever groups differ in common factor means.
