# Comparing latent means using two factor scaling methods : a Monte Carlo study

## Abstract

Social science researchers are increasingly using multi-group confirmatory factor analysis (MG-CFA) to compare different groups' latent variable means. To ensure that a MG-CFA model is identified, two approaches are commonly used to set the scale of the latent variable. The reference indicator (RI) strategy, which involves constraining one loading per factor to a value of one across groups, assumes that the RI has equal factor loadings across groups. The second approach involves constraining each factor's variance to a value of one across groups and, thus, assumes that the factor variances are equal across groups. Latent mean differences may be tested and described using Gonzalez and Griffin's (2001) likelihood ratio test (LRT[subscript k]) and Hancock's (2001) standardized latent mean difference effect size measure ([delta subscript k]), respectively. Applied researchers using the LRT[subscript k] and/or the [delta subscript k] when comparing groups' latent means may not explicitly test the assumptions underlying the two factor scaling methods. To date, no study has examined the impact of violating the assumptions associated with the two scaling methods on latent mean comparisons. The purpose of this study was to assess the performance of the LRT[subscript k] and the [delta subscript k] when violating the assumptions underlying the RI strategy and/or the factor variance scaling method. Type I error and power of the LRT[subscript k] as well as relative parameter bias and parameter bias of the [delta subscript k] were examined when varying loading difference magnitude, factor variance ratio, factor loading pattern and sample size ratio. Rejection rates of model fit indices, including the x² test, RMSEA, CFI, TLI and SRMR, under these varied conditions were also examined. The results indicated that violating the assumptions underlying the RI strategy did not affect the LRT[subscript k] or the [delta subscript k]. However, violating the assumption underlying the factorvariance scaling method influenced Type I error rates of the LRT[subscript k], particularly in unequal sample size conditions. Results also indicated that the four factors manipulated in this study had an impact on correct model rejection rates of the model fit indices. It is hoped that this study provides useful information to researchers concerning the use of the LRT[subscript k] and [delta subscript k] under factor scaling method assumption violations.