Can Unequal Residual Variances Across Groups Mask Differences in Residual Means in the Common Factor Model?

Equality of residual variances across groups is one of the necessary conditions of measurement invariance. The main argument for not applying this restriction in the analysis of empirical data is that unequal residual variances across groups are differences in reliability of the observed variables rather than a violation of measurement invariance. A power study is carried out to investigate the conditions under which violations of measurement invariance can be masked by unequal residual variances across groups. Increasing group differences in residual variance are combined with mean differences in the item-specific residual. Sample sizes needed for rejection of a model with free residual variances across groups are computed for different model sizes and varying group sample size ratios. Increasing group differences in residual variance decreases the power to detect differences in specific means if the residual variances are not held equal across groups. This is especially the case for small mean differences, unequal group sample sizes, and if differences in residual means are accompanied by correlated residuals.