L p Loss Functions in Invariance Alignment and Haberman Linking with Few or Many Groups

The comparison of group means in latent variable models plays a vital role in empirical research in the social sciences. The present article discusses an extension of invariance alignment and Haberman linking by choosing the robust power loss function ρ ( x ) = | x | p ( p > 0 ) . This power loss function with power values p smaller than one is particularly suited for item responses that are generated under partial invariance. For a general class of linking functions, asymptotic normality of estimates is shown. Moreover, the theory of M-estimation is applied for obtaining linking errors (i.e., inference with respect to a population of items) for this class of linking functions. In a simulation study, it is shown that invariance alignment and Haberman linking have comparable performance, and in some conditions, the newly proposed robust Haberman linking outperforms invariance alignment. In three examples, the influence of the choice of a particular linking function on the estimation of group means is demonstrated. It is concluded that the choice of the loss function in linking is related to structural assumptions about the pattern of noninvariance in item parameters.