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Bi-factor and second-order models based on copulas are proposed for itemresponse data, where the items can be split into non-overlapping groups suchthat there is a homogeneous dependence within each group. Our general modelsinclude the Gaussian bi-factor and second-order models as special cases and canlead to more probability in the joint upper or lower tail compared with theGaussian bi-factor and second-order models. Details on maximum likelihoodestimation of parameters for the bi-factor and second-order copula models aregiven, as well as model selection and goodness-of-fit techniques. Our generalmethodology is demonstrated with an extensive simulation study and illustratedfor the Toronto Alexithymia Scale. Our studies suggest that there can be asubstantial improvement over the Gaussian bi-factor and second-order modelsboth conceptually, as the items can have interpretations of latentmaxima/minima or mixtures of means in comparison with latent means, and in fitto data.

Bi-factor and second-order copula models for item response data