  Bi-factor and second-order models based on copulas are proposed for item
response data, where the items can be split into non-overlapping groups such
that there is a homogeneous dependence within each group. Our general models
include the Gaussian bi-factor and second-order models as special cases and can
lead to more probability in the joint upper or lower tail compared with the
Gaussian bi-factor and second-order models. Details on maximum likelihood
estimation of parameters for the bi-factor and second-order copula models are
given, as well as model selection and goodness-of-fit techniques. Our general
methodology is demonstrated with an extensive simulation study and illustrated
for the Toronto Alexithymia Scale. Our studies suggest that there can be a
substantial improvement over the Gaussian bi-factor and second-order models
both conceptually, as the items can have interpretations of latent
maxima/minima or mixtures of means in comparison with latent means, and in fit
to data.
