This paper presents a new class of models for persons-by-items data. The es
sential new feature of this class is the representation of the persons: eve
ry person is represented by its membership to multiple latent classes, each
of which belongs to one latent classification. The models can be considere
d as a formalization of the hypothesis that the responses come about in a p
rocess that involves the application of a number of mental operations. Two
algorithms for maximum likelihood (ML) and maximum a posteriori (MAP) estim
ation are described. They both make use of the tractability of the complete
data likelihood to maximize the observed data likelihood. Properties of th
e MAP estimators (i.e., uniqueness and goodness-of-recovery) and the existe
nce of asymptotic standard errors were examined in a simulation study. Then
, one of these models is applied to the responses to a set of fraction addi
tion problems. Finally, the models are compared to some related models in t
he literature.