Subset choice denotes a situation in which decision makers are offered avai
lable sets from a fixed master set of choice alternatives and each decision
maker is asked to choose a subset of any size from the available set. In t
his paper, we study the relationships between various random utility models
of subset choice. Random utility threshold models of subset choice assume
that there is a (random) utility associated with each available option, and
a (random) utility threshold, such that the decision maker selects those o
ptions in the available set whose utilities are greater than or equal to th
e threshold, ii special case of the random utility threshold model is the l
atent scale model, in which the threshold has a constant value and the rand
om variables associated with the available options are independent of each
other. We show that the size-independent random utility model for approval
voting of Falmagne and Regenwetter (1996) is a random utility threshold mod
el, and develop numerous results relating that model to the class of random
utility threshold models in general, and to the latent scale model in part
icular. Among the features distinguishing some of these models is a closure
property that we call stability under substructures. The size-independent
model is not stable, in the sense that certain marginals of a given size-in
dependent model for n choice alternatives may violate all size-independent
models for n - 1 choice alternatives. In contrast, the general class of ran
dom utility threshold models and also the specific subclass of latent scale
models are stable under substructures.