  Contextuality is a particular quantum phenomenon that has no analogue in
classical probability theory. Given two independent systems, a natural question
is how to represent such a situation as a single test space. In other words,
how separate contextuality scenarios combine into a joint scenario. Under the
premise that the the allowed probabilistic models satisfy the No Signalling
principle, Foulis and Randall defined the unique possible way to compose two
contextuality scenarios. When composing strictly-more than two test spaces,
however, a variety of possible composition methods have been conceived.
Nevertheless, all these formally-distinct composition methods appear to give
rise to observationally equivalent scenarios, in the sense that the different
compositions all allow precisely the same sets of classical and quantum
probabilistic models. This raises the question of whether this property of
invariance-under-composition-method is special to classical and quantum
probabilistic models, or if it generalizes to other probabilistic models as
well, our particular focus being $\mathcal{Q}_1$ models. $\mathcal{Q}_1$ models
are physically important since, when applied to scenarios constructed by a
particular composition rule, coincide with the well-defined "Almost Quantum
Correlations". In this work we see that some composition rules, however, give
rise to scenarios with inequivalent allowed sets of $\mathcal{Q}_1$ models. We
find that the non-trivial dependence of $\mathcal{Q}_1$ models on the choice of
composition method is apparently an artifact of failure of those composition
rules to capture the orthogonality relations given by the Local Orthogonality
principle. We prove that $\mathcal{Q}_1$ models satisfy
invariance-under-composition-method for all the constructive compositions
protocols which do capture this notion of Local Orthogonality.
