  Silbert and Thomas (2013) showed that failures of decisional separability are
not, in general, identifiable in fully parameterized $2 \times 2$ Gaussian GRT
models. A recent extension of $2 \times 2$ GRT models (GRTwIND) was developed
to solve this problem and a conceptually similar problem with the simultaneous
identifiability of means and marginal variances in GRT models. Central to the
ability of GRTwIND to solve these problems is the assumption of universal
perception, which consists of shared perceptual distributions modified by
attentional and global scaling parameters (Soto et al., 2015). If universal
perception is valid, GRTwIND solves both issues. In this paper, we show that
GRTwIND with universal perception and subject-specific failures of decisional
separability is mathematically, and thereby empirically, equivalent to a model
with decisional separability and failure of universal perception. We then
provide a formal proof of the fact that means and marginal variances are not,
in general, simultaneously identifiable in $2 \times 2$ GRT models, including
GRTwIND. These results can be taken to delineate precisely what the assumption
of universal perception must consist of. Based on these results and related
recent mathematical developments in the GRT framework, we propose that, in
addition to requiring a fixed subset of parameters to determine the location
and scale of any given GRT model, some subset of parameters must be set in GRT
models to fix the orthogonality of the modeled perceptual dimensions, a central
conceptual underpinning of the GRT framework. We conclude with a discussion of
perceptual primacy and its relationship to universal perception.
