Identifiability and testability in GRT with Individual Differences
  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. 
Noah H. Silbert and Robin D. Thomas
stat.OT