Identifiability and testability in GRT with Individual Differences

TL;DR

This paper investigates the limitations of GRT models with individual differences, showing that certain assumptions are necessary for identifiability and proposing refinements to improve model validity.

Contribution

It demonstrates the equivalence of models with different assumptions and provides formal proofs on the non-identifiability of means and variances in GRT models, guiding future model specification.

Findings

GRTwIND with universal perception is equivalent to models with decisional failures and perception failures.

Means and variances are not jointly identifiable in 2x2 GRT models.

Parameter fixing is necessary for model identifiability and interpretability.

Abstract

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.