ROC asymmetry is not diagnostic of unequal residual variance in Gaussian signal detection theory

Abstract

Signal detection theory (SDT) is used to analyze yes/no judgment accuracy in many research domains of psychology. SDT yields separate estimates for response bias/criterion (c) and for sensitivity/discriminability (d'). Discrimination performance can be displayed in Receiver Operating Characteristics (ROCs) plotting hit and false alarm rates at various levels of confidence. We provide formal proof and simulations showing that asymmetric ROCs in Gaussian SDT are not exclusively diagnostic of unequal residual variance but may as well result from equal-variance models with c and d' systematically varying across subjects and/or items. Falsely attributing zROC slopes to unequal residual variance while neglecting true group-level variability introduces systematic and unsystematic statistical error. We show that ordinal regression models minimize such errors while estimating all SDT parameters and statistical criteria in a single model.

Does variability in recognition memory scale with mean memory strength or encoding variability in the UVSD model?

“…It also stands that changes in the shape of the z-ROC are not exclusively caused by mnemonic factors (Malmberg & Xu, 2006;Rabe, Lindsay & Kliegl, 2021), and many models reflect this. For instance, previous models have added variability to recognition decision criteria, and have potential to add to discussion about the unequal variance assumption (Benjamin, Diaz, & Wee, 2009).…”

“…It is commonly found that z-ROCs calculated from recognition confidence data are approximately linear, with slopes less than 1 (Glanzer, Kim, Hilford, & Adams, 1999). Since the value of the z-ROC slope has long been presumed to represent the ratio σo / σn in a traditional Gaussian signal detection model (but see Rabe, Lindsay & Kliegl, 2021), a non-unit z-ROC slope necessitates making σo a free parameter with a value typically greater than σn. With this parameterization, the unequal variance signal detection (UVSD) model is defined as having parameters θ = {d, σo, C1, C2, … CI} where I is the highest decision criterion level in terms of strength (Kellen, Klauer, & Bröder, 2013).…”