Fuzzy Signal Detection Theory: ROC Analysis of Stimulus and Response Range Effects J.L. Szalma and P.A. Hancock Department of Psychology and Institute for Simulation and Training University of Central Florida Abstract Prior ROC experiments have found that the Fuzzy Signal Detection Theory (FSDT) meets the normality assumption of traditional Signal Detection Theory (SDT). However, support for the equal variance assumption depended on discrimination difficulty. To further explore fuzzy ROC space we manipulated the number of stimulus categories (range), the difference in magnitude between categories (interval size), and the response set permitted (binary vs. seven categories). Response bias was manipulated via a payoff matrix. Four participants engaged in four temporal discrimination tasks. Results confirmed the FSDT model meets the normality assumption of SDT. The equal variance assumption was met depending on the condition and the participant, possibly because of difficulty in setting stable ‘fuzzy criteria.’ Forcing binary responses resulted in poorer performance relative to conditions in which a range of responses was permitted. Increasing either the range of stimuli (number of categories) or the intercategory interval (20 vs. 80 msec differences) enhanced perceptual sensitivity Acknowledgement This research was supported by a Multidisciplinary University Research Initiative (MURI) program grant from the Army Research Office, P.A. Hancock, Principal Investigator. (Grant# DAAD ). The views expressed in this article are those of the authors and do not necessarily reflect the official policy or position of the Department of the Army, Department of Defense, or the US Government. The authors wish to thank Dr. Sherry Tove, Dr. Elmar Schmeisser, and Dr. Mike Drillings for providing administrative and technical direction for the Grant. References Available Upon Request Elements of Fuzzy Signal Detection Theory Events can belong to the set “signal” (s) to a degree ranging from 0 to 1 Events can belong to the set “response” (r) to a degree ranging from 0 to 1 After defining these sets FSDT measures can be derived. (Parasuraman, Masalonis, & Hancock, 2000) Four Steps for Computation of FSDT Measures 1) Select mapping functions for signal & response dimensions To assign degrees of (s, r) membership to events, all possible states of the world and each possible response must be evaluated using a mapping function that describes the relation between each set and the corresponding real-world variables. 2) Assignment of degrees of membership to the four outcomes using mixed implication functions. H = min (s,r) M = max (s-r, 0) FA = max (r-s, 0) CR = min (1-s, 1-r) 3) Compute fuzzy Hit, Miss, False Alarm, and Correct Rejection Rates HR=Σ(Hi)/Σ(si) for i=1 to N MR = Σ(Mi)/Σ(si) for i =1 to N FAR = Σ(FAi)/ Σ(1-si) for i=1to N CRR = Σ(CRi)/ Σ(1-si) for i= 1 to N 4) Compute detection theory measures traditional SDT equations. Assumptions of SDT  Noise and Signal+Noise distributions are normally distributed Linear ROC  Variances of the two distributions are equal Unit slope Overall Conclusions  Sensitivity is higher for larger intercategory intervals  Sensitivity is higher for larger stimulus ranges  Fuzzy response sets yield more accurate performance assessment than binary response sets (ratings match stimulus level more closely)  Gaussian assumption met  Equal variance assumption – May be met – Depends on effect of instruction set  What is a ‘fuzzy response criterion’? Future Directions Manipulate instruction set Manipulate stimulus distribution (‘signal rate’) Question: What is the structure of the FSDT decision space? Part7s, 7r, Δ=207s, 7r, Δ=8024s, 7r, Δ=207s, 2r, Δ=20 1 EV, A z =.880 d’=1.658 UEV, A z =.911 d a =1.905, b=.341 EV, A z =.929 d’=2.074 EV A z =.794 d’= UEV A z =.788 d a =1.131 b=.744 EV A z =.929 d’=2.078 EV A z =.929 d’=2.073 EV A z =.719 d’ =.82 3 EV A z =.814 d’=1.261 UEV A z =.867 d a =1.576 b=.170 EV A z =.916 d’=1.95* 4 EV, A z =.854 d’=1.491 EV A z =.940 d’=2.198 EV A z =.94 d’=2.200 N A z =.737 d a =.895 b=1.025 Note. EV=equal variance model; UEV=Unequal variance model; N=Neither model fit (A z value is for the UEV model); * Participant never used the prescribed binary categories in the unbiased condition Comparison of Stimulus and Response Range Manipulations Present Experiment: Method Task: Discriminate durations of a 6 by 6 cm light gray square on a gray background Judge the degree to which stimuli were ‘longer’ vs. ‘shorter’ Four Conditions (See Table 1) : 7 stimulus categories response categories (7s7r) Δ=20 7 stimulus categories response categories (7s2r) Δ=20 25 stimulus categories response categories (25s7r) Δ=20 7 stimulus categories response categories (7s7r) Δ=80 Response bias manipulated using payoff matrix 700 trials per condition  Previous experiments have shown that FSDT conforms to the normality assumption  Data regarding equal variance assumption varied according to intercategory intervals  Intercategory intervals confounded with stimulus range (i.e., number of stimulus categories). Table 1. Fuzzy Stimulus and Response: Duration Discrimination ConditionDuration Categories (7s7r) Δ=20 msec (7s2r) Δ=20 msec (25s7r) Δ=20 msec Δ=20 msec intervals680 (7s7r) Δ=80 msec Table 2. Results of FSDT Analysis Results