1. Psy 5037. Psychology of Hearing Psychophysics and Detection Theory Neal Viemeister nfv@umn.edu

2. Psychophysics -Set of tools & theories for measuring & describing perception. Objective vs. subjective psychophysics. Classical psychophysics Gustav Theodore Fechner: Elemente der Psychophysik (1860) (See E. G. Boring, Sensation and perception in the history of experimental psychology. New York: Appleton-Century-Crofts. (1942)

3. Contributions: --Formalization of notion of thresholds: Absolute thresholds (masked thresholds) & difference thresholds (DL, JND) --“Classical” Psychophysical Methods Method of limits Method of Adjustment Method of Constant Stimuli. Psychometric function

4. Psychometric function: Measure of performance (e.g., percent correct, d’) vs. physical stimulus dimension. (Also neurometric function) Issue of “catch trials”. (objective vs. subjective psychophysics) False positives (false alarms) indicate S must be guessing (below threshold). “Correction for guessing” Criterion problem. (Response bias). Issue of sensory threshold Contributions (cont’d) Fundamental formula (Weber’s Law), Measurement formula (Fechner’s Law) From translation of Elemente der Psychophysik (1860): “According to the empirical Weber's law, dg remains constant when db/b remains constant, no matter what absolute values db and b take; and according to the a priori mathematical auxiliary principle the changes dg and db remain proportional to one another so long as they remain very small. The two relations may be expressed together in the following equation: dg = Kdb/b (1) where k [1] is a constant (dependent upon the units selected for g and b). In fact, if one multiplies bd and b by any number, so long as it is the same number for both, the proportion remains constant, and with it also the sensation difference dg. This is Weber's law. If one doubles or triples the value of the variation db without changing the initial value b, then the value of the change dg is also doubled or tripled. This is the mathematical principle. The equation dg = Kdb/b therefore entirely satisfies both Weber's law and this principle; and no other equation satisfies both together. This is to be called the fundamental formula, in that the deduction of all consequent formulas will be based upon it.”1

6. And then: “According to the rule, that the logarithm of a quotient of two numbers may be substituted for the difference of their logarithms,... one can substitute for the above form of the measurement formula the following, which is more convenient for making deductions. g = k(log b/b) (3) From this equation it follows that the sensation magnitude g is not to be considered as a simple function of the stimulus value b, but of its relation to the threshold value b, where the sensation begins and disappears. This relative stimulus value, b/b is for the future to be called the fundamental stimulus value, or the fundamental value of the stimulus. Translated in words, the measurement formula reads: The magnitude of the sensation (g) is not proportional to the absolute value of the stimulus (b), but rather to the logarithm of the magnitude of the stimulus, when this last is expressed in terms of its threshold value(b), i.e. that magnitude considered as unit at which the sensation begins and disappears. In short, it is proportional to the logarithm of the fundamental stimulus value.” [Weber’s Law: ΔI/I= k where ΔI is the JND. Fechner’s Law: S= K log (I/I 0 ) ] Historical importance. Measurement of sensation. Mathematical psychology. Psychophysical Scaling S.S. Stevens (P.A.L. at Harvard) --Ratio scale of sensation --Magnitude estimation/production Stevens power law: S= a I b Transducer function (?). Validity Issue: range effects & other subjective effects

7. Detection Theory A.k.a: Signal Detection Theory (SDT), Theory of Signal Detectability (TSD), Tanner, Swets, and Green (TSG). [Psy 5970 “Detection Theory, Spring semester] SDT is a theoretical framework that uses concepts from communication theory, statistics, and decision theory. Historical: application to psychophysics at U. Mich—Tanner, Swets, Green. Antecedant in Thurstone’s Law of Comparative judgement. Misconceptions: -not thresholds vs. SDT -not normal-normal, equal variance distributions (d’) General conception

8. Fundamental detection task On trial: --Observer makes observation (may be multidimensional) --Observation is mapped onto “decision axis”, i.e. a value of the decision variable (D) is computed. --D is compared to the criterion value of D (D c ) and responds “yes” if D>D c otherwise responds “no” Analysis

9. Measures: p hit = p(Y,sn) = p(Y|sn)p(sn) p-est hit = (#hits/ #trials) “hit rate”= p(Y|sn) (#hit/ # sn trials) “false alarm rate”= p(Y|n) Percent correct= P(C) = p(Y,sn) + p(N,n) = p(Y|sn)p(sn) + [1 – p(Y|n)]p(n) [P(C), p(Y|sn), p(Y|n) reflect both sensory and decision making factors] Describing behavior: Receiver Operating Characteristic (ROC)

10. ROC “isosensitivity curve” -form of ROC determined by input processing (form of underlying distributions of D) -points on ROC determined both by sensitivity (input processing) and criterial (non-sensory) factors -should obtain ROC to evaluate performance. Usually not necessary: empirical ROCs have same general form. Gaussian, equal variance underlying distributions. Single parameter, d’, completely determines ROC of this family. Gaussian ROC’s, d’, etc.

11. --Other SDT based measures, e.g. P(A) --Current trend: 2IFC, 3IFC tracking procedures z-score coordinates

12. Theoretical Contributions of SDT --Alternative to threshold theory. (Discrete vs. continuous decision variables) -- Identification & recognition tasks. --Ideal Observer. Decision variable based on likelihood ratio is (generally) optimal for achieving wide variety of decision goals. --degraded Ideal --Ideal neural observer --internal noise