## Elements of Psychophysical TheoryThis book presents the basic concepts of classical psychophysics, derived from Gustav Fechner, as seen from the perspective of modern measurement theory. The theoretical discussion is elucidated with examples and numerous problems, and solutions to one-quarter of the problems are provided in the text. |

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### Contents

Ordinal Measurement | 13 |

Binary Relations | 14 |

Equivalence Relations Partitions Functions | 18 |

Algebraic Theory Weak Orders | 23 |

Biorders | 30 |

Complements | 38 |

Exercises | 46 |

Extensive Measurement | 50 |

Alternatives to Webers Law | 210 |

Inequalities | 215 |

Fechners Problem Revisited | 217 |

Exercises | 219 |

Psychophysical Methods | 221 |

Adaptive Methods | 223 |

Reference Notes | 231 |

Exercises | 232 |

Construction of a Physical Scale for Length | 51 |

Axioms for Extensive Measurement | 58 |

Representation Theorem | 60 |

Other Empirical Examples | 62 |

Complements and Proofs | 65 |

Reference Notes Further Developments | 71 |

Exercises | 72 |

Functional Equations | 76 |

Cauchy and Related Equations | 81 |

Plateaus Experiment | 90 |

Normal Distribution of Sensory Variables | 92 |

A Functional Inequality | 95 |

Sincov Equations | 97 |

Additive Systems | 101 |

Two Proofs | 103 |

Exercises | 108 |

THEORY | 111 |

Fechners Psychophysics | 113 |

Construction of a Fechnerian Scale | 114 |

Fechners Problem | 118 |

Psychophysical Discrimination Systems | 119 |

Some Necessary Conditions | 122 |

Proofs | 123 |

Reference Notes | 130 |

Models of Discrimination | 133 |

Random Variable Models | 134 |

Thurstones Law of Comparative Judgments | 136 |

Extreme Value Distributions and the Logistic Model | 139 |

BradleyTerryLuce Representation | 143 |

A Model Inconsistent with a Fechnerian Representation | 146 |

Statistical Issues | 147 |

Selected Families | 148 |

Exercises | 149 |

Psychometric Functions | 151 |

Psychometric Families | 156 |

Parallel Psychometric Families | 158 |

Subtractive Families | 161 |

Necessary Conditions for the Existence of a Subtractive Representation | 166 |

Symmetric Families | 167 |

Reference Notes | 170 |

Further Topics on Psychometric Functions | 172 |

Redefining Psychometric Families | 174 |

Ordering the Backgrounds | 180 |

Homomorphic Families | 185 |

Representation and Uniqueness Theorems for Subtractive Families | 191 |

Random Variables Representation | 196 |

Exercises | 197 |

Sensitivity FunctionsWebers Law | 199 |

Sensitivity Functions Weber Functions | 201 |

Linear Psychometric Families Webers Law | 203 |

Signal Detection Theory | 233 |

ROC Graphs and Curves | 234 |

A Random Variable Model for ROC Curves | 236 |

ROC Analysis and Likelihood Ratios | 239 |

ROC Analysis and the Forced Choice Paradigm | 244 |

ROC Analysis of Rating Scale Data | 247 |

The Gaussian Assumption | 249 |

The Threshold Theory | 251 |

Rating Data and the Threshold Theory | 254 |

A General Signal Detection Model | 256 |

Reference Notes | 258 |

Exercises | 259 |

Psychophysics with Several Variables or Channels | 260 |

Probability Summation | 263 |

Two Additive Pooling Rules | 268 |

Additive Conjoint MeasurementThe Algebraic Model | 270 |

Random Additive Conjoint Measurement | 272 |

Probabilistic Conjoint Measurement | 277 |

Bisection | 279 |

Proofs | 281 |

Exercises | 282 |

Homogeneity Laws | 284 |

The Conjoint Webers LawsOutline | 285 |

The Conjoint Webers LawResults | 287 |

The Strong Conjoint Webers Laws | 293 |

The Conjoint Webers Inequality | 298 |

Shift Invariance in Loudness Recruitment | 299 |

Exercises | 303 |

Scaling and the Measurement of Sensation | 305 |

Unidimensional Scaling Methods | 306 |

The KrantzShepard Theory | 311 |

Functional Measurement | 315 |

The Measurement of SensationSources of the Controversy | 317 |

Two Positions Concerning the Scaling of Sensory Magnitudes | 322 |

Why a Psychophysical Scale? | 323 |

Exercises | 324 |

Meaningful Psychophysical Laws | 325 |

Examples | 327 |

Scale Families | 330 |

Meaningful Families of Numerical Codes | 331 |

Isotone and Dimensionally Invariant Families of Numerical Codes | 336 |

An Application in Psychoacoustics | 341 |

Why Meaningful Laws? | 345 |

Exercises | 351 |

353 | |

Answers or Hints to Selected Exercises | 366 |

Author Index | 377 |

381 | |

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### Common terms and phrases

a,b e argument assume assumption Axiom background balanced discrimination binary relation biorder Chapter choice probabilities concatenation conjoint Weber's law considered constant continuous function defined Definition 2.6 denote detection dimensionally invariant discrimination family discrimination system discussed empirical equivalence relation equivalent estimated example Exercise exists experimental fact Falmagne Fechner's function F functional equation holds homomorphic implies independent inequality intensity isotone Krantz Lemma Luce mapping meaningful method metric space noise notation Notice numerical codes obtain open interval open set pa(x pair paradigm parameters particular payoff matrix procedure proof of Theorem Prove psychometric family psychometric functions psychophysicists quadruple condition random representation random variables ratio scale reader real-valued function relation result ROC curve S. S. Stevens satisfying sensation sensitivity function stimulus strictly increasing function subset subtractive Suppose theory trial weak bicancellation weak order Weber function yields

### Popular passages

Page 7 - The aim of natural science," says Mach, "is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time."3 This phenomenalistic conception, as it is called, was already familiar to Goethe. In his posthumous 'Maxims and Reflections...

Page 356 - Falmagne, J.-C. (1978). A representation theorem for finite random scale systems. Journal of Mathematical Psychology, 18, 52-72. Falmagne, J.-C.