Representational Measurement Theory

This chapter focuses on empirical structures, with an ordering attribute, leading to numerical or geometric representations. A historical sketch of physical, behavioral, and social science measurement until 1950 is followed by modern approaches to representational measurement in the behavioral sciences. Among these structures are: difference, additive conjoint, averaging, and non-additive concatenation and conjoint structures. Derived measurement is described by distributive linkages between two structures with a common attribute and is exemplified by physical, utility, and psychophysical examples. Two mathematical results are needed in each case: the existence of structure-preserving numerical representations and their uniqueness. The latter has been analyzed generically. Using these ideas, models of magnitude estimation and production are summarized. The nature of axiomatization is discussed in terms of types of logical languages, the role of certain second-order axioms, and issues of consistency and independence of axioms. Important results about the impossibility of simple axiomatizations of classes of finite structures are described along with mention of some problems. Keywords: axiomatizability; derived measurement; error; invariance; meaningfulness; representational measurement; scale type

[1]  J. C. Falmagne,et al.  Scales and meaningfulness of quantitative laws , 1983, Synthese.

[2]  Fred S. Roberts,et al.  Axiomatic thermodynamics and extensive measurement , 1968, Synthese.

[3]  C. T. Ng,et al.  Functional equations arising in a theory of rank dependence and homogeneous joint receipts , 2003 .

[4]  R. Duncan Luce,et al.  A psychophysical theory of intensity proportions, joint presentations, and matches. , 2002 .

[5]  L. Narens.  Theories of Meaningfulness , 2001 .

[6]  Ehtibar N. Dzhafarov,et al.  Multidimensional Fechnerian Scaling: Basics. , 2001, Journal of mathematical psychology.

[7]  W. Ellermeier,et al.  Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production , 2000, Perception & psychophysics.

[8]  R. Luce.  Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches , 2000 .

[9]  J. Michell.  Measurement in psychology: A critical history of a methodological concept. , 1999 .

[10]  H. Colonius,et al.  Fechnerian metrics in unidimensional and multidimensional stimulus spaces , 1999, Psychonomic bulletin & review.

[11]  Suck.  A Qualitative Characterization of the Exponential Distribution. , 1998, Journal of mathematical psychology.

[12]  Jean-Paul Doignon,et al.  Knowledge spaces , 1998 .

[13]  Jean-Paul Doignon,et al.  An Approval-Voting Polytope for Linear Orders , 1997, Journal of mathematical psychology.

[14]  Regenwetter.  Random Utility Representations of Finite m-ary Relations , 1996, Journal of mathematical psychology.

[15]  Michel Regenwetter,et al.  A random utility model for approval voting , 1996 .

[16]  Louis Narens,et al.  A theory of ratio magnitude estimation , 1996 .

[17]  J. Quiggin.  Generalized expected utility theory : the rank-dependent model , 1994 .

[18]  P. Suppes,et al.  The definability of the qualitative independence of events in terms of extended indicator functions , 1994 .

[19]  L. Thurstone.  A law of comparative judgment. , 1994 .

[20]  A. A. J. Marley,et al.  Aggregation Theorems and the Combination of Probabilistic Rank Orders , 1993 .

[21]  W. Edwards.  Utility Theories: Measurements and Applications , 1992 .

[22]  R. Luce.  Singular points in generalized concatenation structures that otherwise are homogeneous , 1992 .

[23]  Neil A. Macmillan,et al.  Detection Theory: A User's Guide , 1991 .

[24]  P. Fishburn,et al.  Rank- and sign-dependent linear utility models for finite first-order gambles , 1991 .

[25]  J. Michell.  An introduction to the logic of psychological measurement , 1995 .

[26]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[27]  R. Duncan Luce,et al.  On the possible psychophysical laws revisited: remarks on cross-modal matching , 1990 .

[28]  Peter J. Cameron,et al.  Groups of order-automorphisms of the rationals with prescribed scale type , 1989 .

[29]  Peter P. Wakker,et al.  Additive Representations of Preferences , 1989 .

[30]  P. Wakker.  Additive Representations of Preferences: A New Foundation of Decision Analysis , 1988 .

[31]  Peter C. Fishburn,et al.  Nonlinear preference and utility theory , 1988 .

[32]  Theodore M. Alper,et al.  A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness , 1987 .

[33]  Patrick Suppes,et al.  Propensity representations of probability , 1987 .

[34]  T. Ll.  Psychophysical analysis. By L. L. Thurstone, 1927. , 1987 .

[35]  J. Leeuw,et al.  Abstract Measurement Theory. , 1986 .

[36]  Jean-Claude Falmagne,et al.  Statistical issues in measurement , 1985 .

[37]  J. Falmagne.  Elements of psychophysical theory , 1985 .

[38]  D. O'brien.  Plato, weight and sensation : the two theories of the 'Timaeus' , 1984 .

[39]  Jaap Van Brakel,et al.  Foundations of measurement , 1983 .

[40]  Louis Narens,et al.  On the scales of measurement , 1981 .

[41]  Roger N. Shepard,et al.  Psychological relations and psychophysical scales: On the status of “direct” psychophysical measurement ☆ , 1981 .

[42]  Louis Narens,et al.  A general theory of ratio scalability with remarks about the measurement-theoretic concept of meaningfulness , 1981 .

[43]  D. O'brien.  Theories of Weight in the Ancient World: Four Essays on Democritus, Plato and Aristotle. A Study in the Development of Ideas. 1. Democritus: Weight and Size. An Exercise in the Reconstruction of Early Greek Philosophy , 1981 .

[44]  Robert J. Titiev.  A locus problem involving the three-dimensional city-block metric , 1980 .

[45]  P. Suppes.  LIMITATIONS OF THE AXIOMATIC METHOD IN ANCIENT GREEK MATHEMATICAL SCIENCES , 1980 .

[46]  Louis Narens,et al.  Fundamental unit structures: A theory of ratio scalability☆ , 1979 .

[47]  Jean-Claude Falmagne,et al.  Conjoint Weber laws and additivity , 1979 .

[48]  G. Iverson,et al.  Binaural "loudness" summation: probabilistic theory and data. , 1979, Psychological review.

[49]  Jean-Claude Falmagne,et al.  A representation theorem for finite random scale systems , 1978 .

[50]  J. Falmagne.  Random Conjoint Measurement and Loudness Summation. , 1976 .

[51]  S. S. Stevens,et al.  Psychophysics: Introduction to Its Perceptual, Neural and Social Prospects , 1975 .

[52]  P. Fishburn.  Binary choice probabilities: on the varieties of stochastic transitivity , 1973 .

[53]  Robert J. Titiev,et al.  Measurement structures in classes that are not universally axiomatizable , 1972 .

[54]  D. Krantz.  A theory of magnitude estimation and cross-modality matching☆ , 1972 .

[55]  David H. Krantz,et al.  Chapter 3 – Extensive Measurement , 1971 .

[56]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[57]  J. Pfanzagl.  Theory of measurement , 1970 .

[58]  Gustav Theodor Fechner,et al.  Elements of psychophysics , 1966 .

[59]  D. M. Green,et al.  Signal detection theory and psychophysics , 1966 .

[60]  R. Luce,et al.  Simultaneous conjoint measurement: A new type of fundamental measurement , 1964 .

[61]  W. W. Rozeboom,et al.  The untenability of Luce's principle. , 1962, Psychological review.

[62]  E. A. Moody,et al.  The medieval science of weights (scientia de ponderibus) : treatises ascribed to Euclid, Archimedes, Thabit ibn Qurra, Jordanus de Nemore and Blasius of Parma , 1960 .

[63]  Johann Pfanzag.  A general theory of measurement applications to utility , 1959 .

[64]  P. Suppes,et al.  Decision making : an experimental approach , 1959 .

[65]  H. D. Block,et al.  Random Orderings and Stochastic Theories of Responses (1960) , 1959 .

[66]  R. Luce,et al.  On the possible psychophysical laws. , 1959, Psychological review.

[67]  G. Debreu.  Topological Methods in Cardinal Utility Theory , 1959 .

[68]  E. Boring,et al.  The derivation of subjective scales from just noticeable differences. , 1958, Psychological review.

[69]  Patrick Suppes,et al.  Foundational aspects of theories of measurement , 1958, Journal of Symbolic Logic.

[70]  Patrick Suppes,et al.  Introduction To Logic , 1958 .

[71]  R. Luce.  Semiorders and a Theory of Utility Discrimination , 1956 .

[72]  Patrick Suppes,et al.  Outlines of a Formal Theory of Value, I , 1955, Philosophy of Science.

[73]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[74]  L L THURSTONE,et al.  The measurement of values. , 1960, Psychological review.

[75]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[76]  S. S. Stevens.  Mathematics, measurement, and psychophysics. , 1951 .

[77]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[78]  H. Helmholtz,et al.  Counting and measuring , 1930 .

[79]  L. L. Thurstone,et al.  Three psychophysical laws. , 1927 .

[80]  N. Wiener.  A New Theory of Measurement: A Study in the Logic of Mathematics , 1921 .

[81]  W. F. King,et al.  The Theory of Least Squares , 1907 .

[82]  Felix C. Klein,et al.  A comparative review of recent researches in geometry , 1893, 0807.3161.

[83]  J. J. Fourier,et al.  The Analytical Theory of Heat , 1878, Nature.

[84]  J. Maxwell.  A Treatise on Electricity and Magnetism , 1873, Nature.