Abstract
The guiding idea of functional measurement is that measurement theory and substantive theory form an organic unity. Psychological scales are inherent in the statement of quantitative psychological laws, and these laws themselves form the base and frame for psychological measurement. Valid scales thus depend on empirically valid laws. But establishing empirical validity of any law requires appropriate data analysis. Several statistical problems are discussed with respect to simple algebraic laws. To illustrate the necessity for proper tests of goodness of fit for algebraic models, five sets of experimental data are reanalyzed. In each case, the factorial plot and the analysis of variance showed that the data were nonadditive. Nevertheless, an additive model was fit to the data. The correlations between the data and the predictions from the additive model were extremely high, ranging from .964 to .9997. The corresponding observed-predicted scatterplots also gave little sign of the deviations from additivity. These correlation-scatterplot analyses conceal and obscure what the factorial plot and the analysis of variance reveal and make clear. Other topics discussed are accepting and rejecting the null hypothesis, the use of nonmetric smoothing for parameter estimation, and problems of stimulus-response-model generality. An extension of functional measurement is suggested for a practicable error theory for nonmetric analysis.
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Anderson, N. H.,Methods for studying information integration (Technical Report CHIP 43). La Jolla, Calif: Center for Human Information Processing. University of California, San Diego, June 1974.
Anderson, N. H.,Algebraic models for information integration (Technical Report CHIP 45). La Jolla, Calif: Center for Human Information Processing, University of California, San Diego, June 1974.
Anderson, N. H.,Social perception and cognition (Technical Report CHIP 62). La Jolla, Calif: Center for Human Information Processing, University of California, San Diego, July 1976.
References
Anderson, N. H. Applications of an additive model to impression formation.Science, 1962,138, 817–818. (a)
Anderson, N. H. On the quantification of Miller’s conflict theory.Psychological Review, 1962.69, 400–414. (b)
Anderson, N. H. Averaging versus adding as a stimulus-combination rule in impression formation.Journal of Experimental Psychology, 1965,70, 394–400.
Anderson, N. H. Functional measurement and psychophysical judgment.Psychological Review, 1970,77, 153–170.
Anderson, N. H. Test of adaptation-level theory as an explanation of a recency effect in psychophysical integration.Journal of Experimental Psychology, 1971,87, 57–63.
Anderson, N. H. Cross-task validation of functional measurement.Perception & Psychophysics, 1972,12, 389–395.
Anderson, N. H. Algebraic models in perception. In E. C. Carterette & M. P. Friedman (Eds.),Handbook of perception (Vol. 2). New York: Academic Press, 1974. (a)
Anderson, N. H. Cognitive algebra. In L. Berkowitz (Ed.),Advances in experimental social psychology (Vol. 7). New York: Academic Press, 1974. (b)
Anderson, N. H. Cross-task validation of functional measurement using judgments of total magnitude.Journal of Experimental Psychology, 1974,1M, 226–233. (c)
Anderson, N. H. Information integration theory: A brief survey. In D. H. Krantz, R. C. Atkinson, R. D. Luce, & P. Suppes (Eds.),Contemporary developments in mathematical psychology (Vol. 2). San Francisco: Freeman, 1974. (d)
Anderson, N. H. On the role of context effects in psychophysical judgment.Psychological Review, 1975,82, 462–482.
Anderson, N. H. Howfunctional measurement can yield validated interval scales of mental quantities.Journal of Applied Psychology, 1976,61, 677–692. (a)
Anderson, N. H. Integration theory, functional measurement, and the psychophysical law. In H. G. Geissler & Yu. M. Zabrodin (Eds.),Advances in psychophysics, Berlin: VEB Deutscher Verlag, 1976. (b).
Anderson, N. H. Failure of additivity in bisection of length.Perception & Psychophysics, 1977, in press.
Anderson, N. H. &Farkas, A. J. Integration theory applied to models of inequity.Personality and Social Psychology Bulletin, 1975,1, 588–591.
Anderson, N. H., & Shanteau, J. Weak inference with linear models.Psychological Bulletin, 1977, in press.
Binder, A. Further considerations on testing the null hypothesis and the strategy and tactics of investigating theoretical models.Psychological Review, 1963,70, 107–115.
Birnbaum, M. H., &Veit, C. T. Psychophysical measurement: Information integration with difference, ratio, and averaging tasks.Perception & Psychophysics, 1974,15, 7–15.
Bogartz, R. S., &Wackwitz, J. H. Polynomial response scaling and functional measurement.Journal of Mathematical Psychology. 1971,8, 418–443.
Bush, R. R. Estimation and evaluation. In R. D. Luce, R. R. Bush & E. Galanter (Eds.)Handbook of mathematical psychology (Vol.). New York: Wiley, 1963.
Cliff, N. Scaling.Annual Review of Psychology, 1973.24. 473–506.
Curtis, D. W., &Mullin, L. C. Judgments of average magnitude: Analyses in terms of the functional measurement and two-stage models.Perception & Psychophysics, 1975,18, 299–308.
De Leeuw, J., Young, F. W., &Takane, Y. Additive structure in qualitative data: An alternating least squares method with optimal scaling features.Psychometrlka, 1976,41, 471–503.
Durlach, N. I. & Colburn, H. S. Binaural phenomena. In E. C. Carterette & M. P. Friedman (Eds.),Handbook of perception (Vol, 4). New York: Academic Press, in press.
Garner, W. R. Hake, H. W. &Eriksen, C. W. Operationism and the concept of perception.Psychological Review, 1956.63. 149–159.
Graesser, C. C., &Anderson, N. H. Cognitive algebra of the equation: Giftsize = generosity x income.Journal of Experimental Psychology, 1974.103, 692–699.
Grant, D. A. Testing the null hypothesis and the strategy and tactics of investigating theoretical models.Psychological Review, 1962.69, 54–61
Krantz, D. H. Measurement structures and psychological laws.Science, 1972.175, 1427–1435.
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of the data.Journal of the Royal Statistical Society [B], 1965.27, 251–263.
Leon, M., &Anderson, N. H. A ratio rule from integration theory applied to inference judgments.Journal of Experimental Psychology, 1974.102, 27–36.
Marks, L. E.Sensory processes, NewYork: Academic Press, 1974.
Norman, K. L. A solution for weights and scale values in functional measurement.Psychological Review, 1976,83, 80–84.
Oden, G. C.Semantic constraints and ambiguity resolution. Unpublished doctoral dissertation, University of California. San Diego, 1974.
Parducci, A. Thaler, L., &Anderson, N. H. Stimulus averaging and the context for judgment.Perception & Psychophysics. 1968.3, 145–150.
Restle, F.Mathematical models in psychology: An introduction. Baltimore: Penguin Books, 1971
Shanteau, J. Component processes in risky decision making.Journal of Experimental Psychology, 1974.103, 680–691
Shanteau, J. C. &Anderson, N. H. Integration theory applied to judgments of the value of information.Journal of Experimental Psychology, 1972,92, 266–275.
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. II.Psychometrika. 1962.27, 219–246.
Shepard, R. N. Metric structures in ordinal data.Journal of Mathematical Psychology, 1966,3, 287–315.
Sternberg, S. The discovery of processing stages: Extensions of Donders’ method.Acta Psychologica, 1969.30, 276–315.
Stevens, S. S. Issues in psychophysical measurement.Psychological Review, 1971,78, 426–450.
Stevens, S. S. Perceptual magnitude and its measurement. In E. C. Carterette & M. P. Friedman (Eds.),Handbook of perception (Vol. 2). New York: Academic Press, 1974.
Torgerson, W. S. Distances and ratios in psychological scaling.Acta Psychologica, 1961,19, 201–205.
Weiss, D. J. Averaging: An empirical validity criterion for magnitude estimation.Perception & Psychophysics, 1972,12, 385–388.
Weiss, D. J. Funpot: A Fortran program for finding a polynomial transformation to reduce any sources of variance in a factorial design.Behavioral Science, 1973.18, 150.
Weiss, D. J. Quantifying private events: A functional measurement analysis of equisection.Perception & Psychophysics, 1975.17. 351–357.
Weiss, D. J.. &Anderson, N. H. Use of rank order data in functional measurement.Psychological Bulletin, 1972.78, 64–69.
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Preparation of this paper was supported by National Science Foundation Grant BMS 74-19124and by grants from the National Institute of Mental Health to the Center for Human Information Processing, University of California, San Diego.
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Anderson, N.H. Note on functional measurement and data analysis. Perception & Psychophysics 21, 201–215 (1977). https://doi.org/10.3758/BF03214231
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DOI: https://doi.org/10.3758/BF03214231