Polynomial response scaling and functional measurement

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.

Section: Case Of Ordinal Respopsementioning

confidence: 99%

Note on functional measurement and data analysis

Anderson

1977

“…In this case, the assumption of Eq. 10 and the use of the Bogartz and Wackwitz (1971) technique would lead to a scaling of the psychological values of the grades.…”

Section: The Frequency Pn'nciplementioning

confidence: 99%

Using contextual effects to derive psychophysical scales

Birnbaum

1974

207

This paper presents a functional measurement analysis of Parducci's range-frequency theory. The theory provides a basis for (1)finding context-invariant psychophysical scales, (2)establishing the validity of rating scales, and (3) explaining contextual effects in judgment. In an experimental illustration, Ss judged the magnitude of numerals in nine different distributions. All data were used to obtain the psychophysical function for numerals and also to test the model. The large contextual effects were consistent with the model. Quantitative tests of fit supported the equal-interval assumption of the category scale. The psychophysical function appeared nearly linear but with a significant negative acceleration. The data were shown to be qualitatively inconsistent with generalizations of Helson's theory of adaptation level and Johnson's correlation-regression theory. They supported Parducci's range-frequency theory and illustrated how it could be used to factor out contextual effects from effects of stimulus magnitude. Extensions of the range-frequency approach are also discussed.

“…Generally, functional measurement discriminates between model and response scale discrepancies in the following manner: If there is no reason to doubt the validity of the response scale, the test of fit provides the basis for rejecting the model; if there is reason to believe that the response scale is nonlinear (e.g., when the response is on the physical continuum, as in the method of bisection), a search is made for a monotone transformation that will fit the model (Anderson, 1962(Anderson, , 1970Bogartz & Wackwitz, 1971). The decision to rescale the data or to reject the model is based upon subtle theoretical considerations such as the nature of the necessary transformation, the history of previous experiments with the model and response mode, signs of end or ceiling effects, and their dependence upon variations in experimental procedures.…”

Section: Scale Convergence Criterionmentioning

confidence: 99%

Scale convergence as a criterion for rescaling: Information integration with difference, ratio, and averaging tasks

Birnbaum

Veit²

1974

187

Ss lifted pairs of weights simultaneously. one in each hand, and judged either the difference, ratio. or average heaviness of the two weights. Data for the difference and ratio tasks were in general agreement with subtractive and ratio models. but the averaging data showed discrepancies from the constant-weight averaging model similar to those reported in previous psychophysical research. Rescaling was ruled out for the averaging data, because responses to pairs of equal weight were a linear function of subtractive model scale values derived from the difference task data. Scale values for the ratio and difference task data were related exponentially. as were the responses to the pairs, consistent with Torgerson's conjecture that Ss do not distinguish "differences" from "ratios." They appear to use the same composition rule but different output functions, depending on the procedures for responding. The scale convergence criterion can thus prevent inappropriate rescaling when a model fails and can dictate rescaling even when a model fits.