Note on functional measurement and data analysis

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.

Comments on current ratio-setting models for time perception

“…Curtis and Rule (1977) analyzed their data in terms of a two-stage model for psychophysical judgments which has proved extremely useful in other situations (Curtis, Attneave, & Harrington, 1968;Curtis & Mullin, 1975). Blankenship and Anderson (1976) made use of the functional measurement approach (Anderson, 1977) and analyzed their data in terms of a weighted sum integration model. This paper is concerned with the analysis and interpretation of ratio-setting data.…”

Distributional versus error-filled procedures for transformation

“…Birnbaum and Veit concluded that if difference judgments are represented either by a subtractive or ratio model, the additive or parallel-averaging models must be rejected for the size-weight illusion. Anderson's (1977Anderson's ( , 1981Anderson's ( , 1983 reviews of this topic reached conclusions (based on scale-dependent research) different from those of Birnbaum and Veit (1974), although he did not cite their study. Thus, despite a similarity of purpose and terminology, Anderson's (1983) two-operation logic should not be confused with the scalefree test as used by Birnbaum (1974) and by Birnbaum and Veit (1974).…”

Failure of additivity in bisection of length

“…But these correlation-scatterplot statistics are deceiving; they seem to test the model but they do not really do so. This example is but one of many that tell the same story (Anderson, 1976a(Anderson, , 1977Anderson & Shanteau, 1977;Birnbaum, 1973).…”

“…Significant discrepancies need not require rejection of a model (Anderson, 1977), for they could reflect the approximate character of some simplifying assumption. On the other hand, they could also signal inadequacies in the basic assum ptions,…”