Judgments of average magnitude: Analyses in terms of the functional measurement and two-stage models

Subjects made magnitude estimates of the average loudness of pairs of 1,000-Hz tones varying in sound pressure. A test of fit of an averaging model employing an analysis of variance suggested that the judgments were internally consistent. However, estimates of the parameters of a two-stage model based on the assumption that power transformations were imposed in both input and output implied a nonlinear output function, inconsistent with the averaging model. Additional analyses employing a nonmetric scaling solution also suggested that output was nonlinear, indicating that this implication was not an artifact of the strong assumptions of the two-stage model. Large differences were found among the output functions of individual subjects, and it was suggested that these may have inflated the error term in the analysis of variance, reducing its power to detect violations of the additive model. Similar analyses were performed on data from judgments of average grayness collected by Weiss (1972).