Scaling

Scaling methods used in the social sciences typically are statistical procedures that take observed data and extract latent (i.e., abstract) dimensions on which the objects or subjects are ordered. In political science, the primary focus has been on issue/public policy and/or ideological scales using individuals' judgments or observed voting behavior. One important example is the estimation of ideological dimensions for all of congressional history by Keith Poole and Howard Rosenthal. Below, the development of these techniques and their applications in political science are discussed.

Scaling methods used in political science for the most part have their origins in psychology. The original work on scaling in psychology done at the turn of the 20th century was aimed at measuring general ability or intelligence. The pioneers were Karl Pearson and Charles Spearman. Spearman in 1904 used factor analysis to analyze a correlation matrix between test scores of 22 English high school boys for Classics, French, English, Math, Pitch, and Music. Spearman computed a form of rank-order correlation between each pair of skills across the 22 school boys and then extracted a common or general factor (g factor) from the matrix. His method of computing pairwise correlations and calculating the g factor was quickly supplanted by later work.

In particular, Pearson invented the product–moment correlation coefficient that is universally denoted as r, and he should also be credited with the invention of principal components analysis (what we now would think of as straightforward eigenvalue/eigenvector decomposition). Pearson in 1901 called it “the method of principal axes” and stated the problem quite succinctly: “In many physical, statistical, and biological investigations it is desirable to represent a system of points in plane, three, or higher dimensioned space by the ‘best-fitting’ straight line or plane” (p. 559). Remarkably, this also describes the essence of the famous Eckart-Young theorem (1936), which is the foundation of general least squares problems.

Lewis Leon Thurstone thought that Spearman's one-factor theory of intelligence was wrong, and he succeeded in developing a method for extracting multiple factors from a correlation matrix. Thurstone's theory of intelligence postulated seven rather than one primary mental ability, and he constructed tests specific to the seven abilities: verbal comprehension, word fluency, number facility, spatial visualization, associative memory, perceptual speed, and reasoning. Thurstone also developed the law of comparative judgment. Thurstone's law is [Page 2355]more accurately described as a measurement model for a unidimensional subjective continuum. Subjects are asked to make a series of n(n − 1)/2 pairwise comparisons of n stimuli. It is assumed that a subject's response reflects the momentary subjective value associated with the stimulus and that the probability distribution of these momentary values is normally distributed. It is then possible to recover the underlying continuum or scale by essentially averaging across a group of subjects. If the variances of the stimuli (the discriminal dispersions) on the underlying scale are the same (Case 5 of the model), the requirement of the parallel-item characteristic curves in the Rasch model is satisfied. Case 5 of Thurstone's method should yield essentially the same results as the Rasch model for dichotomous data.

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