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References
Clyde Coombs,A Theory of Data (New York: Wiley, 1964), pp. 80–95, 193–98.
A good description of Guttman scaling can be found in Lee F. Anderson, Meredith W. Watts, Jr., and Allen R. Wilcox,Legislative Roll-Call Analysis (Evanston: Northwestern University Press, 1966), pp. 89–121. The original source is found in Samuel A. Stouffer,et al., Measurement and Prediction (Princeton: Princeton University Press, 1950).
This result holds equally in scalogram analysis of multi-category items. Multi-category items involve a “pick 1/n” response task in which the individual picks that response closest to her own opinion. Coombs (1964, pp. 229–236) illustrates scalogram analysis of such items without prior dichotomization. The questions he uses as an example are about perceptions of semifactual information, but we can change the questions so they refer to preferences. The responses in his Table 11.4 (or 11.5) are perfectly scalable. Yet we can easily show that individuals may not have single peaked preferences and that their ideal points need not be at their scale scores. For example, take an individual with response number 8. His responses are 3c, 2c, and 1d, with the alternatives or corresponding midpoints occurring in that order on the continua. But there is nothing that prevents this individual from most preferring 2c, then 3c, and then 1d. Such a preference ordering is clearly not single peaked; nor is this individual's scale score at her ideal point. Further, if there are individuals at each of the points on the continuum, any change in the ordering of alternatives or mid-points destroys the perfect scale pattern. Hence no ordering of the alternatives will assure both perfect scalability and single peaked preferences, even with multi-category items.
For an example showing that the paradox of voting is possible even when legislative votes form a perfect Guttman scale, see Richard G. Niemi and Herbert F. Weisberg,Probability Models of Collective Decision-Making (Columbus: Charles Merrill, 1972), pp. 221–22. Cf. Georges Th. Guilbaud, “Les theories de l'interet general et la probleme logique de l'agregation,”Economie appliquee, V (1952), pp. 501–84 especially 547–48, translated in Paul F. Lararsfeld and Neil W. Henry,Readings in Mathematical Social Science (Chicago: Science Research Associates, 1966); William H. Riker, “Voting and the Summation of Preferences: An Interpretive Bibliographic Review of Selected Developments During the Last Decade,”American Political Science Review, 55 (1961), pp. 900–11, especially pp. 906–07.
See Richard A. Brody and Benjamin I. Page, “Issues in an Election: Vietnam and Presidential Voting in 1968,” paper presented at the 1970 meeting of the International Political Science Association, Munich, Germany, especially pp. 3–4. Voters were asked to place themselves (i.e., their most preferred or “ideal” points) and presidential candidates on a seven-point rating scale running from 1 (labelled “immediate withdrawal”) to 7 (labelled “complete military victory”). Then the respondent's ideal point could be compared to the locations he ascribes to each party's candidates to determine which candidate he regards as closer to him on that scale. But if the preferences are not single peaked, an individual may rationally prefer a candidate who is located farther from his own position over one who is located closer. A person with an “all or nothing” position on the war might have point 7 as his first choice and point 1 as his second. In a choice between candidates perceived to be at positions 1 and 3, this voter would prefer the candidate at point 1 even though that candidate is farther from his or her own first preference at point 7. Thus individual preferences must be single peaked to determine relative preferences for candidates. (Additionally, the preferences must be symmetric. For example, an individual whose primary preference on the Vietnam war is at position 4 and who has a single peaked preference may still prefer immediate withdrawal to complete victory. This would be the case if his or her preference curve sloped very gradually to he left of the most preferred point but very sharply to its right.) See also Page and Brody, “Policy Voting and the Electoral Process: The Vietnam War Issue,”American Political Science Review, 66 (1972), 979–95, footnote 128.
Page and Brody, footnote 9.
David W. Rohde, “Policy Goals, Strategic Choice and Majority Opinion Assignments in the U.S. Supreme Court,”Midwest Journal of Political Science, 16 (1972), 652–82, at p. 666.
This seems to be a reasonable interpretation of Rohde's assertion that “each of the justices is primarily motivated in his decisions by a desire to have the policy made by the Court approximate as closely as possible his preferred position on a given issue .... The amount of utility that a justice derives from a Court decision will be determined by how close the policy expounded in the majority opinion is to his preferred position in that case.”Ibid p. 662.
Andersonet al., op. cit., p. 92. (emphasis ours)
Thus, for example, the well-known representation study of Miller and Stokes correlated how far congressmen and their constituents were willing to go on the dimensions involved, and not necessarily their preferences. We are not arguing there which correlation is theoretically more appropriate — only that there are at least two distinct possible correlations and that either might be profitably pursued. See Warren E. Miller and Donald E. Stokes, “Constituency Influence in Congress,”American Political Science Review, 57 (1963), pp. 45–56.
Coombs,op. cit., “ chap. 5.
The feasibility of obtaining preference orders in survey research is illustrated in Philip E. Converse, “The Problem of Party Distances in Models of Voting Change,” in M. Kent Jennings and L. Harmon Zeigler, ed.,The Electoral Process (Englewood Cliffs, N.J.: Prentice-Hall, 1966), pp. 175–207. Relative ratings accorded to objects have been used to infer preference orders (as in Herbert F. Weisberg and Jerrold G. Rusk “Dimensions of Candidate Evaluation,”American Political Science Review, 64 (1970), pp. 1170–72), though some slippage is inevitable in this approach. As the educational level of the electroate increases, the potential for use of sophisticated measurement techniques on the mass public should increase.
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We would like to thank Michael Coveyou for bringing to our attention one of the examples used in the paper.
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Niemi, R.G., Weisberg, H.F. Single-peakedness and guttman scales: Concept and measurement. Public Choice 20, 33–45 (1974). https://doi.org/10.1007/BF01718176
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DOI: https://doi.org/10.1007/BF01718176