## Stochastic Guttman Order

Louis Guttman conceptualized an ideal scale with the property that knowledge of only the number of items a respondent passes tells the researcher exactly which items the respondent passes. Guttman (1947, 1950) does this by constructing a "scalogram" of the scored responses, in which each column corresponds to a respondent, arranged left to right in decreasing raw score order, and each row corresponds to an item, arranged top to bottom in decreasing order of respondent success. The item on which the respondents are the most successful comes first. When Guttman's ideal is realized, the responses in the top left of the scalogram are all successes and in the bottom right are all failures. There is a distinct diagonal demarcation between success and failure, with no disordering or "inversion" of successes and failures.

In practice, of course, Guttman's ideal is unobtainable, as he well knew. Accordingly he established rules which position a "cutting line" for each respondent at that place in the string of responses which minimizes the number of inversions for that respondent. This is a kind of "least error" fitting of a deterministic ideal to uncertain data. Unfortunately Guttman's rule leads to ambiguous results. If the responses are 1010 to the items ordered in ascending order of difficulty, both 1!010 and 101!0 are "least inversions" placements of the cutting line for one inversion. This has not gone unnoticed and procedures for dealing with this problem have been proposed.

Kenny and Rubin (1977) object to the ambiguity, arbitrariness and lack of clear theoretical basis which underlie the attempts to solve this problem. They build on Guttman's concept of "reproducibility", the proportion of scalable, correctly placed responses in the data. Guttman minimized the inversions one respondent at a time, with ambiguous results. Kenny and Rubin assert that the inversions are more meaningfully reduced by considering all respondents with the same raw score together. This yields the unambiguous result that the cutting line is placed at that point where the observed number of successes would be were the data perfectly scalable. All respondents with the same raw score get the same cutting line, regardless of the number of inversions within each pattern of responses. This is the default in the scalogram programs, SAS and SPSS-X.

The default is well chosen. Ordering by raw score is identical to requiring that raw score be a sufficient statistic. But this requirement leads directly to the Rasch model as the only explanation for the data. Guttman reproducibility, constrained by unambiguity, is equivalent to the Rasch model.

Guttman L 1947. The Cornell technique for scale and intensity analysis. Educational and Psychological Measurement, 7, 274-279

Guttman L 1950. The basis for scalogram analysis. In Stouffer et al. Measurement and Prediction. The American Soldier Vol. IV. New York: Wiley

Kenny DA, Rubin DC 1977. Estimating chance reproducibility in Guttman scaling. Social Science Research, 6, 188-196.

Stochastic Guttman order. Linacre JM. … Rasch Measurement Transactions, 1992, 5:4 p.189

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