Guttman scale

A Guttman scale is a measurement instrument used in psychological and sociological research developed using the scaling technique developed by Louis Guttman in 1944 called Guttman scaling or scalogram analysis. A primary purpose of the Guttman scaling is to ensure that the instrument measures only a single trait (a property called unidimensionality, a single dimension underlies responses to the scale). Guttman's insight was that for unidimensional scales, those who agree with a more extreme test item will also agree with all less extreme items that preceded it.

A perfect Guttman scale
A hypothetical, perfect Guttman scale consists of a unidimensional set of items that are ranked in order of difficulty from least extreme to most extreme position. For example, a person scoring a "7" on a ten item Guttman scale, will agree with items 1-7 and disagree with items 8,9,10. An important property of Guttman's model is that a person's entire set of responses to all items can be predicted from their cumulative score because the model is deterministic.

Deterministic model
An important objective in Guttman scaling is to maximize the reproducibility of response patterns from a single score. A good Guttman scale should have a coefficient of reproducibility (the percentage of original responses that could be reproduced by knowing the scale scores used to summarize them) above .85. Another commonly used metric for assessing the quality of a Guttman scale, is Menzel's coefficient of scalability and the coefficient of homogeneity (Loevinger, 1948; Cliff, 1977; Krus and Blackman, 1988). In order to maximize unidimensionality, misfitting items are re-written or discarded.

Stochastic models
Guttman's deterministic model is brought within a probabilistic framework in item response theory models, and especially Rasch measurement. The Rasch model requires a probabilistic Guttman structure when items have dichotomous responses (e.g. right/wrong). In the Rasch model, the Guttman response pattern is the most probable response pattern for a person when items are ordered from least difficult to most difficult (Andrich, 1985). In addition, the Polytomous Rasch model is premised on a deterministic latent Guttman response subspace, and this is the basis for integer scoring in the model (Andrich, 1978, 2005). Analysis of data using item response theory requires comparatively longer instruments and larger datasets, though, in order to scale item and person locations and evaluate fit of data to the model.

In practice, actual data from respondents do not closely match Guttman's deterministic model. Several probabilistic models of Guttman implicatory scales were developed by Krus (1977) and Krus and Bart (1974).

Applications
The Guttman scale is used mostly when researchers want to design short questionnaires with good discriminating ability. The Guttman model works best for constructs that are hierarchical and highly structured such as social distance, organizational hierarchies, and evolutionary stages.

Unfolding models
A class of unidimensional models that contrast with Guttman's model are unfolding models. These models also assume unidimensionality but posit that the probability of endorsing an item is proportional to the distance between the items standing on the unidimensional trait and the standing of the respondent. For example, items like "I think immigration should be reduced" on a scale measuring attitude towards immigration would be unlikely to be endorsed both by those favoring open policies and also by those favoring no immigration at all. Such an item might be endorsed by someone in the middle of the continuum. Some researchers feel that many attitude items fit this unfolding model while most psychometric techniques are based on correlation or factor analysis, and thus implicitly assume a linear relationship between the trait and the response probability. The effect of using these techniques would be to only include the most extreme items, leaving attitude instruments with little precision to measure the trait standing of individuals in the middle of the continuum.

Example
Here is an example of a Guttman scale - the Bogardus Social Distance Scale:

(Least extreme) (Most extreme)
 * 1) Are you willing to permit immigrants to live in your country?
 * 2) Are you willing to permit immigrants to live in your community?
 * 3) Are you willing to permit immigrants to live in your neighbourhood?
 * 4) Are you willing to permit immigrants to live next door to you?
 * 5) Would you permit your child to marry an immigrant?

E.g., agreement with item 3 implies agreement with items 1 and 2.