Guttman scale

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a Guttman scale (named after Louis Guttman) is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced.

The discovery of a Guttman scale in data depends on their multivariate distribution's conforming to a particular structure (see below).

Hence, a Guttman scale is a hypothesis about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations.

If a Guttman scale is confirmed, the measurement of the attribute is intrinsically one-dimensional; the unidimensionality is not forced by summation or averaging.

For data sets where the categories of all variables are similarly ordered numerically (from high to low or from low to high) with respect to a given attribute, Guttman scale is defined simply thus: Definition: Guttman scale is a data set in which all profile-pairs are comparable.

Data of the four ordinal arithmetic skill variables are hypothesized to form a Guttman scale

In practice, perfect ("deterministic") Guttman scales are rare, but approximate ones have been found in specific populations with respect to attributes such as religious practices, narrowly defined domains of knowledge, specific skills, and ownership of household appliances.

The extent to which a data set conforms to a Guttman scale can be estimated from the coefficient of reproducibility [2][3] of which there are a few versions, depending on statistical assumptions and limitations.

And, to ensure that there is a range of responses (not the case if all respondents only endorsed one item) the coefficient of scalability is used.