Editing Guttman scale (section)

==Deterministic model==
Example 1: Dichotomous variables

A Guttman scale may be hypothesized for the following five questions that concern the attribute "acceptance of social contact with immigrants" (based on the [[Bogardus social distance scale]]), presented to a suitable population:

# Would you accept immigrants as residents in your country? (No=0; Yes=1)
# Would you accept immigrants as residents in your town? (No=0; Yes=1)
# Would you accept immigrants as residents in your neighborhood? (No=0; Yes=1)
# Would you accept immigrants as next-door neighbors? (No=0; Yes=1)
# Would you accept an immigrant as your child's spouse? (No=0; Yes=1)

A positive response by a particular respondent to any question in this list, suggests positive responses by that respondent to all preceding questions in this list. Hence one could expect to obtain only the responses listed in the shaded part (columns 1–5) of Table 1.

{| class="wikitable" style="margin: auto; border: none;"
|+ Table 1. Hypothesized responses to the five social distance variables form a Guttman scale (a cumulative scale)
! 1. Accepts<br/>Immig, in<br/>Country
! 2. Accepts<br/>Immig, in<br/>Town
! 3. Accepts<br/>Immig, in<br/>Neighborhood
! 4. Accepts<br/>Immig, Next<br/>Door
! 5. Accepts<br/>Immig, as<br/>Spouse
! Possible<br/>Scale Score
|-
| {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|0|0}}
|-
| {{shade|25|1}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|0|1}}
|-
| {{shade|25|1}} || {{shade|25|1}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|0|2}}
|-
| {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|25|0}} || {{shade|25|0}} || {{shade|0|3}}
|-
| {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|25|0}} || {{shade|0|4}}
|-
| {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|25|1}} || {{shade|0|5}}
|}

Every row in the shaded part of Table 1 (columns 1–5) is the response ''profile'' of any number (≥ 0) of respondents. Every profile in this table indicates acceptance of immigrants in all senses indicated by the previous profile, plus an additional sense in which immigrants are accepted. If, in a large number of observations, only the profiles listed in Table 1 are observed, then the Guttman scale hypothesis is supported, and the values of the scale (last column of Table 1) have the following properties:

# They assess the strength of the attribute "acceptance of social contact with immigrants";  
# They reproduce the original observations. (For example, a respondent's scale score of 2 implies that that respondent responded positively to questions 1 and 2 and negatively to questions 3, 4, and 5.)

Guttman scale, if supported by data, is useful for efficiently assessing subjects (respondents, testees or any collection of investigated objects) on a one-dimensional scale with respect to the specified attribute. Typically, Guttman scales are found with respect to attributes that are narrowly defined.

While other scaling techniques (e.g., [[Likert scale]]) produce a single scale by summing up respondents' scores—a procedure that assumes, often without justification, that all observed variables have equal weights — Guttman scale avoids weighting the observed variables; thus 'respecting' data for what they are. If a Guttman scale is confirmed, the measurement of the attribute is ''intrinsically'' one-dimensional; the unidimensionality is not forced by summation or averaging. This feature renders it appropriate for the construction of replicable scientific theories and meaningful measurements, as explicated in [[facet theory]].