Editing Guttman scale (section)

== Ordinal variables ==
Given a data set of ''N'' subjects observed with respect to ''n'' ordinal variables, each having any finite number (≥2) of numerical categories ordered by increasing strength of a pre-specified attribute, let ''a<sub>ij</sub>'' be the score obtained by subject ''i'' on variable ''j'', and define the list of scores that subject ''i'' obtained on the ''n'' variables, a<sub>i</sub>=a<sub>i1</sub>...a<sub>in</sub> , to be the ''profile'' of subject ''i''. (The number of categories may be different in different variables; and the order of the variables in the profiles is not important but should be fixed).

Define:

Two profiles, a<sub>''s''</sub> and a<sub>''t''</sub> are ''equal'', denoted a''<sub>s</sub>''=a''<sub>t</sub>'', iff ''a<sub>sj</sub>=a<sub>tj</sub>'' for all ''j''=1...''n''

Profile ''a<sub>s</sub>'' is ''greater'' than Profile ''a<sub>t</sub>'', denoted ''a<sub>s</sub>>a<sub>t</sub>'', iff ''a<sub>sj</sub> ≥ a<sub>tj</sub>'' for all ''j''=1...''n'' and a<sub>''sj'<nowiki/>''</sub> > ''a<sub>tj'</sub>'' for at least one variable, ''j'''.

Profiles ''a<sub>s</sub>'' and ''a<sub>t</sub>'' are ''comparable'', denoted ''a<sub>s</sub>Sa<sub>t</sub>'', iff ''a<sub>s</sub>=a<sub>t</sub>''; or ''a<sub>s</sub>>a<sub>t</sub>''; or ''a<sub>t</sub>>a<sub>s</sub>''

Profiles ''a<sub>s</sub>'' and ''a<sub>t</sub>'' are ''incomparable'', denoted ''a<sub>s</sub>$a<sub>t</sub>'', if they are not comparable (that is, for at least one variable, ''j''', ''a<sub>sj'</sub> > a<sub>tj'</sub>'' and for at least one other variable'', j<nowiki>''</nowiki>'', ''a<sub>tj<nowiki>''</nowiki></sub> > a<sub>sj</sub>''<sub><nowiki>''</nowiki></sub>.

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.

=== Example: Non-dichotomous variables ===
Consider the following four variables that assess arithmetic skills among a population P of pupils:

V1: Can pupil (p) perform addition of numbers? No=1; Yes, but only of two-digit numbers=2; Yes=3.

V2: Does pupil (p) know the (1-10) multiplication table?  No=1; Yes=2.

V3: Can pupil (p) perform multiplication of numbers? No=1; Yes, but only of two-digit numbers=2; Yes=3.

V4: Can pupil (p) perform long division? No=1; Yes=2.

Data collected for the above four variables among a population of school children may be hypothesized to exhibit the Guttman scale shown below in Table 2:

'''Table 2. Data of the four ordinal arithmetic skill variables are hypothesized to form a Guttman scale'''  
<br />
{| class="wikitable"
|V<sub>1</sub>
|V<sub>2</sub>
|V<sub>3</sub>
|V<sub>4</sub>
|<small>Possible</small>

<small>Scale score</small>
|-
|1
|1
|1
|1
|4
|-
|2
|1
|1
|1
|5
|-
|2
|2
|1
|1
|6
|-
|3
|2
|1
|1
|7
|-
|3
|2
|2
|1
|8
|-
|3
|2
|3
|1
|9
|-
|3
|2
|3
|2
|10
|}

The set profiles hypothesized to occur (shaded part in Table 2) illustrates the defining feature of the Guttman scale, namely, that any pair of profiles are comparable. Here too, if the hypothesis is confirmed, a single scale-score reproduces a subject's responses in all the variables observed.

Any ordered set of numbers could serve as scale. In this illustration we chose the sum of profile-scores. According to facet theory, only in data that conform to a Guttman scale such a summation may be justified.