<s>
In	O
the	O
analysis	B-General_Concept
of	I-General_Concept
multivariate	I-General_Concept
observations	I-General_Concept
designed	O
to	O
assess	O
subjects	O
with	O
respect	O
to	O
an	O
attribute	O
,	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
(	O
named	O
after	O
Louis	O
Guttman	O
)	O
is	O
a	O
single	O
(	O
unidimensional	O
)	O
ordinal	O
scale	O
for	O
the	O
assessment	O
of	O
the	O
attribute	O
,	O
from	O
which	O
the	O
original	O
observations	O
may	O
be	O
reproduced	O
.	O
</s>
<s>
The	O
discovery	O
of	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
in	O
data	O
depends	O
on	O
their	O
multivariate	O
distribution	O
's	O
conforming	O
to	O
a	O
particular	O
structure	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
Hence	O
,	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
a	O
hypothesis	O
about	O
the	O
structure	O
of	O
the	O
data	O
,	O
formulated	O
with	O
respect	O
to	O
a	O
specified	O
attribute	O
and	O
a	O
specified	O
population	O
and	O
cannot	O
be	O
constructed	O
for	O
any	O
given	O
set	O
of	O
observations	O
.	O
</s>
<s>
Contrary	O
to	O
a	O
widespread	O
belief	O
,	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
not	O
limited	O
to	O
dichotomous	O
variables	O
and	O
does	O
not	O
necessarily	O
determine	O
an	O
order	O
among	O
the	O
variables	O
.	O
</s>
<s>
A	O
Guttman	B-General_Concept
scale	I-General_Concept
may	O
be	O
hypothesized	O
for	O
the	O
following	O
five	O
questions	O
that	O
concern	O
the	O
attribute	O
"	O
acceptance	O
of	O
social	O
contact	O
with	O
immigrants	O
"	O
(	O
based	O
on	O
the	O
Bogardus	O
social	O
distance	O
scale	O
)	O
,	O
presented	O
to	O
a	O
suitable	O
population	O
:	O
</s>
<s>
If	O
,	O
in	O
a	O
large	O
number	O
of	O
observations	O
,	O
only	O
the	O
profiles	O
listed	O
in	O
Table	O
1	O
are	O
observed	O
,	O
then	O
the	O
Guttman	B-General_Concept
scale	I-General_Concept
hypothesis	O
is	O
supported	O
,	O
and	O
the	O
values	O
of	O
the	O
scale	O
(	O
last	O
column	O
of	O
Table	O
1	O
)	O
have	O
the	O
following	O
properties	O
:	O
</s>
<s>
Guttman	B-General_Concept
scale	I-General_Concept
,	O
if	O
supported	O
by	O
data	O
,	O
is	O
useful	O
for	O
efficiently	O
assessing	O
subjects	O
(	O
respondents	O
,	O
testees	O
or	O
any	O
collection	O
of	O
investigated	O
objects	O
)	O
on	O
a	O
one-dimensional	O
scale	O
with	O
respect	O
to	O
the	O
specified	O
attribute	O
.	O
</s>
<s>
Typically	O
,	O
Guttman	B-General_Concept
scales	I-General_Concept
are	O
found	O
with	O
respect	O
to	O
attributes	O
that	O
are	O
narrowly	O
defined	O
.	O
</s>
<s>
While	O
other	O
scaling	O
techniques	O
(	O
e.g.	O
,	O
Likert	O
scale	O
)	O
produce	O
a	O
single	O
scale	O
by	O
summing	O
up	O
respondents	O
 '	O
scores	O
—	O
a	O
procedure	O
that	O
assumes	O
,	O
often	O
without	O
justification	O
,	O
that	O
all	O
observed	O
variables	O
have	O
equal	O
weights	O
—	O
Guttman	B-General_Concept
scale	I-General_Concept
avoids	O
weighting	O
the	O
observed	O
variables	O
;	O
thus	O
'	O
respecting	O
 '	O
data	O
for	O
what	O
they	O
are	O
.	O
</s>
<s>
If	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
confirmed	O
,	O
the	O
measurement	O
of	O
the	O
attribute	O
is	O
intrinsically	O
one-dimensional	O
;	O
the	O
unidimensionality	O
is	O
not	O
forced	O
by	O
summation	O
or	O
averaging	O
.	O
</s>
<s>
For	O
data	O
sets	O
where	O
the	O
categories	O
of	O
all	O
variables	O
are	O
similarly	O
ordered	O
numerically	O
(	O
from	O
high	O
to	O
low	O
or	O
from	O
low	O
to	O
high	O
)	O
with	O
respect	O
to	O
a	O
given	O
attribute	O
,	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
defined	O
simply	O
thus	O
:	O
</s>
<s>
Definition	O
:	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
a	O
data	O
set	O
in	O
which	O
all	O
profile-pairs	O
are	O
comparable	O
.	O
</s>
<s>
Data	O
collected	O
for	O
the	O
above	O
four	O
variables	O
among	O
a	O
population	O
of	O
school	O
children	O
may	O
be	O
hypothesized	O
to	O
exhibit	O
the	O
Guttman	B-General_Concept
scale	I-General_Concept
shown	O
below	O
in	O
Table	O
2	O
:	O
</s>
<s>
The	O
set	O
profiles	O
hypothesized	O
to	O
occur	O
(	O
shaded	O
part	O
in	O
Table	O
2	O
)	O
illustrates	O
the	O
defining	O
feature	O
of	O
the	O
Guttman	B-General_Concept
scale	I-General_Concept
,	O
namely	O
,	O
that	O
any	O
pair	O
of	O
profiles	O
are	O
comparable	O
.	O
</s>
<s>
According	O
to	O
facet	O
theory	O
,	O
only	O
in	O
data	O
that	O
conform	O
to	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
such	O
a	O
summation	O
may	O
be	O
justified	O
.	O
</s>
<s>
In	O
practice	O
,	O
perfect	O
(	O
"	O
deterministic	O
"	O
)	O
Guttman	B-General_Concept
scales	I-General_Concept
are	O
rare	O
,	O
but	O
approximate	O
ones	O
have	O
been	O
found	O
in	O
specific	O
populations	O
with	O
respect	O
to	O
attributes	O
such	O
as	O
religious	O
practices	O
,	O
narrowly	O
defined	O
domains	O
of	O
knowledge	O
,	O
specific	O
skills	O
,	O
and	O
ownership	O
of	O
household	O
appliances	O
.	O
</s>
<s>
When	O
data	O
do	O
not	O
conform	O
to	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
,	O
they	O
may	O
either	O
represent	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
with	O
noise	O
(	O
and	O
treated	O
stochastically	O
)	O
,	O
or	O
they	O
may	O
have	O
a	O
more	O
complex	O
structure	O
requiring	O
multiple	O
scaling	O
for	O
identifying	O
the	O
scales	O
intrinsic	O
to	O
them	O
.	O
</s>
<s>
The	O
extent	O
to	O
which	O
a	O
data	O
set	O
conforms	O
to	O
a	O
Guttman	B-General_Concept
scale	I-General_Concept
can	O
be	O
estimated	O
from	O
the	O
coefficient	O
of	O
reproducibility	O
of	O
which	O
there	O
are	O
a	O
few	O
versions	O
,	O
depending	O
on	O
statistical	O
assumptions	O
and	O
limitations	O
.	O
</s>
<s>
Guttman	B-General_Concept
scale	I-General_Concept
has	O
been	O
generalized	O
to	O
the	O
theory	O
and	O
procedures	O
of	O
"	O
multiple	O
scaling	O
"	O
which	O
identifies	O
the	O
minimum	O
number	O
of	O
scales	O
needed	O
for	O
satisfactory	O
reproducibility	O
.	O
</s>
<s>
As	O
a	O
procedure	O
that	O
ties	O
substantive	O
contents	O
with	O
logical	O
aspects	O
of	O
data	O
,	O
Guttman	B-General_Concept
scale	I-General_Concept
heralded	O
the	O
advent	O
of	O
facet	O
theory	O
developed	O
by	O
Louis	O
Guttman	O
and	O
his	O
associates	O
.	O
</s>
<s>
This	O
definition	O
of	O
Guttman	B-General_Concept
scale	I-General_Concept
relies	O
on	O
the	O
prior	O
definition	O
of	O
a	O
simple	O
function	O
.	O
</s>
<s>
A	O
Guttman	B-General_Concept
scale	I-General_Concept
may	O
then	O
be	O
defined	O
for	O
a	O
data	O
set	O
of	O
n	O
variables	O
,	O
with	O
the	O
jth	O
variable	O
having	O
kj	O
(	O
qualitative	O
,	O
not	O
necessarily	O
ordered	O
)	O
categories	O
,	O
thus	O
:	O
</s>
<s>
Definition	O
:	O
Guttman	B-General_Concept
scale	I-General_Concept
is	O
a	O
data	O
set	O
for	O
which	O
there	O
exists	O
an	O
ordinal	O
variable	O
,	O
X	O
,	O
with	O
a	O
finite	O
number	O
m	O
of	O
categories	O
,	O
say	O
,	O
1	O
,...,	O
m	O
with	O
m≥	O
maxj(kj )	O
and	O
a	O
permutation	O
of	O
subjects	O
 '	O
profiles	O
such	O
that	O
each	O
variable	O
in	O
the	O
data	O
set	O
is	O
a	O
simple	O
function	O
of	O
X	O
.	O
</s>
