581
Application
of
a
Psychometric
Rating
Model
to
Ordered
Categories
Which
Are
Scored
with
Successive
Integers
David
Andrich
The
University
of
Western
Australia
A
latent
trait
measurement
model
in
which
ordered
response
categories
are
both
parameterized
and
scored
with
successive
integers
is
investigated
and
applied
to
a
summated
rating
or
Likert
ques-
tionnaire.
In
addition
to
each
category,
each
item
of
the
questionnaire
and
each
subject
are
para-
meterized
in
the
model;
and
maximum
likelihood
estimates
for
these
parameters
are
derived.
Among
the
features
of
the
model
which
make
it
attractive
for
applications
to
Likert
questionnaires
is
that
the
total
score
is
a
sufficient
statistic
for
a
subject’s
at-
titude
measure.
Thus,
the
model
provides
a
formal-
ization
of
a
familiar
and
practical
procedure
for
measuring
attitudes.
One
of
the
most
persistent
features
in
the
scoring
of
response
categories
which
are
ordered
is
the
assignment
of
successive
integers
to
the
successive
categories.
This
procedure
is
particularly
evident
in
the
scoring
of
items
of
attitude
questionnaires
constructed
in
the
popular
Likert
tradition
in
which
people
rate
their
degree
of
agreement
or
disagreement
to
statements
or
items.
In
addition,
the
sum-
mary
statistic
indicating
a
person’s
attitude
with
respect
to
a
set
of
items
on
such
a
questionnaire
is
commonly
taken
to
be
the
sum
of
the
integer
scores
assigned
to
the
items.
Although
this
is
the
com-
mon
approach
to
dealing
with
such
ordered
response
categories,
certain
assumptions
of
this
scoring
procedure,
particularly
that
with
respect
to
the
implied
assumption
of
&dquo;equal
distances&dquo;
between
categories,
continues
to
be
questioned.
The
general
approach
for
overcoming
objections
to
the
integer-scoring
procedure
is
to
use
a
re-
sponse
model
which
keeps
track
of
the
category
in
which
a
person
responds.
Two
basic
types
of
models
with
such
a
property
have
been
studied
extensively.
In
the
first
type,
exemplified
by
Samejima
(1969)
and
Kolakowski
and
Bock
(1972),
a
unidimensional
latent
variable
with
distances
between
re-
sponse
categories
is
postulated.
The
boundary
positions
of
the
distances
are
considered
threshold
points
of
the
continuous
latent
variable,
and
the
response
is
supposed
to
be
determined
by
the
interval
in
which
the
value
of
the
latent
variable
falls.
In
the
second
type
of
model,
considered
by
Rasch
(1961,
1968)
and
discussed
in
some
detail
by
Andersen
(1973),
a
multidimensional
parametric
structure
is
postulated
with
respect
to
the
response
categories;
and
after
the
parameter
estimation
is
carried
out,
checks
on
the
possible
reduction
of
the
dimensionality
of
this
structure
are
made.
APPLIED
PSYCHOLOGICAL
MEASUREMENT
Vol.
2,
No. 4 Fall 1978 pp.
581-594
@
Copyright
1978
West
Publishing
Co.
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582
Through
a
special
case
of
this
model,
in
which
a
single
dimension
is
postulated
immediately,
Andersen
(1977)
and
Andrich
(in
press-b)
have
revived
interest
in
the
&dquo;equidistant&dquo;
scoring
of
the
categories.
First,
Andersen
has
shown
that
if
the
sum
of
scores
assigned
to
categories
is
to
provide
a
sufficient
statistic
for
a
person’s
attitude
and
if
certain
plausible
ordering
properties
among
response
patterns
are
to
hold,
then
the
categories
must
be
scored
in
an
&dquo;equidistant,&dquo;
or
equivalently,
integer-
scoring
fashion.
Secondly,
Andrich
has
interpreted
the
model
in
terms
of
thresholds
on
a
latent
con-
tinuum.
This
interpretation
is
set
in
a
similar
context
to
that
of
the
traditional
threshold
formulation
but
is
distinctively
different
from
it.
Thus,
when
it
holds,
the
model
formalizes
in
terms
of
a
latent
trait
approach
the
intuitive
procedure
of
the
Likert-type
questionnaires.
(For
full
details
of
the
statis-
tical
rationale
and
the
interpretation
of
the
model,
the
reader
is
referred
to
the
above
two
papers.
However,
since
they
contain
no
applications,
an
operationalization
and
an
application
of
this
model
are
illustrated
in
the
present
paper.)
The
Rating
Response
Model
The
rating
response
model
takes
the
form
where
is
a
normalizing
factor.
In
Model
1 the
various
variables
are
interpreted
as
follows.
Firstly,
it
is
proposed
that
when
a
person
responds
to
an
item
in
one
of m+1
ordered
response
categories,
the
categories
are
separated
by
m
ordered
thresholds
on
a
latent
continuum.
Secondly,
the
value
x
of
random
variable
X,
where
jcc
{0,
1,
...,
m},
designates
the
number
of
thresholds
passed.
Specifically,
x=0
indicates
a
response
in
the
first
(or
lowest)
category,
in
which
case
no
threshold
is
passed,
while
x=m
indicates
a
response
in
the
last
(or
highest)
category,
in
which
case
all
thresholds
are
passed.
Thirdly,
and
most
importantly,
the
category
coefficients
xx
are
defined
in
terms
of
the
thresholds
T,,,
k=I,
m
as
follows:
Finally, {3
and
d,
respectively,
represent
the
attitude
of
a
person
and
the
affective
value
of
an
item.
Two
related
points
are
stressed
with
respect
to
this
model-points
developed
in
Andrich
(in
press-b).
Firstly,
the
scoring
of
successive
categories
with
successive
integers
depends
on
the
equal
discrimination
at
the
thresholds
and
not
on
equal
distances
between
thresholds.
This
is
in
contrast
to
traditional
formulations
in
which
integer
scoring
is
considered
to
depend
on
equal
distances
between
thresholds.
Secondly,
the
values
of
thresholds
are
estimated.
The
relationship
of
each
category
coeffi-
cient
to
the
thresholds
is
that
except
for
the
first
coefficient
(which
is
always
zero),
it
is
the
sum
of
the
thresholds
up
to
that
category.
The
characteristic
curves
for
the
response
categories
in
relation
to
the
thresholds
are
shown
in
Figure
1
in
the
context
of
the
illustrative
example.
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583
Before
proceeding
with
the
estimation,
it
is
noted
that
when
m=1-that
is,
when
the
response
is
dichotomous-Model
1 reduces
to
Rasch’s
simple
logistic
model
(SLM).
In
the
case
exp
xx
=
[x-l-that
is,
when
the
coefficients
are
specified
in
advance
to
be
binomial-it
reduces
to
the
binomial
logistic
model
(BLM)
described
in
Andrich
(1978,
in
press-a).
Estimation
Because
Model
1 is
a
straightforward
generalization
of
the
SLM
and
the
BLM,
it
is
possible
to
approach
its
parameter
estimation
from
the
two
main
approaches
which
have
been
studied
with
re-
spect
to
these
special
cases.
They
are
the
so-called
conditional
and
unconditional
approaches.
In
the
conditional
approach,
sufficient
statistics
are
identified
for
the
person
and
category
parameters;
and
then
an
equation,
conditional
on
these
statistics
and
containing
only
the
item
parameters,
is
obtained.
With
respect
to
a
sample
of
data,
a
conditional
likelihood
equation
may
be
derived
which
can
be
used
to
obtain
maximum
likelihood
estimates
of
the
item
parameters.
On
the
basis
of
the
item
parameter
estimates
so
obtained,
which
are
then
treated
as
fixed
and
known
quantities,
maximum
likelihood
person
and
threshold
parameters
may
be
estimated.
Although
this
approach,
exemplified
by
Andersen
(1973)
and
Wright
and
Douglas
(1977),
is
ideal
theoretically,
implementation
problems
for
routine
computer
analyses
do
seem
to
arise
in
the
solution
algorithms.
In
particular,
round-off
errors
in
associated
symmetric
functions
seem
to
miti-
gate
against
successful
estimation
with
large
numbers
of
items.
Douglas
(in
press)
has
studied
this
ap-
proach
with
respect
to
the
BLM;
the
problem
is
exaggerated
as
the
number
of
categories
is
also
in-
creased.
In
the
model
of
Equation
1,
these
same
drawbacks
seem
to
exist.
In
the
unconditional
approach,
used
by
Wright
and
Panchapakesan
(1969),
Wright
and
Douglas
(1977),
and
Andrich
(1978),
a
direct
likelihood
function
is
obtained;
and
the
item
and
person
para-
meters
are
effectively
estimated
simultaneously.
While
this
approach
is
not
hampered
by
large
num-
bers
of
items,
it
has
the
weakness
that
the
item
parameters
are
not
consistent.
This
weakness,
dis-
cussed
by
Andersen
(1973)
for
the
SLM,
is
due
to
the
retention
of
the
&dquo;incidental&dquo;
person
parameters
(which
vary
from
person
to
person)
in
the
estimation.
Fortunately,
a
simple
correction
factor,
which
gives
estimates
identical
to
those
obtained
in
the
conditional
approach
in
the
case
of
two
items
and
two
response
categories,
does
exist.
Wright
and
Douglas
provide
a
rationale
for
this
factor
in
the
SLM;
the
factor
will
be
discussed
later
with
respect
to
the
model
studied
in
this
paper.
Unconditional
Estimation
If N
subjects
respond
to I
items
in
one
of m+1
ordered
categories
which
are
successively
scored
from
0
to
m,
then
the
joint
probability
of
the
data
matrix
[xv,],
given
the
parameter
vectors
x,
(3,
d,
is
given
by
~ ~ ~
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584
Expanding,
simplifying,
and
taking
the
logarithm
of
Equation
3
gives
where
Tx
is
the
total
number
of
responses
with
respect
to
all
subjects
and
all
items
which
are
in
the
category
x;
rv
=
~x~,
is
the
total
score
of
subject
v;
S, =
2jc~,
is
the
total
score
of
item
i;
and
L
is
the
likelihood.
It
can
now
be
seen
readily
from
the
logarithmic
form
of
Equation
4,
and
according
to
the
factori-
zation
theorem
of
sufficient
statistics,
that
the
total
person
score r~
is
a
sufficient
statistic
for
(3v.
Simi-
larly,
it
can
be
seen
that
7B
and
S,
are
respectively
sufficient
statistics
for
xx
and
d,.
These
are
key
features
of
the
model,
which
not
only
make
it
relatively
tractable,
but
also
provide
very
clear
connec-
tions
with
more
intuitive
approaches
to
the
analysis
of
questionnaires
with
ordered
response
cate-
gories.
This
is
especially
the
case
with
respect
to
finding
an
index
for
a
person’s
attitude.
Reparameterization
By
differentiating
Equation
4
with
respect
to
each
of
xx,
(3~,
and
d,
and
equating
the
resultant
ex-
pressions
to
zero,
maximum
likelihood
equations
may
be
obtained
directly.
Before
doing
so,
however,
it
proves
expedient
to
anticipate
two
dependencies
in
the
resultant
equations:
one
among
the
item
parameters
and
one
among
the
category
parameters.
These
can
readily
be
seen
from
Equation
3,
since
the
probability
statement
is
unchanged
if
a
constant
is
added
to
each
d&dquo;
providing
it is
also
added
to
each
(3v;
and
it
is
unchanged
if
a
constant
is
added
to
each
xx.
Adding
the
equation
~d,
=
0
provides
an
origin
and
a
simple
method
for
placing
a
constraint
on
the
items.
The
most
efficient
and
useful
method
for
handling
the
indeterminacy
on
the
categories
is
to
reparameterize
them
in
a
way
analogous
to
the
procedure
used
by
Bock
(1972)
for
a
similar
model
in
which
the
categories
are
nominal
rather
than
ordered.
Reparameterizing
according
to
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585
where
the
matrix
B
(called
the
basis
matrix)
takes
the
form
reduces
by
1
the
number
of
parameters
to
be
estimated.
This
overcomes
the
indeterminacy
and
pro-
vides
immediately
an
estimate
of
the
thresholds
rather
than
the
category
coefficients.
This
effect
is
ideal,
since
it
is
the
thresholds
and
not
the
category
coefficients
which
can
be
interpreted
directly.
The
Solution
Equations
With
the
above
general
reparameterization,
the
log
likelihood
takes
the
form
where
bxk
is
the
(x,
k~th
element
of
B.
In
the
estimation,
the
following
relationships,
which
can
be
shown
readily,
help
simplify
the
resulting
expression
considerably:
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