Nonparametric

IRT

analysis

of

Quality-of-Life

Scales

and

its

application

to

the

World

Health

Organization

Quality-of-Life

Scale

-LRB-

WHOQOL-Bref

-RRB-

Background

This

study

investigates

the

usefulness

of

the

nonparametric

monotone

homogeneity

model

for

evaluating

and

constructing

Health-Related

Quality-of-Life

Scales

consisting

of

polytomous

items

and

compares

it

to

the

often-used

parametric

graded

response

model

Introduction

Questionnaires

for

health-related

quality-of-life

-LRB-

HRQoL

-RRB-

measurement

are

important

for

several

reasons

First

they

may

be

used

to

compare

the

mean

level

of

different

patient

groups

with

respect

to

physical

mental

and

social

health

A

researcher

may

want

to

find

out

whether

these

patient

groups

have

different

needs

with

respect

to

for

example

therapy

or

medication

or

whether

different

adaptations

of

their

environment

are

in

order

so

as

to

improve

their

conditions

of

living

Second

HRQoL

questionnaires

are

also

important

for

the

measurement

of

mean

change

either

progress

or

deterioration

of

such

groups

due

to

for

example

therapy

The

researcher

's

interest

then

lies

in

the

effectiveness

of

therapy

with

respect

to

HRQoL

Third

the

total

score

a

patient

obtains

on

an

HRQoL

questionnaire

may

be

used

to

diagnose

this

patient

's

general

level

of

physical

health

and

psychological

well-being

for

example

so

as

to

be

able

to

estimate

the

budget

needed

for

his/her

treatment

during

a

particular

period

To

effectively

measure

HRQoL

we

argue

that

an

instrument

must

meet

two

requirements

The

first

requirement

is

that

it

is

clear

what

the

instrument

measures

one

overall

dimension

of

HRQoL

or

several

dimensions

reflecting

different

aspects

of

HRQoL

If

the

instrument

measures

one

dimension

one

can

use

the

total

score

on

all

items

to

obtain

an

impression

of

the

overall

level

of

HRQoL

If

the

instrument

measures

multiple

dimensions

it

may

be

recommendable

to

determine

total

scores

on

subsets

of

items

-LRB-

e.g.

domain

scores

-RRB-

each

reflecting

a

particular

aspect

of

HRQoL

-LRB-

e.g.

HRQoL

with

respect

to

physical

psychological

and

social

limitations

-RRB-

and

then

assess

individuals

or

compare

groups

on

a

profile

of

scores

These

two

cases

may

be

characterized

as

unidimensional

and

multidimensional

measurement

The

second

requirement

is

that

the

psychometric

properties

of

the

items

are

known

and

found

sufficient

One

important

psychometric

item

property

is

the

item

's

location

on

the

scale

that

quantifies

the

HRQoL

aspect

of

interest

For

example

patients

are

likely

to

experience

fewer

problems

when

engaging

in

activities

like

bathing

and

dressing

than

in

more

demanding

activities

such

as

shopping

and

travelling

The

items

concerning

bathing

and

dressing

require

a

lower

level

of

physical

functioning

than

the

other

two

items

Thus

bathing

and

dressing

are

located

further

to

the

left

-LRB-

at

a

lower

level

of

the

scale

-RRB-

than

shopping

and

travelling

A

good

diagnostic

HRQoL

instrument

contains

items

of

which

the

locations

are

widely

spread

along

the

scale

Such

a

scale

allows

for

measurement

at

varying

levels

of

physical

functioning

and

may

be

used

for

example

for

assessing

mean

differences

between

groups

mean

change

due

to

therapy

and

individual

patients

levels

of

physical

functioning

Another

important

psychometric

item

property

is

the

item

's

discrimination

power

This

is

the

degree

to

which

the

item

distinguishes

patients

with

relatively

low

psychological

well-being

levels

from

patients

with

relatively

high

psychological

well-being

levels

The

higher

the

discrimination

power

the

higher

the

item

's

contribution

to

reliable

measurement

-LRB-

-LSB-

-RSB-

pp.

-RRB-

A

good

diagnostic

instrument

has

items

with

high

discrimination

power

that

each

contributes

effectively

to

reliable

measurement

of

patients

at

different

locations

along

the

scale

Such

an

instrument

picks

up

differences

between

groups

effects

of

therapy

and

individual

levels

of

activity

limitation

Item

response

theory

-LRB-

IRT

-RRB-

models

-LSB-

-RSB-

are

becoming

more

popular

as

statistical

tools

for

scale

construction

in

the

HRQoL

context

IRT

can

be

used

effectively

to

investigate

the

dimensionality

of

an

instrument

and

the

psychometric

properties

of

its

constituent

items

The

goal

of

this

study

is

to

discuss

one

particular

class

of

IRT

models

known

as

nonparametric

IRT

models

-LSB-

-RSB-

and

to

argue

that

this

class

in

particular

provides

a

general

and

flexible

data

analysis

framework

for

studying

the

dimensionality

of

a

set

of

polytomously

scored

items

-LRB-

with

dichotomously

scored

items

as

special

cases

-RRB-

and

ascertaining

ordinal

scales

for

the

measurement

of

HRQoL

aspects

which

contain

items

that

have

varying

locations

and

sufficient

discrimination

power

Over

the

past

few

years

nonparametric

IRT

models

already

have

been

used

occasionally

for

constructing

HRQoL

scales

see

-LSB-

-RSB-

Our

point

of

view

is

that

given

that

the

researcher

has

formulated

desirable

measurement

properties

-LRB-

s

-RRB-

he

should

construct

his/her

scale

by

means

of

an

IRT

model

that

is

as

general

as

possible

while

satisfying

the

desired

measurement

properties

Examples

of

such

properties

are

that

the

items

measure

the

same

dimension

that

the

measurement

level

is

at

least

ordinal

and

that

measurement

values

are

reliable

An

HRQoL

researcher

who

has

constructed

and

pre-tested

a

questionnaire

consisting

of

say

items

is

not

served

well

when

his/her

data

are

analyzed

by

means

of

an

IRT

model

that

is

unnecessarily

restrictive

the

result

of

which

is

that

say

half

of

the

items

are

discarded

We

will

argue

that

the

most

general

IRT

model

that

serves

one

's

purposes

well

often

-LRB-

but

not

always

-RRB-

is

a

nonparametric

IRT

model

Many

questionnaires

are

used

for

assessing

differences

in

HRQoL

between

groups

change

due

to

therapy

and

individual

patients

scale

levels

and

a

general

nonparametric

IRT

model

then

is

the

perfect

choice

for

analyzing

one

's

data

Nonparametric

IRT

models

have

several

advantages

over

more-restrictive

parametric

IRT

models

-LSB-

-RSB-

Nonparametric

IRT

models

-LRB-

-RRB-

are

based

on

less-restrictive

assumptions

thus

they

allow

more

items

into

the

scale

while

maintaining

desirable

measurement

properties

-LRB-

-RRB-

offer

diverse

tools

for

HRQoL

analysis

that

give

ample

information

about

the

dimensionality

of

the

data

and

the

properties

of

the

items

and

-LRB-

-RRB-

provide

patient

measurement

values

and

item

location

and

discrimination

values

which

have

an

interpretation

that

is

close

to

intuition

and

therefore

easy

to

interpret

for

users

of

HRQoL

scales

For

computerized

adaptive

HRQoL

testing

more-restrictive

parametric

IRT

models

such

as

the

Rasch

-LSB-

-RSB-

model

-LRB-

dichotomous

items

-RRB-

and

the

generalized

partial

credit

model

-LSB-

-RSB-

and

the

rating

scale

model

-LSB-

-RSB-

-LRB-

polytomous

items

-RRB-

are

more

appropriate

than

nonparametric

models

Parametric

IRT

models

have

been

used

more

than

nonparametric

IRT

models

especially

in

psychological

and

educational

measurement

and

also

in

HRQoL

research

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

A

reason

for

this

may

be

that

nonparametric

IRT

models

were

developed

later

than

parametric

IRT

models

See

-LSB-

-RSB-

for

reviews

of

nonparametric

IRT

This

paper

is

organized

as

follows

First

we

explain

assumptions

of

IRT

and

compare

parametric

IRT

and

nonparametric

IRT

Second

we

discuss

methods

and

software

from

nonparametric

IRT

that

can

be

used

for

analyzing

the

polytomous

item

scores

obtained

from

HRQoL

questionnaires

Third

we

use

this

software

to

analyze

data

from

the

World

Health

Organization

Quality-of-Life

Scale

-LRB-

WHOQOL-Bref

-RRB-

-LSB-

-RSB-

The

results

are

compared

to

those

obtained

by

means

of

a

parametric

IRT

model

Finally

we

provide

recommendations

for

HRQoL

researchers

on

how

to

use

nonparametric

IRT

methods

for

analyzing

their

data

Definitions

and

assumptions

IRT

models

are

suited

for

the

analysis

of

multi-item

questionnaire

data

that

typically

result

from

HRQoL

questionnaires

The

data

are

discrete

scores

representing

the

responses

of

N

respondents

to

J

items

-LRB-

items

are

indexed

j

j

J

-RRB-

Many

HRQoL

questionnaires

use

items

that

have

three

or

more

ordered

answer

categories

represented

by

three

or

more

ordered

scores

also

called

polytomous

item

scores

For

simplicity

we

assume

that

all

items

have

the

same

number

of

ordered

answer

categories

this

number

is

denoted

by

M

Let

Xj

be

the

random

variable

representing

the

discrete

score

on

item

j

and

let

the

items

be

scored

Xj

M.

For

example

for

an

item

asking

whether

one

is

satisfied

with

one

's

sleep

score

may

represent

very

much

dissatisfied

and

score

M

may

represent

very

much

satisfied

and

the

intermediate

scores

represent

intermediate

levels

of

satisfaction

With

dichotomous

item

scoring

Xj

and

possible

intermediate

satisfaction

levels

are

not

quantified

separately

but

collapsed

into

the

score

categories

and

Usually

researchers

summarize

the

J

item

scores

for

each

patient

by

the

total

score

-LRB-

sometimes

referred

to

as

the

sum

score

-RRB-

which

is

formally

defined

as

Total

score

X

is

an

estimate

of

a

patient

's

true

score

T

this

is

the

expectation

of

X

across

independent

replications

of

the

measurement

procedure

-LRB-

-LSB-

-RSB-

pp.

-RRB-

IRT

models

distinguish

observable

or

manifest

variables

such

as

item

score

Xj

and

total

score

X

from

latent

variables

These

latent

variables

play

the

role

of

summaries

of

the

behavior

that

is

described

by

the

responses

to

the

items

Sometimes

latent

variables

are

interpreted

as

if

they

were

causal

agents

driving

responses

to

items

and

individual

differences

between

patients

We

will

also

use

the

distinction

between

latent

and

manifest

variables

in

our

examples

Thus

we

assume

that

patients

are

characterized

by

either

one

latent

HRQoL

attribute

-LRB-

meaning

that

measurement

is

unidimensional

-RRB-

or

different

HRQoL

attributes

-LRB-

meaning

that

measurement

is

multidimensional

-RRB-

which

together

represent

the

patient

's

latent

physical

mental

or

social

health

For

example

assume

that

measurement

is

unidimensional

and

that

the

latent

variable

is

mental

health

or

psychological

well-being

Psychological

well-being

then

is

an

unobservable

state

in

each

patient

and

inferences

about

it

are

made

on

the

basis

of

the

manifest

responses

reported

by

patients

in

reaction

to

the

items

in

an

HRQoL

questionnaire

Latent

variables

are

denoted

by

notation

θ

If

measurement

is

unidimensional

the

IRT

model

contains

one

latent

variable

θ

and

if

it

is

multidimensional

multiple

latent

variables

are

needed

We

only

consider

unidimensional

IRT

models

here

see

-LSB-

-RSB-

for

a

discussion

of

multidimensional

IRT

models

Several

families

of

IRT

models

for

polytomous

item

scores

have

been

proposed

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

The

family

of

graded

response

models

-LRB-

GRMs

-LSB-

-RSB-

-RRB-

is

suitable

for

analyzing

ordered

item

scores

collected

by

means

of

polytomous

response

scales

-LSB-

-RSB-

Suppose

the

item

How

much

do

you

enjoy

life

has

four

ordered

answer

categories

running

from

Not

at

all

-LRB-

score

-RRB-

to

Very

much

-LRB-

score

-RRB-

thus

Xj

and

M

For

this

item

GRMs

conceptualize

the

response

process

by

means

of

four

conditional

response

probabilities

called

item-step

response

functions

-LRB-

ISRFs

-RRB-

and

denoted

by

P

-LRB-

Xj

m

θ

-RRB-

Pjm

-LRB-

θ

-RRB-

-LRB-

m

M

-RRB-

It

may

be

noted

that

for

m

we

have

that

-LRB-

θ

-RRB-

by

definition

irrespective

of

the

latent

variable

level

thus

this

response

probability

is

uninformative

about

the

response

process

and

may

be

ignored

The

ISRF

describes

the

relationship

between

expressing

at

least

a

particular

minimum

level

of

enjoying

life

-LRB-

i.e.

having

at

least

a

score

of

m

on

the

example

item

-RRB-

and

the

latent

variable

of

psychological

well-being

-LRB-

θ

-RRB-

Figure

shows

an

example

of

the

ISRFs

of

two

items

having

four

answer

categories

each

-LRB-

i.e.

M

-RRB-

The

solid

lines

denote

the

ISRFs

for

one

item

and

the

dashed

lines

denote

the

ISRFs

for

the

other

item

Two

things

are

noteworthy

First

the

ISRFs

of

different

items

may

intersect

but

based

on

the

cumulative

character

of

the

definition

of

the

ISRFs

the

ISRFs

of

the

same

item

can

not

intersect

Second

the

ISRFs

have

been

drawn

as

monotone

curves

with

rather

irregular

shapes

Such

shapes

are

typical

of

a

nonparametric

GRM

-LSB-

-RSB-

and

found

not

to

prevent

a

set

of

items

from

having

favourable

measurement

properties

as

we

will

see

shortly

Figure

also

shows

monotone

ISRFs

but

now

having

smooth

S-shapes

typical

of

a

parametric

GRM

This

parametric

GRM

nearly

has

the

same

measurement

properties

as

the

nonparametric

GRM

but

because

it

assumes

smooth

ISRFs

-LRB-

Fig.

-RRB-

instead

of

irregular

ISRFs

-LRB-

Fig.

-RRB-

it

is

more

restrictive

and

often

leads

to

the

rejection

of

more

items

-LRB-

and

thus

shorter

scales

-RRB-

than

its

nonparametric

counterpart.Fig

of

item

step

response

functions

of

-LRB-

a

-RRB-

a

nonparametric

item

response

model

and

-LRB-

b

-RRB-

a

parametric

item

response

modelFig

of

-LRB-

a

-RRB-

flat

ISRFs

and

-LRB-

b

-RRB-

non-monotone

ISRFs

GRMs

have

the

next

three

assumptions

as

a

point

of

departure

The

first

assumption

is

unidimensionality

-LRB-

UD

-RRB-

that

is

each

model

assumes

that

one

latent

variable

θ

summarizes

the

variation

in

the

J

item

scores

in

the

questionnaire

Assumption

UD

implies

that

respondents

can

be

ordered

meaningfully

by

means

of

a

single

number

The

second

assumption

is

local

independence

-LRB-

LI

-RRB-

that

is

if

we

condition

on

θ

the

J

item

scores

are

statistically

independent

An

implication

of

LI

is

that

in

a

subgroup

of

patients

who

have

the

same

θ

value

all

covariances

between

item

scores

are

The

third

assumption

is

monotonicity

-LRB-

M

-RRB-

that

is

the

ISRFs

are

each

assumed

to

be

monotone

increasing

functions

in

θ

-LRB-

see

Figs.

and

b

-RRB-

Applied

to

the

latent

variable

of

psychological

well-being

-LRB-

θ

-RRB-

and

the

items

from

the

WHOQOL-Bref

assuming

UD

LI

and

M

means

that

we

hypothesize

that

-LRB-

-RRB-

only

psychological

well-being

drives

responses

to

items

and

has

a

systematic

effect

on

individual

differences

in

item

scores

and

total

scores

-LRB-

UD

-RRB-

-LRB-

-RRB-

given

a

fixed

level

of

psychological

well-being

relationships

between

concrete

aspects

of

psychological

well-being

such

as

represented

by

the

items

How

much

do

you

enjoy

life

and

Are

you

able

to

concentrate

are

explained

completely

-LRB-

i.e.

the

covariance

between

these

items

conditional

on

θ

equals

-RRB-

-LRB-

LI

-RRB-

and

-LRB-

-RRB-

the

higher

the

level

of

psychological

well-being

the

higher

the

probability

that

one

enjoys

life

and

is

able

to

concentrate

Parametric

and

nonparametric

graded

response

models

Parametric

Graded

Response

Model

The

ISRFs

of

the

parametric

GRM

-LSB-

-RSB-

are

defined

by

logistic

functions

that

have

the

following

parameters

δjm

the

location

parameter

of

the

mth

ISRF

of

item

j

-LRB-

i.e.

Pjm

-LRB-

θ

-RRB-

-RRB-

on

the

scale

of

θ

αj

the

slope

parameter

or

discrimination

power

of

item

j

The

meaning

of

these

parameters

is

explained

after

the

ISRF

of

the

parametric

GRM

is

introduced

This

ISRF

is

defined

as

Figure

shows

the

logistic

ISRFs

of

two

items

-LRB-

M

-RRB-

The

two

items

are

denoted

by

j

-LRB-

solid

ISRFs

-RRB-

and

k

-LRB-

dashed

ISRFs

-RRB-

For

each

item

the

three

location

parameters

are

also

shown

For

item

j

by

definition

we

have

that

and

for

item

k

by

definition

An

ISRF

's

location

parameter

is

the

value

of

θ

for

which

the

probability

of

having

an

item

score

of

at

least

m

equals

that

is

Pjm

-LRB-

θ

-RRB-

Pkm

-LRB-

θ

-RRB-

m

M.

Figure

also

shows

that

the

slopes

of

the

ISRFs

of

the

same

item

are

equal

-LRB-

mathematically

they

must

be

equal

or

the

ISRFs

would

intersect

this

is

impossible

given

the

cumulative

definition

of

the

ISRFs

-RRB-

but

also

that

the

ISRFs

of

item

j

are

steeper

than

the

ISRFs

of

item

k.

Steepness

of

slopes

is

evaluated

as

follows

For

ISRF

m

of

item

j

consider

the

point

with

coordinates

-LRB-

δjm

-RRB-

This

is

the

point

in

which

the

slope

of

a

logistic

ISRF

is

steepest

and

this

steepest

slope

is

taken

to

be

typical

for

the

whole

ISRF

Parameter

αj

expresses

this

maximum

steepness

-LRB-

but

is

not

exactly

equal

to

it

-RRB-

In

the

example

in

Fig.

we

have

that

αj

αk

Nonparametric

Graded

Response

Model

Instead

of

choosing

a

parametric

function

nonparametric

IRT

models

typically

define

order

restrictions

on

the

ISRFs

The

nonparametric

GRM

better

known

as

the

monotone

homogeneity

model

-LRB-

MHM

-RRB-

for

polytomous

items

-LSB-

-RSB-

assumes

UD

LI

and

M

that

is

for

any

pair

of

θs

say

θa

and

θb

the

MHM

assumes

that

Thus

the

ISRF

is

monotone

non-decreasing

in

θ

see

Fig.

for

examples

of

ISRFs

that

are

monotone

but

not

logistic

This

assumption

says

that

a

higher

level

of

psychological

well-being

induces

a

higher

probability

of

obtaining

at

least

an

item

score

of

m

-LRB-

i.e.

a

higher

item

score

-RRB-

The

ISRFs

of

different

items

can

have

any

monotone

form

and

be

very

different

Requiring

monotone

ISRFs

only

is

less

restrictive

than

requiring

monotone

logistic

ISRFs

thus

the

MHM

is

a

more

general

model

for

describing

the

data

than

the

GRM

-LRB-

henceforth

we

call

the

nonparametric

GRM

by

its

better

known

name

-LRB-

in

fact

acronym

-RRB-

MHM

and

the

parametric

GRM

simply

the

GRM

-RRB-

Unlike

the

GRM

the

MHM

does

not

provide

numerical

estimates

of

the

latent

variable

θ

Instead

the

MHM

allows

that

total

score

X

orders

patients

stochastically

on

latent

variable

θ

in

almost

all

practical

measurement

situations

-LSB-

-RSB-

This

means

that

for

two

total

scores

X

denoted

v

and

w

-LSB-

-RSB-

This

inequality

says

that

as

the

total

score

increases

the

mean

θ

also

increases

-LRB-

or

stays

the

same

-RRB-

Thus

groups

of

patients

that

have

higher

total

scores

on

average

also

have

higher

latent

variable

values

This

result

may

not

seem

spectacular

at

first

sight

but

it

-LRB-

-RRB-

ascertains

an

ordinal

scale

for

patient

measurement

-LRB-

-RRB-

using

only

observable

total

scores

-LRB-

without

requiring

the

actual

estimation

of

θ

-RRB-

For

the

psychological

well-being

example

if

the

MHM

fits

the

data

ordering

patients

by

means

of

the

total

score

by

implication

orders

them

on

the

latent

variable

θ

Also

the

MHM

does

not

provide

numerical

estimates

of

the

item

parameters

δ

and

α

Instead

a

distinction

can

be

made

between

drawing

information

about

item

functioning

from

estimates

of

the

complete

ISRFs

and

item

parameters

typical

of

the

MHM

Estimates

of

the

complete

ISRFs

provide

much

information

about

the

exact

relationship

between

the

item

scores

and

the

latent

variable

-LSB-

-RSB-

ISRFs

that

are

relatively

flat

or

fail

to

be

monotone

can

be

studied

in

much

detail

so

as

to

reveal

why

they

dysfunction

Figure

shows

three

relatively

flat

ISRFs

of

a

hypothetical

item

-LRB-

M

-RRB-

This

item

does

not

distinguish

low

θ

and

high

θ

patients

well

It

may

be

noted

that

the

ISRFs

do

not

all

need

to

be

flat

simultaneously

but

given

that

they

can

not

intersect

if

one

ISRF

is

flat

others

are

likely

to

be

relatively

flat

as

well

and

the

item

as

a

whole

contributes

little

to

the

reliable

ordering

of

patients

on

θ

Figure

shows

three

non-monotone

ISRFs

of

another

hypothetical

item

Each

shows

relative

good

distinction

between

low

θ

values

and

between

above-average

θ

values

but

bad

distinction

just

below

the

middle

of

the

scale

Again

non-intersection

of

the

ISRFs

of

the

same

item

implies

that

often

several

ISRFs

simultaneously

show

such

disturbing

non-monotonicities

For

the

example

item

one

may

conclude

that

when

the

questionnaire

contains

few

items

which

are

effective

in

the

high

θ

area

this

item

may

be

retained

to

cover

this

area

even

though

this

would

be

at

the

expense

of

measurement

quality

just

below

the

middle

of

the

scale

For

each

item

the

MHM

framework

provides

M

location

parameters

and

a

scalability

coefficient

which

provides

information

about

item

discrimination

The

location

parameters

are

the

proportions

of

the

population

of

interest

which

have

at

least

a

score

m

on

item

j

and

which

are

denoted

by

πjm

m

M.

For

the

same

item

due

to

non-intersection

of

ISRFs

we

have

that

πjM

whereas

in

the

GRM

item

location

parameters

are

ordered

oppositely

δjM

The

item

scalability

coefficient

Hj

-LRB-

e.g.

-LSB-

-RSB-

pp.

-LSB-

-RSB-

chap

-RRB-

summarizes

the

discrimination

power

of

an

item

across

its

M

ISRFs

The

numerical

Hj

value

is

determined

by

the

interplay

of

the

slope

and

the

location

of

the

ISRFs

of

all

J

items

and

the

distribution

of

the

latent

variable

θ

-LSB-

-RSB-

and

it

expresses

how

well

item

j

separates

patients

given

the

ISRFs

of

item

j

relative

to

the

other

items

ISRFs

and

the

distribution

of

θ

Mathematically

holding

constant

this

distribution

and

the

location

of

all

items

ISRFs

coefficient

Hj

is

higher

when

the

slopes

of

ISRFs

of

item

j

are

steeper

-LSB-

-RSB-

Nonparametric

versus

parametric

graded

response

models

Relationships

Among

Models

The

MHM

shares

assumptions

UD

LI

and

M

with

the

GRM

but

the

MHM

is

less

restrictive

with

respect

to

the

shape

of

the

ISRFs

Thus

the

MHM

is

more

general

than

the

GRM

or

equivalently

the

GRM

is

a

special

case

of

the

MHM

This

hierarchy

implies

that

if

the

GRM

fits

the

data

by

implication

the

more

general

MHM

also

fits

but

if

the

MHM

fits

the

data

this

does

not

imply

that

the

GRM

also

fits

Fit

of

the

GRM

then

needs

to

be

investigated

separately

Because

of

the

hierarchical

relationship

for

any

data

set

the

MHM

fits

as

least

as

many

items

as

the

GRM

-LRB-

Table

-RRB-

Table

of

monotone

homogeneity

model

-LRB-

MHM

-RRB-

and

graded

response

model

-LRB-

GRM

-RRB-

Nonparametric

IRT

-LRB-

MHM

-RRB-

Parametric

IRT

-LRB-

GRM

-RRB-

Restrictiveness

of

modelsLow

many

items

admitted

to

the

scaleHigh

fewer

items

admitted

to

the

scaleInterpretation

of

parametersIntuitively

appealingMore-complicatedParameters

-LRB-

typical

range

-RRB-

Person

levelT

X

θ

-LRB-

θ

-RRB-

ISRF

locationπjmδjm

-LRB-

δjm

-RRB-

ISRF

discriminationHjαj

-LRB-

αj

-RRB-

Data

analysisExploratory

data

as

point

of

departureConfirmatory

model

as

hypothesisApplicationsComparing

groupsMeasuring

changeDiagnosing

patientsComparing

groupsMeasuring

changeDiagnosing

patientsConstructing

item

banksAdaptive

testing

Patient

and

Item

Parameters

The

MHM

and

the

GRM

provide

the

following

patient

and

item

parameters

-LRB-

also

see

Table

-RRB-

For

patient

measurement

the

MHM

uses

total

score

X

to

order

patients

on

latent

variable

θ

Because

total

score

X

has

an

easy

interpretation

and

moreover

in

many

IRT

models

X

and

θ

tend

to

correlate

extremely

high

suggesting

a

strong

linear

relationship

-LSB-

-RSB-

total

score

X

may

be

preferred

in

practice

Total

score

X

is

the

sum

of

the

rating

scale

scores

on

the

J

items

whereas

estimates

of

θ

are

expressed

on

a

logit

scale

which

does

not

have

a

straightforward

interpretation

for

users

of

HRQoL

scales

In

general

the

ordinal

relationship

of

θ

with

total

score

X

-LRB-

which

can

be

approximated

well

by

a

linear

relation

-RRB-

enables

users

to

switch

between

scales

and

use

the

one

that

suits

their

goals

best.Because

item

location

δjm

is

expressed

on

the

same

scale

as

latent

variable

θ

it

also

has

an

interpretation

in

logits

For

many

users

proportion

πjm

the

proportion

of

patients

who

have

at

least

an

item

score

of

m

has

an

easier

interpretation.Item

discrimination

αj

gives

the

maximum

slope

of

the

logistic

ISRF

irrespective

of

the

locations

of

the

other

ISRFs

of

item

j

and

the

other

items

in

the

questionnaire

and

irrespective

of

the

distribution

of

θ

Thus

information

on

ISRF

slopes

is

absolute

in

the

sense

that

a

particular

αj

value

does

not

provide

information

on

the

item

's

suitability

for

measurement

in

a

particular

group

-LRB-

characterized

by

a

particular

distribution

of

θ

-RRB-

by

means

of

a

set

of

J

items

-LRB-

characterized

by

particular

location

and

slope

parameters

-RRB-

On

the

other

hand

item

scalability

coefficient

Hj

depends

explicitly

on

the

interplay

of

the

distribution

of

θ

the

spread

of

locations

of

the

ISRFs

and

the

slopes

of

the

ISRFs

In

particular

keeping

two

of

these

factors

fixed

Hj

tends

to

increase

in

the

third

This

dependence

on

the

distribution

of

θ

and

the

item

properties

informs

the

researcher

precisely

how

well

item

j

separates

patients

with

low

and

high

θ

values

in

the

particular

group

of

patients

under

consideration

using

the

particular

set

of

items

The

difference

between

absolute

slope

information

-LRB-

αj

-RRB-

and

relative

slope

information

-LRB-

Hj

-RRB-

is

illustrated

as

follows

Two

data

sets

of

size

N

and

five

items

-LRB-

J

-RRB-

-LRB-

M

for

each

item

-RRB-

were

generated

using

the

item

parameters

in

Table

The

first

data

set

came

from

a

hypothetical

clinical

population

with

θ

N

-LRB-

-RRB-

-LRB-

i.e.

low

psychological

well-being

level

-RRB-

and

the

second

data

set

came

from

a

hypothetical

healthy

population

with

θ

N

-LRB-

-RRB-

-LRB-

i.e.

high

psychological

well-being

level

-RRB-

It

may

be

noted

that

for

each

of

the

five

items

αj

by

definition

irrespective

of

ISRF

location

parameters

and

the

θ

distribution

The

Hj

values

in

the

clinical

group

were

computed

and

found

to

range

from

to

In

the

healthy

group

the

Hj

values

were

found

to

be

smaller

for

items

and

-LRB-

-RRB-

In

the

clinical

group

the

ISRFs

of

these

three

items

were

located

more

closely

to

the

middle

of

the

θ

distribution

-LRB-

Fig.

shows

this

for

Item

-RRB-

such

that

higher

Hj

values

resulted

but

in

the

healthy

group

the

ISRFs

were

located

further

in

the

lower

tail

of

the

θ

distribution

-LRB-

Fig.

-RRB-

resulting

in

lower

Hj

values

-LRB-

is

considered

minimally

acceptable

-LSB-

-RSB-

chap

-RRB-

Thus

the

items

and

are

well

suited

for

measurement

in

the

clinical

group

but

not

in

the

healthy

population

despite

their

overall

discrimination

power

expressed

by

αj

for

J

For

item

was

a

little

higher

in

the

healthy

group

because

the

second

and

third

ISRFs

discriminate

particularly

well

at

higher

ranges

of

θ

For

item

the

location

parameters

of

the

ISRFs

were

widely

spread

across

the

θ

distribution

resulting

in

good

discrimination

both

at

lower

and

higher

levels

of

θ

in

both

distributions

We

conclude

that

item

scalability

coefficient

Hj

has

the

advantage

that

it

takes

the

item

-LRB-

and

not

the

individual

ISRF

-RRB-

as

a

unit

and

depends

simultaneously

on

the

θ

distribution

the

slopes

of

the

ISRFs

and

the

spread

of

the

locations

of

the

ISRFs

Thus

Hj

informs

the

researcher

whether

item

j

discriminates

well

in

the

group

under

consideration

using

the

particular

set

of

items

whereas

item

discrimination

αj

provides

information

about

the

discrimination

power

irrespective

of

the

patient

group

under

consideration

and

the

item

properties

of

item

j

and

the

other

items

in

the

scale.Table

of

item

parameters

of

the

graded

response

model

-LRB-

GRM

-RRB-

and

Item

H

Values

in

two

different

populations

with

normally

distributed

latent

traitsItem

jItem

parameters

GRMHjα

N

-LRB-

-RRB-

θ

N

-LRB-

-RRB-

Note

Hj

values

are

based

on

simulated

data

for

sample

size

N

Fig.

ISRFs

of

the

same

item

-LRB-

Item

-RRB-

relative

to

two

different

distributions

of

the

latent

variable

Confirmatory

and

exploratory

data

analysis

In

general

before

IRT

models

are

accepted

as

reasonable

descriptions

of

the

data

their

goodness-of-fit

to

these

data

must

be

investigated

and

assessed

In

general

goodness-of-fit

research

is

different

for

parametric

and

nonparametric

IRT

-LRB-

with

the

GRM

and

the

MHM

as

special

cases

respectively

-RRB-

In

a

parametric

IRT

analysis

the

model

often

serves

as

null-hypothesis

and

it

is

tested

whether

this

null-hypothesis

must

be

rejected

or

may

be

supported

by

the

data

Nonparametric

IRT

analysis

in

general

takes

the

data

as

point

of

departure

and

-LRB-

-RRB-

instead

of

positing

a

unidimensional

or

multidimensional

latent

variable

structure

analyzes

the

data

to

find

its

true

dimensionality

and

-LRB-

-RRB-

instead

of

positing

a

logistic

or

other

functional

shape

estimates

the

ISRFs

from

the

data

so

as

to

diagnose

the

items

functioning

-LSB-

-RSB-

This

research

strategy

renders

nonparametric

IRT

a

more

flexible

data-analysis

tool

than

parametric

IRT

One

could

also

characterize

this

distinction

as

confirmatory

-LRB-

parametric

IRT

-RRB-

versus

exploratory

-LRB-

nonparametric

IRT

-RRB-

-LRB-

Table

-RRB-

Application

of

parametric

and

nonparametric

IRT

If

a

nonparametric

IRT

model

such

as

the

MHM

fits

the

data

the

result

is

a

scale

on

which

patients

can

be

ordered

by

means

of

the

total

score

X

This

total

score

has

a

strong

linear

correlation

with

latent

variable

θ

Such

a

scale

suffices

in

many

applications

Examples

are

the

comparison

of

groups

the

measurement

of

change

due

to

therapy

and

the

establishment

of

the

patient

's

psychological

well-being

level

as

low

medium

or

high

-LRB-

Table

-RRB-

The

practical

advantage

of

nonparametric

IRT

models

over

parametric

IRT

models

is

that

the

scales

they

produce

contain

more

items

thus

reducing

the

risk

of

wasting

items

that

have

non-logistic

but

monotone

ISRFs

that

discriminate

well

in

-LRB-

part

of

-RRB-

the

group

under

consideration

-LRB-

e.g.

Fig.

-RRB-

Such

items

contribute

well

to

reliable

measurement

In

addition

rejection

of

such

items

may

also

harm

the

coverage

of

the

latent

attribute

If

a

fitting

parametric

IRT

model

is

obtained

for

a

set

or

a

subset

of

the

items

one

has

a

parsimonious

description

of

the

item

characteristics

and

one

can

use

the

estimated

item

parameters

to

scale

the

items

and

the

estimated

θs

as

interval

level

measures

to

locate

patients

on

this

scale

If

a

large

set

of

items

also

known

as

an

item

bank

-LSB-

-RSB-

is

available

and

if

a

parametric

IRT

model

fits

the

item

bank

parametric

IRT

models

have

the

advantage

that

the

patient

's

θ

can

be

assessed

using

different

sets

of

items

from

the

item

bank

This

may

be

useful

for

the

measurement

of

change

when

change

is

so

large

that

the

set

of

items

that

was

used

initially

no

longer

captures

the

higher

or

lower

θ

levels

needed

for

the

second

measurement

thus

necessitating

the

use

of

other

items

Another

application

of

parametric

IRT

is

computerized

adaptive

testing

-LRB-

CAT

-RRB-

which

selects

items

that

match

the

patient

's

θ

level

well

from

a

huge

item

bank

so

as

to

optimize

accuracy

of

θ

measurement

In

principle

CAT

requires

different

item

sets

for

different

θ

values

-LRB-

Table

-RRB-

Nonparametric

IRT

analysis

in

practice

Software

for

nonparametric

IRT

analysis

Several

programs

are

available

for

data

analysis

using

nonparametric

IRT

but

not

each

can

handle

polytomous

item

scores

We

briefly

discuss

the

programs

MSP

-LRB-

-LSB-

-RSB-

also

see

-LSB-

-RSB-

-RRB-

and

-LSB-

-RSB-

Both

programs

are

used

regularly

and

together

they

provide

the

researcher

with

a

clear

and

informative

picture

of

-LRB-

-RRB-

the

dimensionality

of

the

data

-LRB-

-RRB-

the

-LRB-

lack

of

-RRB-

monotonicity

of

the

ISRFs

and

-LRB-

-RRB-

estimates

of

item

locations

and

item

discrimination

Program

MSP

uses

the

MHM

as

the

main

analysis

model

but

another

nonparametric

model

not

discussed

here

is

also

included

in

the

program

Basically

the

program

consists

of

two

parts

The

first

part

of

MSP

is

an

algorithm

for

exploring

the

dimensionality

of

the

data

-LRB-

-LSB-

-RSB-

chap

-LSB-

-RSB-

chap

-RRB-

The

algorithm

uses

item

scalability

coefficient

Hj

to

select

items

Items

that

are

related

to

the

same

latent

variable

θ

are

selected

one

by

one

on

the

basis

of

their

Hj

value

Suppose

that

one

latent

variable

drives

the

responses

to

one

subset

of

the

items

another

latent

variable

drives

the

responses

to

another

subset

of

items

and

so

on

Then

the

algorithm

selects

mutually

exclusive

clusters

of

items

each

of

which

is

driven

by

a

different

latent

variable

The

second

part

of

MSP

provides

several

statistical

tools

for

exploring

the

shape

of

the

ISRFs

-LSB-

-RSB-

This

is

most

useful

after

the

dimensionality

of

the

data

has

been

ascertained

For

example

due

to

their

strong

positive

tendency

the

non-monotone

ISRFs

in

Fig.

may

have

an

Hj

value

that

is

high

enough

for

the

items

to

be

selected

in

a

unidimensional

cluster

but

the

non-monotonicity

also

may

distort

parts

of

the

ordinal

scale

defined

by

the

items

in

the

cluster

MSP

estimates

ISRFs

by

means

of

a

number

of

discrete

points

that

are

connected

to

form

a

jagged

curve

Figure

shows

four

discrete

ISRFs

of

the

same

item

-LRB-

i.e.

M

-RRB-

each

estimated

by

means

of

eleven

points

The

researcher

can

manipulate

the

number

of

estimated

points

If

more

points

are

estimated

from

the

same

data

more

details

of

the

ISRF

become

visible

-LRB-

i.e.

bias

is

reduced

-RRB-

but

because

for

each

point

fewer

data

are

available

accuracy

decreases

In

statistics

this

is

known

as

the

bias-accuracy

trade-off

and

it

is

advisable

to

try

several

options

to

reach

a

good

decision

MSP

tests

observable

deviations

from

monotonicity

for

significance

Figure

also

shows

a

mean

ISRF

-LRB-

dashed

curve

-RRB-

which

we

may

call

the

item

score

function

-LRB-

ISF

-RRB-

and

which

is

not

standard

output

of

MSP

This

function

must

be

monotone

nondecreasing.Fig

-LRB-

a

-RRB-

Discrete

estimates

of

four

ISRFs

-LRB-

solid

curves

from

MSP

-RRB-

of

Item

-LRB-

Satisfied

doing

daily

activities

from

Physical

Health

and

Well

Being

domain

-RRB-

and

the

corresponding

discrete

item-score

function

-LRB-

dashed

curve

-RRB-

-LRB-

b

-RRB-

also

estimated

as

a

continuous

curve

-LRB-

from

-RRB-

Program

-LSB-

-RSB-

can

be

used

for

studying

the

shape

of

the

ISF

Unlike

MSP

produces

continuous

estimates

of

response

functions

-LRB-

-LSB-

-RSB-

here

only

the

ISFs

-RRB-

and

like

MSP

shows

graphical

displays

of

these

estimates

that

can

be

manipulated

with

respect

to

bias

and

accuracy

and

also

here

it

is

advisable

to

try

several

options

The

quality

of

the

decision

can

be

improved

by

using

the

confidence

envelopes

for

the

estimated

ISFs

for

statistical

testing

Figure

shows

an

example

of

an

estimated

ISF

-LRB-

solid

curve

-RRB-

and

its

confidence

envelopes

-LRB-

dashed

curves

-RRB-

which

were

estimated

by

means

of

Research

strategies

for

nonparametric

IRT

analysis

MSP

provides

a

method

for

investigating

the

dimensionality

of

the

data

and

MSP

and

both

can

be

used

to

investigate

assumption

M

The

investigation

of

dimensionality

and

monotonicity

serves

to

identify

the

items

that

together

constitute

an

ordinal

patient

scale

for

the

same

latent

variable

For

investigating

dimensionality

MSP

offers

the

researcher

the

possibility

to

set

a

positive

lower

bound

c

on

Hj

Under

the

MHM

the

lowest

admissible

value

is

c

MSP

's

default

is

c

-LRB-

-LSB-

-RSB-

chap

-LSB-

-RSB-

-RRB-

This

default

value

ascertains

a

lower

bound

on

the

overall

discrimination

power

of

the

items

-LRB-

but

researchers

are

free

to

choose

a

higher

value

-RRB-

and

as

a

result

item

clusters

consist

only

of

sufficiently

discriminating

items

that

measure

the

same

latent

variable

Thus

MSP

aims

to

produce

unidimensional

scales

that

allow

accurate

patient

measurement

estimates

the

ISFs

-LRB-

e.g.

Fig.

-RRB-

by

means

of

the

nonparametric

regression

method

known

as

kernel

smoothing

-LRB-

e.g.

-LSB-

-RSB-

chap

-LSB-

-RSB-

-RRB-

The

availability

of

confidence

envelopes

for

the

continuous

ISF

estimates

provides

detailed

information

of

-LRB-

lack

of

-RRB-

monotonicity

for

each

item

provides

these

estimates

irrespective

of

the

dimensionality

of

the

data

Thus

a

good

research

strategy

is

to

first

investigate

item-set

dimensionality

by

means

of

MSP

and

then

use

MSP

and

to

study

the

ISRFs

and

the

ISFs

in

dimensionally

distinct

clusters

See

-LSB-

-RSB-

for

another

method

for

assessing

the

shape

of

these

curves

A

real-data

example

The

World

Health

Organization

Quality-of-Life

Scale

The

WHOQOL-Bref

was

developed

for

assessing

individuals

perception

and

feelings

of

their

daily

life

The

questionnaire

starts

with

two

items

which

ask

for

global

estimates

of

one

's

quality

of

life

and

then

continues

with

items

covering

four

domains

-LRB-

a

-RRB-

physical

health

and

well-being

-LRB-

seven

items

-RRB-

-LRB-

b

-RRB-

psychological

health

and

well-being

-LRB-

six

items

-RRB-

-LRB-

c

-RRB-

social

relations

-LRB-

three

items

-RRB-

and

-LRB-

d

-RRB-

environment

-LRB-

eight

items

-RRB-

The

two

general

items

were

left

out

of

the

analysis

In

agreement

with

their

numbering

in

the

WHOQOL-Bref

the

other

items

were

numbered

from

to

Examples

of

items

are

Do

you

have

enough

energy

for

daily

life

-LRB-

physical

domain

-RRB-

How

much

do

you

enjoy

life

-LRB-

psychological

domain

-RRB-

How

satisfied

are

you

with

your

personal

relationships

-LRB-

social

domain

-RRB-

How

safe

do

you

feel

in

your

daily

life

-LRB-

environmental

domain

-RRB-

Each

item

uses

a

five-point

rating

scale

-LRB-

i.e.

Xj

-RRB-

the

higher

the

item

score

the

better

one

's

quality

of

life

on

the

specific

domain

covered

by

the

item

The

data

were

collected

by

undergraduate

psychology

students

of

Tilburg

University

as

part

of

a

course

Research

Practical

in

the

academic

year

Students

were

instructed

to

strive

for

a

sample

of

participants

equally

distributed

across

both

sexes

and

the

following

age

categories

and

more

than

years

The

final

sample

consisted

of

N

respondents

from

the

Dutch

population

Of

these

respondents

were

women

mean

age

was

years

-LRB-

SD

-RRB-

had

completed

community

college

or

university

had

completed

vocational

school

had

high

school

at

most

and

had

only

elementary

school

or

less

N

cases

had

missing

item

scores

Missing

values

were

estimated

using

two-way

imputation

Comparable

to

an

analysis-of-variance

layout

this

method

uses

both

a

person

effect

and

an

item

effect

for

estimating

a

missing

score

-LRB-

for

details

see

-LSB-

-RSB-

-RRB-

MSP

and

were

used

to

analyze

these

data

and

construct

one

or

more

scales

thus

illustrating

the

possibilities

of

the

MHM

For

the

sake

of

comparison

we

ran

a

principal

component

analysis

and

a

GRM

scale

analysis

on

the

data

Results

Sample

statistics

of

item

and

scale

scores

Table

shows

that

the

mean

item

scores

ranged

from

-LRB-

Item

Satisfied

with

sex

life

-RRB-

to

-LRB-

Item

Moving

around

well

-RRB-

The

mean

X

scores

were

-LRB-

physical

domain

-RRB-

-LRB-

psychological

domain

-RRB-

-LRB-

social

domain

-RRB-

and

-LRB-

environmental

domain

-RRB-

Correlations

between

the

domain

scores

ranged

from

-LRB-

between

physical

and

social

domains

-RRB-

to

-LRB-

between

physical

and

environmental

domains

-RRB-

Table

from

MSP

item

selection

procedure

-LRB-

Item

clusters

item

Hj

values

and

total

H

-RRB-

and

Item

Hj

values

and

total

H

for

each

content

domainMSP

item

selection

procedureHj

per

content

domainc

c

health

and

due

to

Experiencing

Satisfied

with

Moving

around

Satisfied

doing

daily

Need

medical

treatment

for

daily

Satisfied

work

Scale

Psychological

health

and

Being

able

to

Satisfied

with

Acceptance

physical

Experiencing

negative

Life

Scale

Social

relationship

with

other

Satisfied

with

sex

Satisfied

support

from

Scale

safe

in

daily

Satisfied

living

Enough

financial

Satisfied

getting

adequate

health

Availability

information

needed

in

daily

Opportunities

Healthy

Satisfied

with

transport

in

daily

Scale

aReversely

scored

items

Dimensionality

analysis

Principal

components

analysis

Dimensionality

was

explored

by

means

of

a

principal

components

analysis

using

polychoric

correlations

The

ratio

of

the

first

to

the

second

eigenvalue

of

the

polychoric

correlation

matrix

was

A

ratio

of

is

taken

as

evidence

of

considerable

strength

of

the

first

dimension

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

The

first

factor

explained

of

the

variance

A

confirmatory

factor

analysis

of

the

four

a

priori

domain

scales

of

the

WHOQOL-bref

improved

fit

over

the

one-factor

model

-LRB-

P

-RRB-

However

the

factors

correlated

from

-LRB-

physical

and

social

domain

-RRB-

to

-LRB-

psychological

and

social

domain

-RRB-

The

explorative

factor

analysis

in

conjunction

with

the

confirmative

factor

analysis

justifies

the

assumption

of

a

general

HRQoL

dimension

underlying

each

scale

Monotone

homogeneity

model

analysis

Next

MSP

was

used

treating

all

items

as

a

fixed

scale

The

MHM

does

not

allow

negatively

correlating

items

in

one

scale

Item

-LRB-

Need

medical

treatment

for

daily

functioning

-RRB-

and

Item

-LRB-

Satisfied

with

sex

life

-RRB-

correlated

negatively

but

not

significantly

-LRB-

P

-RRB-

thus

all

items

were

used

for

analysis

The

item

Hj

values

-LRB-

not

tabulated

-RRB-

ranged

from

-LRB-

Item

Satisfied

with

sleep

-RRB-

to

-LRB-

Item

Satisfied

with

living

conditions

and

Item

Enough

energy

for

everyday

life

-RRB-

The

total-scale

H

coefficient

was

equal

to

The

results

suggest

that

the

items

tend

to

cover

one

latent

HRQoL

aspect

which

however

induces

only

weak

general

association

between

the

items

Thus

in

addition

to

this

common

aspect

it

seems

reasonable

to

also

look

for

more-specific

HRQoL

aspects

that

are

covered

by

subsets

of

items

Dimensionality

was

investigated

by

means

of

the

MSP

search

algorithm

using

several

c

values

starting

with

-LRB-

default

-RRB-

and

then

increasing

c

with

steps

of

in

each

next

analysis

round

We

only

report

results

for

c

and

c

-LRB-

other

values

did

not

reveal

interesting

results

-RRB-

For

c

one

scale

consisting

of

items

-LRB-

H

-RRB-

and

one

scale

consisting

of

items

-LRB-

H

-RRB-

were

found

-LRB-

Table

-RRB-

The

four

remaining

items

were

not

selected

because

their

Hj

values

were

under

-LRB-

i.e.

and

and

-RRB-

The

scale

had

a

rather

heterogeneous

content

The

scale

asked

about

distraction

due

to

pain

-LRB-

Item

-RRB-

and

the

need

for

medical

treatment

for

daily

functioning

-LRB-

Item

-RRB-

Thus

their

high

scalability

may

be

explained

by

the

use

of

palliative

medicines

For

c

five

scales

were

found

consisting

of

and

items

respectively

-LRB-

Table

-RRB-

The

first

scale

consisted

of

items

from

the

physical

domain

and

the

psychological

domain

The

other

scales

consisted

of

items

from

one

domain

Scale

again

covered

Item

and

Item

Scales

and

covered

environmental-domain

aspects

Scale

contained

all

social-domain

items

Thus

for

default

c

of

the

items

were

selected

in

one

scale

The

pattern

of

item

selection

for

higher

c

values

such

as

c

showed

that

the

item

set

progressively

crumbled

into

many

smaller

scales

while

other

items

remained

unselected

Sijtsma

and

Molenaar

-LRB-

-LSB-

-RSB-

pp.

see

also

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

argued

that

this

typical

pattern

of

results

gives

evidence

that

the

set

constitutes

a

unidimensional

scale

The

total-scale

H

equaled

giving

evidence

of

weak

scalability

-LRB-

-LSB-

-RSB-

p.

-RRB-

Most

of

the

item

Hj

values

were

between

and

also

suggesting

a

weak

relationship

with

the

latent

variable

-LRB-

-LSB-

-RSB-

p.

-RRB-

Finally

it

was

investigated

whether

the

four

a

priori

identified

item

domains

could

be

considered

as

separate

scales

Table

-LRB-

last

column

-RRB-

shows

that

the

total-scale

H

values

ranged

from

-LRB-

environmental

domain

-RRB-

to

-LRB-

social

domain

-RRB-

Thus

based

on

Mokken

's

classification

of

scales

-LSB-

-RSB-

the

four

a

priori

item

domains

constituted

weak

to

medium

scales

Two

items

had

H

values

just

smaller

than

c

-LRB-

i.e.

Item

-LRB-

-RRB-

Satisfied

with

sleep

and

Item

-LRB-

-RRB-

Satisfied

with

yourself

-RRB-

The

content

domains

may

be

considered

as

unidimensional

clusters

of

items

measuring

distinct

aspects

of

HRQoL

each

of

which

are

related

to

a

more

general

underlying

HRQoL

construct

Because

of

their

conceptual

clarity

the

remaining

analyses

were

done

on

the

a

priori

defined

item

domains

Monotonicity

assessment

in

each

item

domain

For

each

item

domain

MSP

was

used

to

assess

the

ISRFs

shapes

First

ISRFs

were

estimated

accurately

-LRB-

i.e.

many

cases

were

used

to

estimate

separate

points

of

the

ISRFs

-RRB-

but

at

the

expense

of

possible

bias

-LRB-

i.e.

only

few

points

were

estimated

-RRB-

Second

ISRFs

were

estimated

with

little

bias

-LRB-

i.e.

many

points

were

estimated

-RRB-

but

at

the

expense

of

accuracy

-LRB-

i.e.

few

cases

were

used

to

estimate

each

point

-RRB-

For

the

physical

domain

the

first

analysis

-LRB-

high

accuracy

more

risk

of

bias

-RRB-

revealed

four

items

of

which

one

or

more

ISRFs

showed

minor

violations

of

monotonicity

but

of

these

violations

were

significant

-LRB-

level

one-tailed

test

because

only

sample

decreases

are

tested

as

violations

increases

support

monotonicity

-RRB-

The

second

analysis

-LRB-

more

inaccuracy

less

bias

-RRB-

revealed

that

for

all

seven

items

one

or

more

ISRFs

showed

one

or

more

local

decreases

but

them

were

significant

Figure

shows

the

local

nonsignificant

decreases

in

the

ISRFs

for

Item

-LRB-

Distraction

due

to

pain

-RRB-

Fig.

ISRFs

of

Item

-LRB-

Distraction

due

to

pain

from

Physical

Health

and

Well

Being

domain

-RRB-

showing

nonsignificant

violations

of

assumption

M

and

rejected

by

the

GRM

-LRB-

a

-RRB-

Results

from

MSP

-LRB-

including

the

ISF

-RRB-

-LRB-

b

-RRB-

results

from

-LRB-

ISF

and

confidence

envelopes

-RRB-

-LRB-

c

-RRB-

and

results

from

-LRB-

GRM

-RRB-

For

the

psychological

domain

for

both

analyses

-LRB-

i.e.

high

accuracy

versus

little

bias

-RRB-

two

items

were

found

which

had

ISRFs

showing

significant

local

decreases

For

example

Fig.

shows

for

Item

-LRB-

Being

able

to

concentrate

-RRB-

that

the

estimate

of

ISRF

-LRB-

θ

-RRB-

P

-LRB-

θ

-RRB-

-LRB-

the

second

curve

from

the

top

-RRB-

has

several

local

decreases

the

largest

of

which

was

significant

-LRB-

significance

level

P

-RRB-

The

estimate

of

ISRF

-LRB-

θ

-RRB-

P

-LRB-

θ

-RRB-

shows

two

small

decreases

they

were

not

significant

For

the

social

domain

and

the

environmental

domain

no

significant

violations

of

the

monotonicity

assumption

were

found

It

can

be

concluded

that

assumption

M

holds

for

each

of

the

four

scales.Fig

ISRFs

of

Item

-LRB-

Being

able

to

concentrate

-RRB-

showing

significant

violations

of

assumption

M

and

rejected

by

the

GRM

-LRB-

a

-RRB-

Results

from

MSP

-LRB-

including

the

ISF

-RRB-

-LRB-

b

-RRB-

results

from

-LRB-

ISF

and

confidence

envelopes

-RRB-

and

-LRB-

c

-RRB-

results

from

-LRB-

GRM

-RRB-

Next

was

used

to

investigate

assumption

M

for

the

ISFs

Several

sample

sizes

were

used

for

estimating

curve

fragments

of

the

ISFs

and

balancing

the

bias-accuracy

trade-off

Figure

shows

the

estimated

ISF

of

Item

-LRB-

Distraction

due

to

pain

-RRB-

The

confidence

envelopes

show

that

the

local

decrease

of

the

estimated

ISF

can

be

ignored

safely

For

the

estimated

ISF

of

Item

-LRB-

Being

able

to

concentrate

-RRB-

Fig.

suggests

a

violation

of

assumption

M

for

high

latent

variable

levels

-LRB-

i.e.

θ

-RRB-

With

MSP

the

user

specifies

the

minimum

number

of

observations

used

for

estimating

each

point

of

an

ISRF

but

controls

the

bias-accuracy

trade-off

by

means

of

a

bandwidth

parameter

also

to

be

specified

by

the

user

but

without

being

able

to

control

the

number

of

observations

for

estimating

separate

curve

fragments

The

effect

may

be

that

in

particular

at

the

lower

end

and

higher

end

of

the

scale

the

ISF

is

estimated

very

inaccurately

Combining

the

results

from

MSP

-LRB-

no

significant

decreases

of

the

ISRFs

at

the

higher

end

of

the

θ

scale

-RRB-

and

-LRB-

a

smooth

monotone

increasing

ISF

in

the

middle

of

the

θ

scale

-RRB-

we

may

conclude

that

assumption

M

holds

for

item

scores

Thus

the

expected

item

score

monotonically

increases

in

the

latent

variable

Comparison

of

the

MHM

with

the

GRM

The

GRM

was

fitted

using

-LSB-

-RSB-

Estimation

problems

occurred

for

item

scoring

because

some

score

categories

were

-LRB-

almost

-RRB-

empty

This

was

resolved

by

combining

scores

and

into

score

and

re-scoring

the

items

Table

provides

the

estimated

slope

parameters

-LRB-

αj

-RRB-

and

the

three

location

parameters

-LRB-

δjm

j

-RRB-

and

also

the

estimated

item

Hj

coefficients

-LRB-

also

in

Table

last

column

-RRB-

and

the

nonparametric

location

parameters

-LRB-

πjm

j

-RRB-

The

location

parameters

indicate

that

the

items

are

relatively

popular

-LRB-

highly

endorsed

-RRB-

Table

of

monotone

homogeneity

model

-LRB-

MHM

-RRB-

scale

analysis

and

estimated

item

parameters

from

the

graded

response

model

-LRB-

GRM

-RRB-

MHMGRM

health

and

due

to

painb

Experiencing

energya

Satisfied

with

sleep

Moving

around

well

Satisfied

doing

daily

activities

Need

medical

treatment

for

daily

functioninga

Satisfied

work

capacity

Scale

value

Psychological

health

and

life

Being

able

to

concentrate

Satisfied

with

yourself

Acceptance

physical

appearance

Experiencing

negative

feelingsa

Life

meaningful

Scale

value

Social

relationship

with

other

people

Satisfied

with

sex

life

Satisfied

support

from

others

Scale

value

safe

in

daily

life

Satisfied

living

conditions

Enough

financial

resources

Satisfied

getting

adequate

health

care

Availability

information

needed

in

daily

life

Opportunities

leisure

Healthy

environment

Satisfied

with

transport

in

daily

life

Scale

value

a

Reversely

scored

items

b

For

the

GRM

analysis

items

were

recoded

by

collapsing

item

scores

and

into

item

score

Several

methods

are

available

for

assessing

the

fit

of

the

GRM

but

many

are

problematic

and

no

generally

accepted

standard

goodness-of-fit

method

for

the

GRM

is

presently

available

-LRB-

-LSB-

-RSB-

pp.

-RRB-

We

investigated

goodness-of-fit

of

the

GRM

by

means

of

posterior

predictive

assessment

-LSB-

-RSB-

which

can

provide

graphical

and

numerical

evidence

about

model

fit

Model

fit

was

investigated

at

the

level

of

items

The

most

interesting

results

were

found

for

items

from

the

physical

and

psychological

domains

Results

are

only

discussed

for

these

domains

For

the

physical

domain

Item

-LRB-

Distraction

due

to

pain

-RRB-

and

Item

-LRB-

Being

able

to

concentrate

-RRB-

showed

significant

misfit

-LRB-

P

-RRB-

In

Fig.

the

curve

made

up

by

dots

connected

by

straight

line

pieces

represents

the

estimated

nonparametric

ISF

of

Item

the

solid

curve

represents

the

expected

ISF

of

Item

under

the

GRM

and

the

dotted

curves

represent

the

confidence

envelopes

The

GRM

rejects

this

item

Thus

modeling

the

jagged

pattern

of

the

estimated

ISF

by

means

of

logistic

ISRFs

having

the

same

slopes

would

do

injustice

to

the

data

However

it

is

noteworthy

that

Item

has

good

measurement

properties

-LRB-

Table

e.g.

-RRB-

under

the

more

general

MHM

and

from

this

model

's

perspective

it

might

be

retained

in

the

scale

Item

-LRB-

Being

able

to

concentrate

-RRB-

was

a

popular

item

-LRB-

Table

also

see

Fig.

upper

two

ISRFs

-RRB-

As

a

result

the

GRM

could

not

be

estimated

accurately

item

parameters

were

estimated

very

inaccurately

-LRB-

standard

errors

-RRB-

Figure

shows

evidence

of

misfit

at

θ

for

which

the

observed

ISF

fell

outside

the

confidence

interval

This

means

that

the

GRM

gives

biased

results

for

θ

The

nonparametric

estimates

of

the

ISRFs

were

monotone

This

provides

evidence

that

the

MHM

adequately

fitted

Item

However

which

is

rather

low

A

reason

to

keep

this

item

in

the

scale

is

that

it

may

help

measuring

differences

at

the

lower

and

middle

ranges

of

the

θ

scale

which

are

the

most

relevant

ranges

for

measuring

HRQoL

Summary

of

the

scale

properties

The

WHOQOL-Bref

is

most

often

used

in

scientific

research

-LRB-

e.g.

epidemiological

studies

and

clinical

trails

-RRB-

and

by

health

professionals

-LRB-

e.g.

to

assess

treatment

efficacy

-RRB-

-LSB-

-RSB-

The

nonparametric

MHM

analyses

revealed

that

the

scales

have

adequate

properties

for

comparing

groups

on

the

underlying

HRQoL

aspects

Each

of

the

four

domains

of

the

WHOQOL-Bref

constitutes

a

unidimensional

scale

each

scale

measuring

a

different

aspect

of

HRQoL

in

addition

to

a

weak

common

HRQoL

attribute

This

justifies

reporting

both

separate

domain

scores

and

possibly

an

overall

HRQoL

score

The

scalability

results

showed

that

the

domain

scales

are

weak

to

moderate

with

scalability

coefficients

H

ranging

from

to

The

test-score

reliabilities

of

the

four

domain

scores

were

and

respectively

The

rank

correlations

between

sum

score

X

and

the

estimated

θ

from

the

GRM

varied

from

-LRB-

physical

health

-RRB-

to

-LRB-

environment

-RRB-

-LRB-

Pearson

correlations

ranged

from

-LRB-

physical

health

-RRB-

to

-LRB-

environment

-RRB-

-RRB-

Thus

X

and

estimated

θ

carry

nearly

the

same

rank

order

-LRB-

and

numerical

-RRB-

information

This

interesting

result

further

justifies

the

use

of

the

nonparametric

MHM

for

scale

analysis

and

the

use

of

X

for

-LRB-

at

least

-RRB-

ordinal

measurement

of

persons

Because

the

item-score

distributions

were

severely

skewed

to

the

left

the

lower

response

categories

and

were

ineffective

for

HRQoL

measurement

in

the

general

population

The

locations

of

the

ISRFs

for

the

higher

response

categories

and

were

well

spread

along

the

θ

scale

The

ISRFs

discrimination

power

as

reflected

by

the

Hj

values

often

was

in

the

weak

to

medium

range

Thus

the

higher

response

categories

are

modestly

informative

across

a

wide

range

on

the

θ

scale

The

relatively

short

WHOQOL-Bref

may

also

be

considered

for

use

as

a

tool

for

assessing

HRQoL

at

the

individual

level

in

clinical

and

medical

settings

For

example

the

WHOQOL-Bref

may

be

used

to

evaluate

whether

a

patient

's

HRQoL

has

improved

after

taking

medication

An

interesting

feature

of

a

fitting

IRT

model

is

that

psychometric

properties

can

be

evaluated

conditionally

on

the

latent

variable

For

example

the

measurement

error

of

X

can

be

evaluated

at

different

values

of

the

latent

variable

provides

graphical

information

about

the

standard

error

of

measurement

based

on

the

MHM

model

For

example

for

the

physical-health

domain

estimated

a

standard

error

of

measurement

ranging

from

for

X

to

for

X

Thus

to

be

significant

at

the

significance

level

differences

between

two

observed

X

scores

have

to

be

larger

than

for

X

and

larger

than

for

X

-LRB-

e.g.

see

-LSB-

-RSB-

p.

-RRB-

This

is

a

substantial

standard

error

of

measurement

relative

to

the

length

of

scale

This

relatively

large

measurement

error

appears

to

be

consistent

with

the

observed

Hj

values

which

indicate

weak

to

moderate

scalability

For

the

other

three

content

domains

the

standard

error

of

measurement

was

also

substantial

Thus

caution

has

to

be

exercised

when

drawing

conclusions

about

differences

and

changes

in

individual

levels

of

HRQoL

based

on

observed

X

scores

from

the

WHOQOL-Bref

and

any

other

HRQOL

measure

see

-LSB-

-RSB-

Discussion

This

study

explained

how

the

nonparametric

monotone

homogeneity

model

contributes

to

the

construction

of

scales

for

the

measurement

of

HRQoL

The

MHM

is

more

general

than

parametric

IRT

models

-LSB-

-RSB-

such

as

the

much-used

parametric

graded

response

model

-LSB-

-RSB-

but

also

the

partial

credit

model

-LSB-

-RSB-

and

the

generalized

partial

credit

model

-LSB-

-RSB-

Hemker

et

al.

-LSB-

-RSB-

showed

that

all

known

parametric

IRT

models

for

polytomous

items

are

special

cases

of

the

nonparametric

MHM

This

means

that

any

item

set

satisfying

the

requirements

of

a

parametric

IRT

model

for

polytomous

items

also

satisfies

the

requirements

of

the

nonparametric

MHM

Given

the

greater

generality

and

flexibility

of

the

nonparametric

MHM

which

results

in

longer

scales

and

because

X

and

estimated

θ

carry

the

same

rank

order

information

-LRB-

based

on

the

approximate

stochastic

ordering

property

of

θ

given

X

-RRB-

the

nonparametric

MHM

is

highly

suited

for

person

measurement

In

an

HRQoL

context

often

little

is

known

about

the

psychometric

properties

of

new

questionnaires

A

typical

nonparametric

MHM

analysis

explores

the

dimensionality

of

the

data

by

capitalizing

on

model

assumptions

such

as

monotonicity

-LRB-

MSP

-RRB-

and

studies

the

shapes

of

the

ISRFs

and

the

ISFs

in

order

to

learn

more

about

the

-LRB-

mal

-RRB-

functioning

of

individual

items

-LRB-

MSP

and

-RRB-

This

results

in

scales

on

which

groups

can

be

compared

and

changes

monitored

without

making

unduly

restrictive

assumptions

about

the

data

The

properties

of

any

IRT

model

only

hold

for

the

application

at

hand

when

the

model

fits

the

data

In

case

of

misfit

the

structure

of

the

model

does

not

match

the

structure

of

the

data

One

cause

of

misfit

is

that

the

data

are

multidimensional

while

the

model

assumes

unidimensionality

Another

cause

of

misfit

is

that

the

real

ISRFs

may

not

be

monotone

or

that

they

are

monotone

but

fail

to

have

the

logistic

shape

assumed

by

many

parametric

models

Other

causes

of

misfit

such

as

a

multiple-group

structure

as

in

differential

item

functioning

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

or

person

misfit

-LSB-

-RSB-

were

not

considered

here

When

the

MHM

fits

the

data

the

researcher

may

decide

to

also

investigate

goodness

of

fit

of

the

GRM

or

other

parametric

IRT

models

for

polytomous

items

The

choice

of

a

parametric

model

may

be

based

on

the

flexibility

of

the

model

For

example

the

partial

credit

model

only

has

item

location

parameters

but

assumes

the

slopes

of

the

response

functions

to

be

the

same

within

and

between

items

whereas

the

generalized

partial

credit

model

also

allows

for

varying

slope

parameters

between

items

just

as

the

GRM

If

one

pursues

a

parametric

IRT

analysis

misfit

may

be

a

good

reason

to

resort

to

a

nonparametric

IRT

model

and

still

have

an

ordinal

patient

scale

If

CAT

is

pursued

one

of

the

parametric

models

is

a

better

option

provided

the

model

fits

the

data

well

In

an

HRQoL

context

CAT

indeed

could

prove

to

be

successful

because

patients

have

a

definitive

interest

in

providing

truthful

answers

-LRB-

in

the

educational

context

in

which

CAT

originated

CAT

requires

that

items

be

kept

secret

This

requires

item

banks

often

containing

hundreds

of

calibrated

items

-RRB-

As

a

result

in

HRQoL

measurement

CAT

presently

meets

with

a

growing

interest

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

nonparametric monotone homogeneity model