In
te
r
n
ation
a
l Jou
rn
al
o
f E
v
al
u
a
t
i
on
a
n
d
R
e
se
arc
h
in
Ed
u
c
ation
(
IJERE
)
V
o
l.7,
N
o.2,
Ju
n
e
20
1
8
,
pp. 142~
151
ISSN
: 2252-
88
22,
D
O
I
:
10.11
59
1
/ijer
e
.
v
7
.
i
2
.12
9
0
0
142
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
ERE
Item Response Theory: An Introdu
ction to Late
n
t Trait M
o
dels
to Test and Item Development
Ad
o
A
bdu Bichi
1
,
R
o
h
aya T
a
lib
2
Measure
m
ent & Eval
uation,
Faculty
of
Educati
on,
Univers
iti
Tekn
olog
i
M
a
la
y
s
ia
,
Ma
la
y
s
ia
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
A
pr 23,
2
0
1
8
Re
vise
d Ma
y
2
7
, 201
8
Ac
ce
p
t
ed
M
a
y
3
1
,
2
018
Tes
t
i
ng
in
e
d
u
cat
io
nal
s
y
stem
p
erfo
rm
a
n
u
m
ber
of
f
u
n
ct
ions,
th
e
resul
t
s
f
r
o
m
a
t
est
can
b
e
us
ed
t
o
m
a
k
e
a
n
um
ber
of
d
eci
sio
n
s
i
n
e
duca
t
ion.
I
t
i
s
th
eref
ore
w
e
ll accepted
i
n
t
he
e
d
u
cat
io
n
lit
era
t
ure
th
a
t
,
t
e
sti
ng
i
s
a
n
i
mp
or
ta
n
t
elem
en
t
o
f
e
du
c
a
t
i
o
n
.
To
e
ffect
iv
ely
ut
ilize
t
h
e
t
e
sts
in
e
du
ca
ti
o
n
al
p
ol
icies
and
q
u
al
ity
a
s
s
u
r
ance
i
t
s
v
a
li
d
i
ty
a
nd
r
el
iability
e
s
t
imat
es
a
r
e
n
ece
s
s
ary
.
Th
ere are tw
o g
e
neral
l
y
acceptab
l
e fram
e
w
o
rks
us
ed in
eval
uati
n
g
t
h
e
quality
of
t
es
t
i
n
e
d
u
cat
ion
a
l
and
ps
ycho
lo
gi
cal
m
eas
urem
ent
s
,
thes
e
ar
e;
C
las
s
ical
Tes
t
T
heo
r
y
(CTT
)
and
Item
R
es
pon
se
T
heo
r
y
(IRT).
Th
e
estim
a
tes
o
f
t
e
s
t
it
e
m
s
vali
dity
a
n
d
r
el
iab
ilit
y
d
epen
d
on
a
p
ar
ti
cu
lar
m
easu
r
em
e
nt
m
od
e
l
us
ed.
It
i
s
v
ital
f
o
r
a
test
d
ev
e
l
o
p
er
t
o
be
f
ami
l
iar
wit
h
t
he
diff
erent
t
e
st
dev
e
lo
pm
ent
and
it
e
m
anal
y
sis
m
e
th
ods
i
n o
r
der t
o
faci
litat
e
th
e
dev
e
lo
pm
ent
of
a new tes
t. T
h
e
CT
T
i
s a trad
ition
a
l ap
pro
ach w
hi
c
h
w
as w
ide
ly criti
ci
se
i
n
th
e
m
eas
urem
ent
c
o
mm
un
ity
f
o
r
it
s
sh
ortcom
in
gs
s
uch
a
s
s
am
pl
e
dep
e
nd
en
cy
o
f
co
ef
fici
e
n
t
m
easures
a
n
d
e
s
t
i
m
ates
o
f
meas
urem
ent
e
rror
.
H
o
w
e
v
e
r
,
t
h
e
I
R
T
i
s
a
m
o
d
e
r
n
a
p
p
r
o
a
c
h
w
h
i
c
h
p
r
o
v
i
d
e
s
s
o
l
u
t
i
o
n
s
to
m
os
t
o
f
the
CTT’s
i
d
e
n
tif
i
e
d
shor
t
c
oming
s
.
This
p
aper
t
her
e
f
o
re,
provid
es
a
com
p
rehens
iv
e
ov
ervi
ew
o
f
th
e
IRT and
its proced
ures
a
s applied
t
o
te
s
t
i
te
m
dev
e
lo
pm
ent
an
d
a
n
aly
s
i
s
.
Th
e
pap
e
r
co
nclu
des
w
ith
s
o
m
e
sug
g
es
t
i
ons
f
o
r
tes
t
d
ev
elop
ers
a
n
d
t
e
st
s
pecialists
a
t
all
lev
e
ls
t
o
ado
p
t
IRT
f
or
its
i
denti
f
ied
cruci
a
l
th
eoret
i
cal
a
n
d
e
m
p
i
r
ical
g
ain
s
o
ver
CTT.
I
RT
b
ased
p
ar
am
eter
est
i
m
a
tes
sh
ould
be
s
up
erior
an
d
rel
i
abl
e
t
h
a
n
CT
T
b
a
s
e
d
p
a
ram
e
ter
est
i
m
a
tes.
W
i
t
h
t
h
ese
f
eatu
r
es,
IRT
can
h
el
p
res
o
lv
e
t
h
e
prob
le
ms
a
ss
oc
ia
te
d
with
tes
t d
e
si
gn
bas
ed o
n CTT
K
eyw
ord
:
C
l
as
si
c
a
l
T
e
st
T
h
e
o
r
y
Item
Respo
nse
Theory
R
e
l
i
a
b
i
lit
y
V
a
li
di
ty
Co
pyri
gh
t © 2
018 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
Scien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Roha
ya Ta
l
ib,
Fa
cult
y
o
f
E
duc
ati
o
n,
Un
iv
e
r
sit
i
Tekn
o
lo
gi
M
al
ay
sia
,
81
3
10 Jo
h
o
r B
a
hru,
Ma
l
a
y
sia
.
Em
ail:
roha
yat
a
l
i
b
@
u
t
m.
my
1.
I
N
TR
OD
U
C
TI
O
N
A
sse
ssm
e
nt
i
n
e
duc
a
t
i
o
n
sys
t
em
s
e
r
ves
a
num
ber
o
f
p
urp
o
ses,
w
hi
c
h
i
nc
lu
de
s
impr
o
v
in
g
in
struc
tio
na
l
pla
n
nin
g
,
a
c
t
i
ng
a
s
a
m
ec
han
i
sm
t
o
c
h
a
nge
i
ns
truc
t
i
ona
l
c
o
nte
n
t,
m
easurin
g
le
arne
rs’
profic
ie
nc
y,
c
o
m
pa
riso
n
o
f
s
tude
nt
p
e
rform
anc
e
s
or
a
ch
ie
vem
e
nt
da
t
a
,
plac
em
en
t
of
s
t
ude
n
t
s,
d
ete
r
min
i
ng
a
stude
n
t
f
a
t
e
(sho
u
l
d
he
/s
he
b
e
reta
ine
d
o
r
p
r
omo
t
ed)
and
h
o
l
d
in
g
s
c
ho
o
l
s
an
d
e
d
ucat
ors
acc
o
unta
b
le
.
Ac
co
rd
ing
to
G
u
r
sk
i
[
1
],
t
e
s
ts
m
i
g
ht
b
e
h
e
lp
ful
in
e
v
a
l
u
a
t
in
g
t
he
e
ffec
t
i
v
ene
s
s
of
t
he
i
n
s
t
r
uc
tio
n.
T
he
qua
l
i
t
y
of suc
h
e
v
a
l
ua
t
i
o
n
s de
pen
d
s to a l
a
r
ge e
x
t
en
t on t
h
e na
t
u
r
e
and
q
ua
l
i
t
y
of t
h
e c
o
l
l
e
c
t
ed
i
nform
a
tio
n dur
i
n
g
t
h
e
assessm
ent
set
tin
g.
T
hro
u
g
h
the
de
ca
de
s,
t
he
f
ac
e
o
f
t
es
t
i
n
g
h
a
s
u
nder
g
o
n
e
m
a
n
y
c
h
a
nge
s.
F
rom
oral
t
o
st
a
ndard
ize
d
t
est
i
n
g
,
t
o
a
u
t
h
e
n
tic
a
sse
ss
m
e
nt,
up
t
o
t
he
p
rese
n
t
da
y
i
t
h
a
s
c
on
t
i
n
u
e
d
to
c
ha
n
g
e
w
i
t
h
educ
a
t
i
o
na
l
p
o
lic
y an
d
prac
tic
e
s
.
H
i
g
h
-sta
ke
s
te
stin
g
is
u
se
d
for
t
h
e
pur
pose
s
o
f
pr
ov
i
d
i
n
g
r
e
sul
ts
t
hat
ha
ve
i
m
por
tan
t
c
ons
eq
ue
nce
s
su
ch
a
s,
l
i
c
en
si
ng
,
ce
rti
f
i
c
at
io
n
s
o
r
c
r
ed
enti
al
in
g
.
V
a
l
i
d
i
t
y
i
s
t
h
e
c
o
rne
r
stone
u
p
o
n
wh
ich
a
l
l
me
asurem
en
t
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
I
t
e
m
Re
sponse
T
h
e
o
ry:
An Int
r
od
uc
t
i
o
n
t
o
L
a
t
e
n
t
T
r
ai
t
Mo
del
s
to
T
e
st …
(Ado
Abd
u
Bic
h
i)
14
3
syste
m
s
are
bu
il
t
.
I
n
e
d
ucat
i
o
nal
m
e
a
s
urem
ent
s
,
the
val
i
di
ty
o
f
i
n
f
e
r
e
n
c
e
s
d
r
a
w
n
f
r
o
m
t
e
s
t
r
e
s
u
l
t
s
m
u
s
t
b
e
sou
nd
a
n
d
w
e
l
l
-
g
ro
un
de
d
on
p
ri
nci
p
les
a
n
d
em
pirica
l
e
v
id
e
n
ce
w
h
ic
h
s
h
ou
l
d
b
e
a
b
l
e
t
o
w
i
t
h
sta
nd
cr
i
tic
ism
[2].
It
i
s
c
lea
r
t
ha
t
q
u
a
lit
y
ass
e
ssme
n
t
is
l
ike
l
y
t
o
l
ea
d
t
o
i
mpr
ove
me
nts
i
n
st
u
de
n
t
l
ear
n
i
n
g
[
3].
To
da
y
tes
t
in
g
is
e
ver
y
w
h
ere
i
n
o
ur
e
duc
a
t
iona
l
sys
t
e
m
;
a
n
d
w
i
th
a
dv
anc
e
men
t
s
in
i
t
s
d
esi
g
n
and
tech
n
o
l
o
gy,
c
o
u
p
l
e
d
w
it
h
t
h
e
a
d
ve
nt
o
f
the
“
A
ge
o
f
A
c
c
o
u
n
t
ab
i
l
i
ty”
,
t
e
s
t
i
n
g
i
s
an
i
m
porta
nt
e
le
me
nt
o
f
educ
a
t
i
o
n.
H
o
w
ever
,
for
tes
t
qua
l
ity
t
o
be
e
sta
b
l
i
s
h
e
d
c
ert
a
in
c
r
i
t
e
r
i
a
o
n
t
h
e
a
r
e
a
s
o
f
t
e
s
t
d
e
s
i
g
n
,
t
e
s
t
a
n
a
l
y
s
i
s
tech
n
i
q
u
es
a
n
d
t
es
t
sc
or
e
i
n
terpr
e
ta
t
i
on
m
u
s
t
b
e
m
e
t.
I
n
an
y
of
fi
c
i
al
t
es
t
,
m
o
r
e
e
s
p
eci
al
ly
i
n
l
a
rg
e
s
c
a
l
e
assessm
ent,
t
h
e
q
ues
tio
n
of
r
e
l
i
a
b
ili
t
y
a
nd
v
a
l
i
d
it
y
is
o
f
grea
t
c
oncer
n.
E
duca
t
io
na
l
ass
e
ssme
n
t
sh
o
u
l
d
f
o
llow
the e
s
t
a
b
lis
hed
criteria
and
g
u
i
de
line
s
o
f va
li
d
and r
e
l
i
a
b
le t
est de
vel
o
pme
n
t
.
The
proce
ss
of
t
es
t
de
v
e
lopm
ent
i
n
cl
u
d
es
f
i
v
e
d
i
ffe
ren
t
s
t
e
ps,
i
n
clu
d
i
ng
t
e
s
t
co
n
cep
tu
ali
z
a
t
io
n
,
t
es
t
c
o
nst
r
u
c
ti
on
,
t
e
st
t
ry
-out
,
a
n
al
ysi
s
,
a
n
d
re
v
i
si
on
[
4
]
.
As
c
a
n
be
s
e
e
n,
item
a
n
al
ys
i
s
f
o
llow
s
a
n
in
itia
l
tr
y-
ou
t
o
f
the
tes
t
.
Th
e
g
o
al
o
f
it
e
m
a
n
a
ly
sis
me
t
h
o
d
s
is
t
o
m
a
xim
i
z
e
t
h
e
p
s
yc
h
o
m
e
tric
qua
l
ity
o
f
score
s
f
r
o
m
t
h
e
te
st
[5].
T
he
re
a
re
e
duca
t
io
na
l
m
e
th
o
d
s
of
a
s
s
e
ssi
ng
t
h
e
i
t
em
s
i
n
a
n
e
wl
y
cre
a
t
ed
t
e
s
t
,
w
h
i
ch
h
elp
i
n
a
c
h
i
e
vi
ng
ac
cura
cy
a
n
d
e
la
b
o
r
a
t
i
on
o
f
t
h
e
tes
t
r
esu
lts.
It
i
s
im
por
tan
t
f
o
r
a
t
est
de
v
e
lope
r
t
o
b
e
fam
iliar
w
i
th
t
he
di
ffe
re
nt
t
e
s
t
d
e
ve
lo
pm
en
t
a
n
d
i
t
em
a
nal
y
s
i
s
m
e
tho
d
s
i
n
o
r
d
er
t
o
faci
l
ita
te
t
he
d
e
v
e
l
o
p
me
nt
o
f
a
ne
w
test.
Th
e
s
e
f
r
amewo
r
k
s
a
re
C
l
a
ssi
c
a
l
T
est
Th
eory
(
C
TT)
a
n
d
I
t
e
m
R
e
spo
n
s
e
Th
eo
ry
(
IR
T).
Th
e
e
s
t
i
m
at
es
o
f
val
i
d
it
y a
n
d re
li
a
b
ili
t
y
of
t
e
st
i
t
e
ms
d
epe
nds
on
a
part
icu
l
ar
me
asure
m
e
n
t
mode
l u
s
ed.
T
h
i
s
p
a
p
e
r
t
h
e
r
e
f
o
r
e
d
i
s
c
u
s
s
e
s
t
h
e
I
R
T
f
r
a
m
e
w
o
r
k
,
i
t
s
a
s
s
u
m
p
t
i
o
ns
its
a
p
p
l
ica
t
i
o
n
i
n
t
he
deve
l
opm
en
t
o
f
t
e
s
t
w
h
ic
h
in
c
l
ude
s
val
i
dit
y
a
nd
rel
i
ab
i
l
i
t
y
i
n
IRT,
item
a
n
al
ysis
a
n
d
I
t
e
m
s
s
elec
t
i
o
n
.
U
s
in
g
IRT
t
o
d
e
v
e
l
o
p
t
es
t
item
s
w
il
l
o
v
e
r
com
e
t
he
l
im
itat
i
o
n
s
o
f
t
h
e
C
T
T
.
I
R
T
c
a
n
p
r
o
d
u
c
e
i
t
e
m
s
t
a
t
i
s
t
i
c
s
in
de
pen
d
e
n
t
o
f
e
xam
i
nee
s
a
m
p
les
a
nd
p
e
r
s
on
s
t
a
t
i
s
t
i
c
s
i
nde
pe
nde
nt
o
f
th
e
pa
rt
ic
u
l
ar
s
e
t
o
f
i
t
ems
adm
i
n
i
s
t
ere
d
[
6].
2.
ITEM
R
E
S
PON
S
E THEO
RY
Item
r
e
s
ponse
the
o
ry
o
r
t
h
e
l
a
te
n
t
t
rai
t
m
odel
s
p
ro
vi
d
e
a
r
ich
st
a
t
is
tica
l
t
o
o
l
f
or
a
na
lys
i
s
o
f
e
d
u
cati
o
n
a
l
t
e
st
a
n
d
p
sy
ch
ol
og
i
cal
m
e
a
su
re
me
n
t
s
c
a
l
e
.
The
IR
T
m
et
ho
ds
w
e
r
e large
l
y d
e
vel
o
ped
i
n
t
he
196
0s
t
h
ro
ugh
1
980
s,
t
h
oug
h
,
a
s
Boc
k
[
7
]
n
ot
ed
i
n
h
i
s
"Bri
e
f
H
i
s
t
o
ri
c
a
l
R
e
v
i
e
w
of
I
t
e
m
Res
p
o
n
se
T
he
ory"
.
The
fo
u
nda
t
i
o
n
f
or
t
hese
m
o
d
el
s
be
ga
n
w
ith
T
h
u
rstone
i
n
t
h
e
19
2
0
s.
In
h
is
p
ape
r
t
i
t
l
e
d
“
A
M
e
t
ho
d
o
f
S
ca
lin
g
P
s
ycho
l
o
gica
l
and
E
duc
at
i
o
n
a
l
Te
sts.”
H
e
p
r
o
vide
s
a
tec
h
n
i
que
f
o
r
p
l
a
c
i
n
g
t
h
e
i
t
e
m
s
o
f
t
h
e
[
8
]
t
e
s
t
o
f
chi
l
d
ren’s
m
e
nta
l
d
e
v
e
l
o
p
me
nt
o
n
a
n
a
ge
-gra
ded
sca
l
e
.
A
not
her
w
o
r
k
t
h
a
t
s
e
r
v
e
s
a
s
a
b
a
s
i
s
f
o
r
l
a
t
e
r
deve
l
opm
en
t
o
f
I
RT
w
as
t
he
[
9]
b
oo
k
e
n
t
i
t
l
e
d
“
S
t
a
t
is
ti
cal
T
he
or
ies
o
f
M
enta
l
T
e
s
t
S
co
r
e
s”.
They
p
r
o
vide
a
rig
o
rou
s
a
nd
un
i
f
ie
d
st
a
tis
tic
al
t
r
eatm
e
n
t
o
f
c
l
a
s
sica
l
te
st
t
h
e
o
r
y
,
partic
u
l
arl
y
t
he
c
hap
t
ers
w
r
itte
n
b
y
Birnba
um
i
n
thi
s
b
o
ok.
S
om
e
of
t
he
r
ece
n
t
c
o
l
l
e
c
t
io
n
a
n
d
te
xts
on
t
h
e
d
e
v
e
l
op
me
nt
a
nd
a
ppli
c
a
t
io
n
of
I
RT
inc
l
ude
: [10]
;
[11]
; [1
2]
; [1
3];
[14]
; a
n
d [15].
H
a
m
b
let
on
&
Jone
s
[6]
D
e
sc
ribe
d
Item
R
es
po
nse
T
h
e
o
ry
a
s
a
ge
n
e
r
al
s
ta
ti
st
ica
l
t
heor
y
a
b
out
e
x
ami
n
ee
it
e
m
a
n
d
t
e
s
t
p
e
rforma
n
c
e
an
d
how
p
e
rfo
rman
c
e
rel
a
t
e
s
t
o
t
he
a
bi
l
i
t
i
es
t
ha
t
a
r
e
m
e
a
s
ured
by
t
h
e
it
e
m
s
i
n
t
he
t
e
s
t.
I
tem
respons
es
can
e
i
t
h
e
r
b
e
di
scr
e
t
e
o
r
c
o
n
t
i
n
uou
s
an
d
c
a
n
be
d
i
c
hot
o
m
ousl
y
o
r
p
o
l
y
chot
o
m
ousl
y
s
c
o
red;
i
t
e
m
sc
o
r
e
c
a
t
e
gori
e
s
ca
n
b
e
o
rde
r
ed
o
r
u
norde
r
e
d;
t
her
e
can be
o
n
e
a
b
i
lit
y
or
m
an
y
abi
l
i
t
i
es
u
n
d
erly
i
ng
test
p
e
rfor
m
a
n
c
e
;
an
d
t
h
e
r
e
ar
e
ma
ny
w
a
y
s
(i.e
.,
m
odel
s
)
in
w
hic
h
t
he
r
elat
i
ons
hi
p
betw
ee
n
item
re
spo
n
se
s
a
n
d
t
h
e
u
n
d
e
r
l
y
i
ng
a
b
i
l
i
t
y
or
a
bi
l
ities
c
a
n
be
s
pec
i
fie
d
.
Wi
thi
n
t
he
g
e
n
e
r
al
I
RT
fra
m
e
w
ork,
m
any
m
ode
ls
s
h
a
ve
b
ee
n
for
m
ula
t
e
d
a
nd
a
p
p
lie
d
to
r
e
a
l
t
e
st
d
a
t
a
.
A
cc
or
ding
to
[
1
6
]
“
I
tem
Re
s
p
o
n
se
T
he
ory
(IRT),
bas
e
d
on
l
a
t
e
n
t
t
r
a
it
t
h
e
o
r
y
,
i
n
c
o
rpora
t
e
s
m
e
a
sur
e
m
e
nt
a
ss
um
p
tio
ns
a
b
o
u
t
e
xa
m
i
ne
e
it
e
m
a
nd
te
st
p
erfor
m
a
n
ce,
a
nd
h
ow
p
e
rfor
m
anc
e
r
elates
t
o
k
n
o
wl
e
dge
a
s
m
e
a
s
ur
ed
b
y
t
h
e
i
t
em
s
o
n
a
tes
t
.
W
ithin th
e
g
e
n
eral
IRT
fram
e
wor
k
, m
a
ny mode
ls ha
v
e
bee
n
for
mu
la
te
d
.
F
am
ous nam
es associa
t
e
d
with t
hes
e
vari
ous
s
cor
i
n
g
m
o
d
els
a
r
e
d
i
c
h
o
t
om
o
u
s,
b
in
omia
l,
P
oiss
on,
r
at
i
ng
sc
ale,
f
a
c
e
t
s,
m
ult
i
no
m
i
al
l
og
i
t
,
or
po
lyt
o
m
o
us.
T
h
ese
sc
ori
n
g
mode
l
s
h
a
n
dle
i
t
e
m
r
e
spon
se
s
t
h
at
a
r
e
d
i
s
c
r
ete
or
c
ont
in
u
ous
a
nd
d
i
c
h
o
t
omo
u
s
or
po
lyt
o
m
o
us
s
core
d” (
P.
1).
The
c
h
ara
c
t
eri
s
t
i
c
s
o
f
Item
R
esp
onse
Mo
de
l
s
,
a
s
s
um
m
a
rised
b
y
[
17
]
are
,
f
i
r
st
,
an
I
RT
m
odel
m
u
st
sp
e
c
i
f
y
t
h
e
rela
t
i
on
sh
i
p
b
e
t
w
e
en
t
h
e
o
b
s
e
r
ved
re
spon
s
e
a
nd
und
e
rl
yin
g
u
no
b
s
e
r
va
b
l
e
c
o
ns
truct.
S
eco
n
d
l
y
,
t
h
e
mode
l
m
u
s
t
p
r
o
v
i
de
a
w
ay
t
o
estima
t
e
sc
ores
on
t
h
e
ab
i
l
i
t
y
.
T
hi
rd,
the
exa
m
inee
’s
s
c
o
res
wil
l
be
the
ba
sis
for
est
i
ma
t
i
o
n
o
f
t
h
e
u
nde
r
l
yi
ng
con
s
truc
t.
F
ina
l
l
y
,
a
n
I
RT
m
o
d
el
a
ss
um
es
t
h
a
t
t
h
e
perf
orm
a
nce
o
f
a
n
e
x
a
m
i
n
e
e
ca
n
be
e
ntire
l
y
pr
edi
c
te
d
or
e
xp
la
ine
d
by
o
n
e
o
r
mor
e
a
bi
li
t
i
e
s
.
In
item
re
sp
o
n
se
t
he
ory,
it
is
o
fte
n
a
ss
um
ed
tha
t
a
n
e
x
am
ine
e
h
as
s
ome
la
te
nt,
u
n
o
b
s
er
vab
l
e
t
r
ai
t
(
a
lso
c
a
l
l
e
d
t
h
e
a
bil
i
t
y)
,
w
h
i
c
h
ca
nn
o
t
b
e
st
ud
ie
d
direc
t
l
y
.
The
p
u
r
p
o
s
e
of
I
RT
i
s
t
o
p
rop
o
se
m
odels
t
ha
t
p
e
rm
i
t
t
o
l
i
nk
thi
s
l
a
t
e
n
t
tr
ai
t
to
s
ome
o
b
se
r
v
ab
le
cha
r
ac
t
e
ris
tics
of
t
he
e
xam
i
ne
e,
e
spe
c
ia
l
l
y
h
i
s/he
r
fa
cu
l
t
i
e
s
t
o
c
o
rr
ectl
y
a
nsw
e
rin
g
t
o
a
set
of
q
ues
t
i
o
n
s
t
ha
t
f
o
rm
a test [18].
Item
R
esp
o
n
se
T
he
ory,
item
pa
ram
e
te
rs
i
n
c
lu
de
d
i
f
f
i
c
u
l
t
y
(
l
oc
at
i
on),
discr
i
m
i
na
ti
o
n
(
sl
ope),
a
n
d
pse
u
d
o
-g
uess
i
n
g
(low
er
a
s
y
m
p
tote
).
T
hree
m
ost
c
o
m
m
onl
y
use
d
I
RT
m
ode
ls
a
re
;
o
n
e-pa
ram
e
ter
l
o
g
i
s
t
ic
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
2,
June
2
0
18 :
1
4
2
– 1
51
14
4
mode
l
(1P
L
M
or
R
as
ch
m
o
d
e
l
)
,
tw
o-
para
me
t
e
r
l
o
gis
tic
m
ode
l
(
2
P
L
M
)
and
three
param
e
ter
logis
t
ics
model
(3P
L
M
)
.
1PM
p
o
sse
s
se
s
on
l
y
i
tem
d
i
f
f
i
c
u
l
t
y
p
ar
am
eter
(
b)
,
th
e
2
P
L
M
in
a
dd
i
tio
n
t
o
(
b)
p
o
ssess
a
s
econ
d
para
me
ter
kno
w
n
a
s
discrim
i
na
t
i
on
par
a
m
e
ter
(
a
),
w
hich
a
ll
ow
s
t
he
i
tem
s
t
o
d
i
ffere
n
tia
te
o
r
discr
i
mi
n
a
te
t
h
e
e
x
ami
n
ees
o
f
di
ff
e
r
e
n
t
abi
liti
e
s
.
Th
e
3
P
LM
i
n
add
i
tio
n
t
o
t
he
(
b)
a
nd
(a
)
c
on
ta
in
s a th
i
r
d par
a
m
e
ter,
kno
w
n
as
the
pse
u
d
o
-c
ha
nce
para
me
t
e
r
(c
).
A
s
note
d
b
y
[1
0]
t
he
p
se
ud
o-c
h
a
n
ce
o
r
gue
ssin
g
p
a
r
a
m
eter
c
orr
e
sponds
t
o
the
l
o
w
e
r
asym
pt
o
t
e
of
t
he
i
tem
c
h
ara
c
t
e
rist
ic
c
ur
ve
w
h
i
c
h
r
epr
esent
s
t
he
p
r
o
ba
bi
lit
y
tha
t
l
o
w
-ab
i
l
i
t
y
exa
m
ine
e
s
w
il
l
answ
er
an
it
e
m
c
orr
ectl
y
in
a
test
a
n
d
prov
i
de
a
n
est
i
m
a
te
o
f
the
g
u
ess
i
n
g
par
a
m
e
ter
2.1.
Ass
u
m
p
t
ions of
Item R
es
p
o
nse Theory
Whe
n
i
de
nt
i
f
y
i
ng
t
h
e
m
a
jor
assumpt
i
ons
o
f
t
h
e
Ite
m
R
es
po
nse
T
h
e
ory
st
a
t
e
d
t
ha
t,
t
he
f
i
r
st
assum
p
ti
o
n
,
[
1
9]
s
ta
tes
tha
t
i
f
t
h
e
e
x
a
m
i
n
ee
k
n
o
w
s
t
he
c
or
rec
t
a
n
swe
r
t
o
th
e
it
e
m
,
h
e
/
s
he
w
il
l
go
di
rec
t
l
y
t
o
answ
e
r
it
c
o
r
r
ec
t
l
y,
t
h
i
s
ass
u
m
p
ti
on
rela
t
e
s
to
a
n
y
t
e
s
t
the
o
ry
.
Wi
t
h
o
u
t
th
is
a
ssu
m
p
tio
n,
t
h
e
re
m
ay
not
b
e
a
go
o
d
r
e
a
so
n
f
o
r
te
st
in
g.
T
h
e
t
w
o
o
t
h
er
s
t
r
ong
a
ss
um
p
t
io
ns
o
f
IR
T
are
Un
id
i
m
en
sio
n
a
li
t
y
a
nd
L
o
c
al
in
de
pen
d
e
n
c
e
.
These
a
s
sum
p
tio
ns
a
r
e
p
ar
am
ou
nt
a
nd
s
ho
u
l
d
ho
l
d
i
r
r
espec
t
i
v
e
of
t
he
l
a
t
e
n
t
tra
i
t
mode
l
use
d
.
Th
is
m
e
a
ns te
s
t
data
c
a
n
o
n
l
y
be va
l
id
f
or
l
a
t
e
n
t tra
it m
ode
l e
stim
ati
o
n on
l
y
if
the
s
e
a
s
su
m
p
tio
ns a
re
m
et.
i.
U
n
idime
n
sio
n
alit
y
U
n
i
d
i
m
en
si
o
n
a
l
i
t
y
s
t
ate
t
h
a
t
t
he
re
i
s
on
l
y
one
a
b
i
l
i
t
y
b
e
i
n
g
m
e
asu
r
ed
.
An
ot
he
r
re
se
arc
h
er
[
19
]
furt
her
ex
pla
i
n
th
a
t
,
the
t
h
eo
ry
o
f
l
a
te
n
t
t
r
a
it
as
sume
s
t
h
at
a
s
e
t
o
f
trai
ts
u
nde
rl
ies
te
st
p
erfor
m
a
n
ce
.
The
exa
m
ine
e
’
s
ab
i
lit
y
i
n
a
s
et
o
f
un
i
d
i
m
en
si
o
n
a
l
l
a
t
en
t
space
c
an
b
e
r
ep
rese
nt
e
d
b
y
a
v
ect
o
r
o
f
abi
lit
y
sco
r
e
s
a
s
(i.e
.,
θ
1,
θ
2,
θ
3….
.
θ
n)
.
The
Item
r
e
s
ponse
m
odels
t
ha
t
assume
a
s
i
n
g
l
e
l
a
t
e
n
t
a
b
i
l
i
t
y
i
s
r
e
f
e
r
r
e
d
t
o
a
s
un
idim
en
si
o
n
a
l
.
T
h
is
a
ssum
p
tio
n
me
ans
t
h
a
t
t
he
ite
ms
m
ea
sure
o
n
l
y
one
a
rea
of
a
b
ili
ty
o
r
k
now
l
e
dge
.
The
con
d
i
t
i
on
of
u
ni
d
i
me
nsio
na
li
ty
doe
s
not
por
ten
d
t
ha
t
t
h
e
i
t
e
m
s
m
u
st
c
orre
late
p
o
s
it
ive
l
y
w
i
t
h
e
ac
h
o
t
h
e
r
.
A
n
it
e
m
m
a
y
ne
g
a
tive
l
y
corr
ela
t
e
w
i
t
h
o
t
h
e
r
s i
t
e
m
a
nd ca
n still
be
un
i
dime
nsi
o
nal.
The
ass
u
mp
ti
o
n
o
f
U
n
i
d
im
en
si
ona
l
ity
r
eq
u
i
res
tha
t
a
l
l
i
te
m
s
on
a
t
e
st
m
e
a
sure
a
s
in
gle
la
te
nt
t
ra
i
t
and
v
i
ola
t
ion
of
t
h
i
s
un
i
d
im
e
n
sio
n
a
l
i
t
y
w
o
uld
l
e
a
d
t
o
se
r
i
o
u
s
m
isle
a
d
i
ng
resu
lt.
Th
e
assum
p
tio
n
ca
n
b
e
sat
i
sf
i
e
d if a sing
l
e d
o
mi
nan
t
fac
tor un
derl
ie respo
nses.
U
n
i
d
i
mens
ion
a
lit
y
IR
T
an
a
l
y
s
i
s
a
ssu
mes
t
h
e
p
r
e
s
en
ce
of
a
dom
i
n
a
n
t
abi
lit
y
or
t
rai
t
t
h
a
t
in
fl
ue
n
c
es
t
es
t
perf
o
r
ma
nce
-
w
h
ich
is
c
al
le
d
u
n
i
d
i
m
e
ns
io
na
lit
y
[1
3].
In
ot
her
w
o
rds,
u
ni
d
i
me
nsio
na
li
ty
r
efe
r
s
tha
t
t
here
e
xi
st
a
s
in
gle
l
a
t
e
n
t
tra
i
t
var
i
a
b
l
e
t
o
e
x
p
l
ai
n
the
varia
b
i
l
i
t
y
o
f
obs
erve
d sc
ore
a
s
w
ell a
s
assu
m
p
t
i
o
n
for th
e
t
es
t deve
lopm
e
n
t i
n
class
i
c
a
l
t
est
the
o
ry.
ii
.
L
o
ca
l
I
n
de
pe
n
d
e
n
c
e
The
ass
u
mp
ti
o
n
o
f
l
o
ca
l
in
d
e
pe
n
d
e
n
ce
m
ea
ns
t
ha
t,
t
h
e
p
ro
ba
b
ili
t
y
o
f
a
n
e
x
am
i
n
ee
get
t
i
n
g
i
te
m
corr
ectl
y
is no
t
a
f
fec
t
e
d
b
y
t
h
e answer g
i
v
en
t
o ot
he
r i
t
em
s in
t
he
t
e
s
t
.
I
t
n
ece
ssitat
e
s
tha
t
exc
lud
i
n
g
t
he
a
bi
l
i
t
y
there
i
s
n
o
re
la
tio
ns
hi
p
be
tw
ee
n
t
h
e
test
ite
m
re
sponse
s
o
t
h
er
th
an
t
he
r
el
at
io
nship
d
e
t
e
rmi
n
e
d
b
y
t
h
e
a
b
ilit
y
or
o
t
h
er
m
ode
l
para
me
t
e
rs.
F
o
r
exam
ple,
i
f
t
h
e
r
e
s
p
o
n
se
s
t
o
o
ne
item
struc
t
ural
ly
c
ons
tra
i
n
t
h
e
p
o
ss
ib
le
answe
r
s
to
o
t
h
er
item
s
,
t
h
en
t
he
i
tem
s
a
re
n
o
t
l
ocal
l
y
i
nd
e
p
en
d
e
n
t
.
I
f
t
h
e
s
e
a
s
s
u
m
p
t
i
o
n
s
a
r
e
m
e
t
,
a
n
I
R
T
mode
l
ca
n
b
e
s
ucc
e
ss
fu
lly
e
mplo
ye
d
[
20].
Loc
a
l
i
nde
pe
n
d
e
n
ce
m
e
a
ns
t
he
p
er
form
ance
on
d
i
ffe
r
en
t
i
t
em
s
i
s
in
de
pen
d
e
n
t
b
u
t
c
o
nd
iti
o
n
al
on
the
stu
d
e
n
t
’
s
a
b
il
it
y
a
nd
d
o
es
n
o
t
m
ea
n
tha
t
i
tem
s
do
n
o
t
c
o
r
r
ela
t
e
w
ith
e
ac
h
ot
her.
sugge
sts
that,
t
h
e
r
e
is
n
o c
o
rrela
ti
on b
e
t
w
een
t
es
t
i
t
e
m
s wh
e
n
p
erson
‘
s ab
i
l
ity
l
ev
el
i
s c
o
n
t
roll
e
d
[
1
3
]
.
Item
Resp
o
nse
Theor
y
-
t
h
e
ge
n
e
r
aliz
ed
mod
el
)
(
)
(
1
)
1
(
g
g
g
g
b
Da
b
Da
g
g
g
e
e
c
c
P
(1)
W
h
ere:
ag
= gra
d
i
e
n
t
o
f
t
h
e
ICC at
t
he
po
i
nt
q
(
item
di
scrim
i
na
tio
n)
bg
= the
ab
i
lit
y
leve
l a
t
w
hi
c
h
a
g
is m
aximiz
e
d
(item
diff
icu
l
t
y)
cg
=
p
ro
ba
bi
lit
y of l
ow
pe
r
so
ns
c
or
rec
t
l
y
a
n
s
w
e
ring a
que
sti
o
n
(or
endor
sin
g
)
g
Item
Resp
o
nse
Theor
y
Mod
els
S
c
huma
c
ker
[1
6]
s
umm
a
r
i
se
d
the
mode
ls
w
hen he
sai
d;
IRT
m
odels
di
f
f
er
de
pen
d
i
n
g
on
whe
t
her
t
h
e
rel
a
ti
ons
hi
p
b
e
t
w
e
e
n
i
t
em
pe
rf
orm
a
nce
an
d k
n
owle
dg
e
i
s
c
o
nsi
d
ere
d
a o
n
e-
, t
w
o
-
or t
h
ree-
p
a
ram
e
ter
log
i
s
t
ic
func
tio
n.
D
iffe
re
nt
IRT
p
a
ram
e
te
riz
a
t
i
o
n
m
o
del
s
a
d
j
us
t
f
o
r d
iffe
re
n
t
i
t
e
m
pr
o
p
e
r
t
i
es
l
e
a
d
i
n
g
to
di
ffere
nt a
b
i
lit
y e
s
tim
a
t
io
n.
1-p
a
ram
e
ter (
1
-PL
)
IRT
adj
us
t
s
fo
r
ite
m
dif
fic
u
l
t
y
; 2-
par
am
e
t
er (2-
P
L
)
IRT
accou
n
t
s
f
o
r d
i
f
f
i
cu
l
t
y
an
d d
i
scr
i
m
i
na
t
i
o
n
o
f
a
n
it
e
m
; and 3-
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
I
t
e
m
Re
sponse
T
h
e
o
ry:
An Int
r
od
uc
t
i
o
n
t
o
L
a
t
e
n
t
T
r
ai
t
Mo
del
s
to
T
e
st …
(Ado
Abd
u
Bic
h
i)
14
5
par
am
e
t
er (
3
-
P
L
)
IRT
take
s
in
to
acc
oun
t
the
e
f
f
e
c
t
o
f
i
t
e
m
gue
ssi
n
g,
d
i
f
f
ic
ul
ty
a
n
d
di
sc
ri
m
i
n
a
t
i
on.
A po
p
u
lar
o
n
e-p
a
ram
e
te
r
m
ode
l, dev
e
l
o
ped
by
G
e
or
g
e
Rasc
h, i
s
al
so c
o
m
m
o
nl
y
use
d
whe
r
e
i
t
e
m
diff
ic
u
l
t
y
p
r
o
v
ides
a
n
un
b
i
a
s
ed,
ef
fic
i
e
n
t
,
suff
ic
ien
t
,
an
d co
ns
iste
n
t
est
i
m
a
te of
se
p
a
ra
te
per
s
o
n
a
nd
item
cal
i
b
ra
ti
o
n
s (P.
1)
T
h
e
On
e-
Paramet
er
Logistic
M
od
el (ite
m
difficu
lty)
The
1-
pa
ram
e
ter
mode
l
e
x
p
l
a
i
n
s
t
he
r
e
l
at
ions
h
i
p
be
tw
ee
n
the
ab
i
l
i
t
y
a
nd
pro
b
ab
i
l
i
t
y
of
a
c
orr
ect
respo
n
se
o
n
t
h
e
item
i
n
t
er
ms
o
f
t
h
e
item
di
ffic
ult
y
.
A
n
i
tem
’
s
d
i
f
fic
u
lt
y
par
a
m
e
ter
(b)
is
t
he
p
o
i
nt
on
th
e
a
b
ili
ty
s
cal
e
c
o
rre
sp
ond
ing
to
t
h
e
l
o
c
atio
n
o
n
t
h
e
i
t
e
m
chara
c
t
erist
i
c
cur
v
e
(I
CC)
w
he
r
e
t
he
p
ro
ba
bi
l
ity
o
f
a
corr
ect
r
espon
s
e
i
s
0.5 [21].
P
θ
e
/1
e
(2)
W
h
ere:
P (θ)
=
a
b
il
ity
o
f a
st
ud
e
n
t
a
nd
a(θ)
=
diffic
u
l
t
y
leve
l
o
f item
a
n
d
e=
2.73
=
discr
i
m
i
nat
i
on
in
de
x
b(θ) =
1
i
n t
h
is
m
ode
l.
In
1
P
L
M
i
t
e
m
d
i
s
c
ri
mi
n
a
tio
n
is
t
ak
en
a
s
1
a
nd
thi
s
m
a
y
not
b
e
o
f
gr
eat
u
ti
lit
y
w
h
ere
s
h
arp
me
asure
m
e
n
t
i
s
r
e
q
u
i
r
e
d
e
.
g.
e
xa
mi
nee
s
w
i
t
h
eq
ua
l
ra
w
s
c
or
e
i
n
a
t
e
s
t
w
i
l
l
h
a
v
e
e
q
u
a
l
I
R
T
s
c
o
r
e
a
n
d
t
h
u
s
may fail to produce r
a
nking
Th
e
Tw
o
–P
ar
ame
t
er
Lo
g
i
stic
Mod
el (
i
t
em
d
ifficu
lty a
n
d d
i
scr
i
m
i
n
a
tion)
The
2-P
a
rame
ter
m
odel
ma
k
e
s
use
of
t
he
b
pa
ramet
e
r
(i
t
e
m
di
ff
i
c
ul
ty
)
j
u
st
a
s
in
t
h
e
1
P
L
M,
a
nd
add
i
tio
n
ad
d
a
n
e
lem
e
n
t
t
ha
t
in
d
i
cate
s
h
ow
w
il
l
s
a
n
item
sepa
ra
tes
s
t
ude
nt
s
i
n
t
o
d
iffer
e
nt
a
b
i
l
i
t
y
l
e
vel
s
t
h
i
s
para
me
ter
is
c
alle
d
it
e
m
d
iscri
m
i
n
a
t
i
on
(a
).
T
he
i
tem
discr
i
m
i
n
at
i
on
(
a
)
p
a
r
a
m
e
ter
used
i
n
t
h
e
2P
LM
i
s
equa
l
to t
he
s
l
o
pe
o
f the
i
t
em
cha
rac
t
e
r
is
t
i
c
s
c
urve
w
he
n i
t
is at
i
t
s
s
t
e
e
p
e
st
[
21
].
P
θ
1
1
e
.
(3)
T
h
e
T
h
re
e-Par
a
mete
r
L
o
g
i
s
t
ic M
ode
l
(
Diffic
u
lty, discr
i
m
inati
o
n,
a
nd
g
uessing
)
The
3P
L
m
ode
l
b
u
i
l
d
s
u
p
on
the
tw
o-
pa
ram
e
ter
m
o
de
l
b
y
a
dd
i
ng
pse
ud
o-
c
h
ance
-
l
eve
l
p
a
r
am
e
t
e
r
c
.
The
c
p
a
r
amet
er
i
s
t
h
e
v
a
lu
e
o
f
t
h
e
l
o
w
e
r
a
sy
mp
tot
e
o
f
th
e
i
t
e
m
ch
ar
a
c
t
e
r
i
s
t
i
c
c
u
r
v
e
a
n
d
i
s
i
n
d
i
c
a
t
i
v
e
o
f
t
h
e
pro
b
ab
i
lit
y
t
h
a
t
a
n
exam
inee
w
ith
a
v
ery
l
o
w
abil
ity
s
core
w
ou
l
d
a
n
swer
a
n
i
t
e
m co
rre
ctl
y
.
P
θ
c
1c
1e
.
(4)
A
ll
IR
T
mode
ls
a
re
d
e
r
ive
d
t
o
ge
ne
rate
i
te
m
c
h
ara
c
terist
ic
c
urv
es.
A
n
i
t
e
m
c
h
ara
c
teris
t
i
c
c
urve
p
l
o
t
s
the
pro
b
ab
i
lit
y
t
h
a
t
a
n
e
x
am
ine
e
w
ill
re
sp
o
nd
c
o
rr
ectl
y
t
o
an
i
tem
so
le
ly
a
s
a
func
ti
o
n
o
f
t
h
e
test’s
l
a
t
e
n
t
tra
i
t
[2
1].
[17
]
n
o
t
ed
“
The
ma
in
d
i
ffe
re
nc
e
t
o
b
e
f
o
u
n
d
i
n
cur
r
ent
l
y
po
pul
a
r
i
t
e
m
re
sp
on
se
m
o
d
e
l
s
i
s
i
n
t
h
e
ma
them
at
i
c
al
f
orm
of
P
i
(
θ
)
,
t
h
e
I
C
C
.
I
t
i
s
u
p
t
o
t
h
e
t
e
s
t
d
e
v
e
l
o
p
e
r
o
r
I
R
T
u
s
e
r
t
o
c
h
o
ose
o
n
e
of
t
he
m
any
ma
them
at
i
c
al f
unc
t
i
o
n
s
to
s
erve
as
t
h
e
fo
r
m
o
f
the
ICCs”
.
Th
e
v
a
lu
e
s
o
n
t
h
e
X-ax
i
s
o
f
an
I
C
C
re
p
r
e
s
ent
t
h
e
l
a
t
e
n
t
t
rait
,
usua
ll
y
ra
n
g
i
n
g
fr
om
-
3
to
+
3.
T
h
e
Y
-
axi
s
r
e
p
rese
nts
t
h
e
pro
b
ab
i
lit
y
of
a
n
e
x
am
ine
e
’s
s
uc
cess.
A
s
th
e
l
a
t
e
nt
t
ra
i
t
in
cre
a
se
s,
t
h
e
p
ro
b
a
bi
lit
y
of
t
he
exa
m
ine
e
re
spon
d
i
n
g
c
or
rec
tly
w
il
l
i
n
cre
a
se
but
w
i
t
h
d
i
m
i
n
i
s
h
in
g
ret
u
rns.
I
n
the
i
r
disc
uss
i
o
n
o
f
i
t
e
m
cha
r
ac
t
e
ris
tic
c
urves,
[
2
1
]
dis
c
usse
d
tw
o
i
n
t
e
rpr
e
tat
i
ons
t
h
a
t
the
y
c
onsi
d
e
r
a
cc
eptab
l
e.
T
he
f
irs
t
i
nter
pre
t
a
t
io
n
of
a
c
orre
ct
r
espo
nse
is
“
the
pr
oba
b
ili
t
y
t
ha
t
a
r
a
nd
omly
c
hos
e
n
me
mbe
r
o
f
a
h
o
moge
n
e
ou
s
su
bp
o
p
u
l
a
tio
n
w
ill
r
e
sp
on
d
c
o
rre
ctl
y
t
o
a
n
i
t
e
m”
(
p.
3
41
)
.
A
s
e
c
ond
i
n
te
rpre
t
a
t
i
on
is
t
h
a
t
the
proba
b
i
l
ity
r
e
p
rese
n
t
s
th
e
pro
b
ab
i
lit
y
of
a
spec
i
f
i
c
e
x
a
m
i
n
e
e
r
espo
nd
ing
corr
ect
ly f
or
a s
u
b
-
po
pul
a
t
i
o
n
of
i
t
e
ms.
In t
h
i
s s
t
u
d
y
, only a few
of
t
h
e
m
odels tha
t (
a
) assum
e
a
singl
e a
b
il
i
t
y un
d
e
r
l
ies te
st per
form
ance
, (b)
ca
n
be
a
p
p
l
i
e
d
t
o
d
i
ch
o
t
omo
u
s
l
y
s
c
o
red
da
ta
,
and
(c
)
assu
me
t
he
r
e
l
at
ionsh
i
p
b
e
t
w
ee
n
it
em
p
e
r
f
o
rma
n
c
e
a
nd
abi
l
i
t
y
i
s
give
n
b
y
a
one
-,
t
w
o
-,
o
r
three-param
e
ter
logistic
f
unc
t
i
o
n
w
il
l
be
c
o
n
s
i
d
e
re
d.
T
ypic
a
lly,
tw
o
assum
p
ti
o
n
s
a
r
e ma
de i
n spec
i
f
yin
g
IRT
m
o
d
el
s:
O
ne re
l
a
t
e
s
t
o
t
he d
ime
n
si
ona
l st
r
u
ct
ure of the
t
es
t da
ta, an
d
the o
t
her
re
l
a
te
s to t
he
m
athe
m
a
tica
l
form
of
t
he i
tem
c
h
ara
c
te
ris
t
i
c
fu
n
c
t
i
o
n
or c
urve
(
deno
te
d
ICC).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
2,
June
2
0
18 :
1
4
2
– 1
51
14
6
IRT
- I
t
e
m
C
h
a
ra
c
t
eri
sti
c
Cur
v
e
s
A
n
I
C
C
i
s
a
p
l
o
t
o
f
t
h
e
r
e
s
p
o
n
d
e
n
t
s
’
a
b
i
l
i
t
y
(
l
i
k
e
l
i
n
e
s
s
t
o
e
n
d
or
se)
ov
er
t
he
p
ro
bab
i
lit
y
o
f
t
he
m
corr
ectl
y
a
nsw
e
rin
g
t
he
q
u
e
s
t
i
o
n
(e
n
dors
i
n
g
).
T
he
h
ig
her
t
h
e
a
b
ili
t
y
,
th
e
hi
g
h
er
t
he
c
hanc
e
t
h
a
t
t
he
y
w
i
ll
respo
nd
c
o
rr
ectl
y
.
The
pro
b
a
bil
ity
o
f
a
c
o
rrec
t
r
espo
nse
i
s
d
e
t
er
mine
d
b
y
t
he
i
tem
’
s
di
f
f
icu
l
ty
a
nd
t
h
e
exa
m
ine
e
'
s
a
b
il
i
t
y.
T
his
pr
oba
b
i
l
i
t
y
c
a
n
be
s
ee
n
as
ill
us
trate
d
u
s
i
n
g
i
t
e
m
c
h
a
r
a
c
t
e
r
i
s
t
i
c
c
u
r
v
e
(
I
C
C
)
in F
ig
ure
1.
F
i
gure
1.
I
tem c
h
ara
c
teris
t
ic c
urve
Fr
om
t
his
I
C
C
abo
v
e
,
w
e
c
a
n
o
b
se
rve
tha
t
a
s
the
exam
i
n
ee's
a
b
i
l
it
y
i
n
c
r
e
a
ses,
t
he
p
r
o
bab
i
l
i
t
y
o
f
a
corr
ect
r
espon
s
e
i
nc
rea
s
es; t
h
is
i
s
w
h
at yo
u
w
ou
l
d
e
x
p
ec
t i
n
pr
a
c
t
i
ce.
As
g
i
v
en
e
a
r
li
er
t
h
e
i
t
e
m
d
i
ffic
u
lty
(
a
-
v
a
l
u
e)
m
ea
su
res
t
h
e
d
i
f
fic
u
l
t
y
o
f
an
sw
er
i
ng
a
n
i
te
m
corr
ectly.
The
pr
ece
d
i
ng
d
isc
u
ssi
o
n
a
n
d
e
q
u
a
t
io
n
s
u
gge
sts
tha
t
t
he
p
ro
ba
b
i
li
ty
o
f
e
n
d
o
r
s
in
g
a
n
i
t
e
m
corr
ec
t
l
y
or
a
corr
ect
r
espo
nse
is
0
.
5
f
or
a
n
y
e
xam
i
nee
w
hose
a
b
i
lit
y
is
e
qua
l
t
o
t
h
e
va
l
u
e
of
t
he
d
iff
i
c
u
lty
para
me
terpra
ct
ice.
F
igure
2 and
Figure
3
show
t
h
e ICCs
o
f two
d
iffer
e
n
t
i
tem
s
Di
f
f
icu
l
t
Item
v
s.
E
asy
Item
F
i
gure
2. I
CC i
n
case
of
a
diff
i
c
u
lt
ite
m
F
i
gure
3.
ICC
in
case of an
eas
y
i
t
e
m
F
i
gure
2 a
n
d F
i
g
u
r
e
3 s
how
t
he IC
C
s o
f
tw
o
di
f
fere
n
t
i
t
e
ms
,
w
i
t
h
dif
f
erent i
t
em di
f
ficult
y parameters
and
pro
b
a
b
il
it
y
o
f
e
nd
orsi
n
g
an
item
c
o
r
r
ec
tly
o
r
a
c
o
rrec
t
r
es
po
nse
.
B
y
co
mpari
n
g
t
h
ese
t
w
o
ICC
s
,
we
c
a
n
se
e
t
h
at
t
h
e
i
t
e
m
d
i
ffi
c
ult
y
p
aramet
er
d
et
ermi
n
e
s
t
h
e
lo
c
a
t
i
on
of
t
he
I
C
C
.
F
r
om
F
i
gu
re
2
t
he
p
r
o
ba
bi
lit
y
of
them
c
or
rec
t
l
y
a
nsw
e
rin
g
t
h
e
que
st
i
on
(e
nd
o
r
sing)
i
s
h
i
gh
er
a
t
0.
9
4
,
i
n
o
rder
t
o
g
e
t
a
0.94
pr
oba
bil
i
t
y
o
f
a
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
I
t
e
m
Re
sponse
T
h
e
o
ry:
An Int
r
od
uc
t
i
o
n
t
o
L
a
t
e
n
t
T
r
ai
t
Mo
del
s
to
T
e
st …
(Ado
Abd
u
Bic
h
i)
14
7
corr
ect
r
espo
nse
for
t
h
is
i
t
em
,
t
h
e
exam
inee
m
ust
have
h
ig
her
a
b
i
l
i
t
y
s
k
i
l
l
l
e
v
e
l
o
f
a
b
o
u
t
5
t
o
b
e
a
b
l
e
t
o
g
e
t
the
i
t
e
m
,
th
is
s
ig
n
i
fies
t
ha
t
,
t
h
e
item
i
s
d
iffic
u
l
t
,
bec
a
u
s
e
i
t
ca
n
o
n
l
y
b
e
a
n
sw
er
ed
by
a
h
i
gher
a
b
ili
t
y
s
t
u
den
t
s.
O
n
t
he
o
t
h
er
h
a
nd,
F
i
g
ure
3
the
pr
o
b
ab
i
l
i
t
y
o
f
a
n
sw
eri
n
g
the
qu
es
tio
n
c
o
rrec
t
l
y
(
en
do
rsi
n
g
)
i
s
h
i
gh
e
r
a
t
a
b
o
u
t
0.94,
i
n
orde
r
t
o
g
e
t
a
0
.9
4
proba
bil
i
t
y
o
f
a
c
orr
ect
r
e
s
pons
e
f
o
r
thi
s
i
tem
,
t
he
e
xaminee
must
h
ave
moder
a
te
a
b
ili
ty
s
ki
ll
l
e
v
el
o
f
a
b
ou
t
0
.
7
t
o
b
e
a
b
l
e
t
o
ge
t
t
h
e
it
e
m
c
o
r
re
ctl
y
,
th
i
s
s
ig
ni
fies
t
ha
t,
t
he
i
te
m
is
e
as
y,
b
ec
ause
it
c
a
n
o
n
l
y
be
a
nsw
e
r
e
d by
an
e
xam
i
ne
e
w
ith
m
ode
rate
a
b
i
l
i
t
y
l
e
ve
ls
2.2.
V
al
id
i
t
y an
d
R
e
li
ab
i
lity
i
n
Ite
m R
e
sp
on
se Th
e
ory
(IR
T
)
a)
V
a
li
di
ty
i
n
IR
T
The
m
eani
n
g
of
v
al
idi
t
y
a
n
d
r
elia
b
ili
t
y
i
n
IRT
differ
fr
om
t
ha
t
o
f
CTT.
T
he
I
RT
f
oc
use
s
on
th
e
c
h
ara
c
t
e
rs
o
f
th
e
it
e
m
.
A
v
a
l
i
d
ity
i
n
IRT
me
an
s
to
w
h
a
t
ext
e
n
t
in
d
i
vi
dua
l
exa
m
ine
e
s
a
nd
t
est
item
s
h
a
v
e
a
go
o
d
r
a
nki
ng
i
n
the
ab
i
l
i
t
y
w
hic
h
t
he
t
est
i
t
e
m
s
m
easure
,
t
h
i
s
m
e
a
n
s
t
h
e
abi
lit
y
of
a
ny
tes
t
t
o
r
a
n
k
a
n
in
div
i
dua
l a
cco
r
d
ing
to
t
h
e
ir a
bil
i
t
y
as
w
e
ll a
s
r
ank
the i
t
em
s
a
ccordi
ng
to
t
he
ir le
v
e
l
o
f d
i
ff
i
c
u
lty
[
2
2
].
b)
Re
li
a
b
i
l
i
t
y i
n
IRT
: Ite
m
a
nd Te
st
I
nfor
ma
t
i
on F
unc
t
i
o
n
s
Th
e
re
li
ab
il
i
t
y
i
n
I
R
T
m
e
a
n
s
t
o
w
h
at
e
xt
en
t
t
h
e
s
c
o
r
e
s
a
re
i
nd
e
pe
nde
n
t
o
f
gro
u
p
s
(sam
ple
s
)
a
s
w
ell
a
s
from
the
i
tem
s
, in o
the
r
w
ords
t
he char
a
c
t
e
r
istic
s o
f
te
s
t ite
m
s
i
s
not
a
ffected
b
y
the
sam
p
les
from
which
the
y
w
e
r
e
es
tim
ated,
a
nd
e
v
en
i
f
sam
e
i
te
m
s
w
e
r
e
adm
i
ni
stere
d
t
o
d
i
f
f
er
ent
gro
u
p
i
t
p
ro
vi
de
t
he
s
a
m
e
score
and
rank
i
ng.
S
i
m
i
la
rl
y,
r
elia
bi
l
ity
a
c
c
o
rd
i
n
g
t
o
I
RT
i
s
a
n
i
tem
a
n
d
te
st
i
nf
orm
a
ti
on
o
r
,
t
h
e
d
e
gree
t
o
w
h
ic
h
a
n
in
ves
tiga
t
or
o
r
re
sear
cher
can
b
e
c
e
r
t
ai
n
of
a
p
er
so
n
’
s
loc
a
ti
o
n
al
o
ng
θ.
T
he
a
m
o
un
t
o
f
i
tem
i
n
f
o
rm
atio
n
is
pro
port
i
ona
te
t
o
the
sta
n
dard
e
rror
of
e
st
ima
t
e
(S
E
E
)
for
ea
ch
possible
θ
[
2
3
]
.
A
smal
ler
SEE
indi
c
a
tes
a
st
r
o
nger
ce
rta
i
nty
i
n
t
he
e
s
t
i
m
ate
of
θ
a
n
d
t
her
e
fore
m
ore
i
n
fo
r
m
a
t
io
n
a
b
ou
t
i
n
d
i
v
i
dua
l
s
w
ith
t
hat
part
i
c
ular
θ
v
a
lu
e
.
B
y
rul
e
,
an
i
t
e
m
p
r
ovi
d
e
s
it
s
hi
gh
e
s
t
amou
nt
o
f
i
n
fo
rma
t
i
o
n
n
e
ar
i
t
s
d
if
fi
cu
lty
v
a
l
u
e
(
“b
”)
b
eca
u
s
e
there
is t
he
l
e
a
s
t am
ou
nt o
f
v
a
ria
b
il
i
t
y
(er
r
or)
nea
r
thi
s va
lu
e
[24].
F
i
g
u
r
e
4
s
how
s tes
t
i
n
f
orm
a
ti
o
n
func
tio
n
F
i
gure
4. T
est Infor
m
a
tio
n
F
unct
i
o
n
The
t
h
re
e
IRT
m
o
de
l
s
a
re
u
se
d
to
e
va
lua
t
e
t
h
e
va
l
i
d
i
t
y
a
n
d
reli
ab
i
lit
y
of
i
tem
s
t
est
acc
ordi
ng
t
o
t
h
e
three
par
a
m
e
te
rs. The
abi
li
ty
o
f
th
e
e
x
amin
ee, l
ev
e
l
o
f
it
em d
i
ffic
u
l
t
y
an
d
a
bi
l
i
t
y
o
f item
to d
iscrim
i
n
a
t
e
2.3.
A
pp
l
i
c
a
ti
o
n
of
IR
T in Te
s
t D
e
vel
o
p
m
e
n
t Proc
ess
The
pe
nd
u
l
um
s
w
i
n
g
i
n
te
st
d
eve
l
opm
en
t
t
echn
i
que
s
i
s
f
rom
C
l
a
s
s
i
cal
T
es
t
Th
eo
ry
(
C
TT)
t
o
It
em
Re
s
p
o
n
se
T
he
ory
(
I
RT)
.
A
c
c
ord
i
n
g
t
o
[2
5],
ap
pl
i
c
a
t
ion
o
f
I
RT
i
n
t
es
t
de
ve
lo
pm
ent
pro
c
ess
is
a
r
ec
e
n
t
tren
d
w
h
ic
h
m
a
rks
a
depa
rt
ure
fro
m
t
he
t
ra
dit
i
o
n
a
l
pra
c
t
i
c
e
o
f
bas
i
n
g
t
e
s
t
d
e
v
e
l
o
p
m
e
n
t
o
n
C
T
T
.
W
i
t
h
I
R
T
,
i
t
e
m
s
a
r
e
c
a
l
i
b
r
at
ed
w
i
t
h
out
r
ef
e
r
en
c
e
t
o
t
h
e
s
a
mp
l
e
i
n
t
e
rms
of
t
h
e
t
r
ai
t
l
e
vel
or
a
b
i
li
ty
l
eve
l
o
f
an
i
n
d
i
v
i
dua
l
referr
ed
t
o
as
t
heta
(
ɵ)
a
n
d
i
te
m
pa
ram
e
te
r
e
s
t
i
m
a
tes.
T
he
item
p
ara
m
e
t
er
e
st
i
m
at
es
a
re
i
t
e
m
di
sc
ri
m
i
n
a
ti
o
n
p
o
w
e
r
(
p
a
ra
me
t
e
r
a
)
,
i
t
e
m
di
ffi
c
ul
t
y
(
p
a
ra
me
te
r
b
)
a
nd
gu
e
s
si
ng
(
param
e
ter
c).
Para
m
e
te
r
‘a’
indicates
degree
to
w
h
i
c
h
a
p
erson’s
re
sp
onse
t
o
a
n
i
t
em
r
ela
t
e
s
o
r
varies
w
ith
h
i
s
/
her
tra
i
t
le
ve
l
or
a
bil
i
t
y
;
pa
ram
e
t
e
r
‘b’
in
dic
a
t
e
s
am
ou
nt
o
f
tra
i
t
i
n
t
he
i
t
e
m;
w
h
ile
p
ar
am
eter
‘
c
’
i
nd
i
ca
tes
pro
b
a
bil
i
t
y
t
ha
t
a
p
e
rson
w
h
o
doe
s
no
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
2,
June
2
0
18 :
1
4
2
– 1
51
14
8
pos
se
ss
t
he
t
r
a
it
w
ill
r
e
s
p
o
n
d
t
o
a
n
i
t
e
m
corr
ectl
y
.
IRT
is
c
on
si
der
e
d
to
b
e
a
p
o
w
e
rful
m
e
th
o
d
o
f
item
selec
t
i
o
n,
w
hic
h
p
ro
v
i
de
s
d
i
ff
er
en
t
e
s
t
i
m
at
es
o
f
erro
r
o
f
m
e
a
s
u
re
me
nt
a
t
e
a
c
h
ab
i
lit
y
l
e
v
e
l.
I
ts
a
pp
lic
a
b
ili
ty
i
n
deve
l
o
p
i
ng
b
e
tter
tes
t
s,
i
t
e
m
bia
s
,
d
i
ffe
r
en
ti
al
i
tem
fu
nc
t
i
o
n
i
n
g
,
i
t
em
b
a
n
ki
n
g
a
nd
t
ai
l
o
red
tes
tin
g
has
be
e
n
st
r
e
ssed
[2
6; 2
7; 2
5].
2.4.
I
t
em
A
na
lysis In
I
RT
Whe
n
e
m
p
l
o
ying
it
e
m
r
espo
nse
t
h
eor
y
,
ite
m
a
n
alysis
c
o
n
s
i
sts
of
(
a
)
determ
in
ing
sa
m
p
le-i
nvar
i
a
n
t
it
e
m
p
ara
m
e
t
e
r
s
usi
n
g
rela
ti
v
e
ly
c
ompl
ex
m
a
t
he
ma
t
i
c
a
l
t
ec
hn
i
que
s
a
n
d
la
rge
sam
p
l
e
s
iz
es,
and
(b)
ut
iliz
in
g
go
o
dne
ss-of-f
it
c
riteria
to
d
e
t
ect
i
tem
s
t
ha
t
do
no
t
fi
t
t
h
e
sp
e
c
i
f
ied
r
e
sp
o
n
se
m
ode
l.
T
h
e
p
rope
rt
y
o
f
s
am
ple
in
varia
n
c
e
i
nhe
rent
w
i
t
h
i
n
I
RT
m
eans
t
h
a
t
t
e
s
t
de
ve
lo
pe
rs
do
n
o
t
n
eed
a
r
ep
re
sen
t
a
t
iv
e
sa
m
p
l
e
o
f
the
e
x
ami
n
ee
p
opu
l
a
ti
on
to
c
a
l
ib
rat
e
t
est
it
e
m
s.
T
h
e
y
do
,
h
o
we
v
e
r,
n
e
e
d
a
h
et
e
r
og
e
n
eous
a
nd
l
a
r
g
e
e
xa
mi
n
ee
s
a
m
p
l
e
t
o
e
n
s
u
r
e
p
r
o
p
e
r
i
t
e
m
p
a
r
a
m
e
t
e
r
e
s
t
i
m
a
t
i
o
n
.
H
o
w
e
v
e
r
,
t
h
e
t
es
t
de
ve
lo
per
using
IRT
i
s
f
a
c
e
d
w
i
t
h
a
di
ffe
re
nt
p
r
o
b
l
e
m
.
Bec
a
use
I
R
T
r
e
qu
ires
l
arge
r
sa
mp
le
s
iz
es
t
o
o
b
ta
i
n
g
o
o
d
ite
m
para
me
ter
es
tima
t
e
s
,
th
e
te
st
deve
l
ope
r
m
u
st
e
nsure
t
h
at
t
he
e
xam
i
nee
s
a
m
p
l
e
i
s
of
s
u
f
f
i
c
i
e
n
t
s
i
z
e
to
g
u
a
ra
n
t
e
e
a
ccu
ra
t
e
i
t
e
m
c
a
lib
ra
ti
on
.
P
oor
i
tem
s
a
r
e
u
su
al
ly
i
de
nt
i
f
ied
t
h
ro
ug
h
a
c
onsi
d
er
at
ion
of
t
h
e
i
r
di
scrimination
indice
s
(the
v
al
ue
o
f
a
i
w
i
l
l
be
a
l
ow
pos
it
i
v
e
or
e
ve
n
ne
g
a
tive
)
a
n
d
d
iffi
cul
t
y
i
n
d
i
ce
s
(
ite
m
s
s
h
o
u
l
d
ne
it
her
be
v
e
r
y
dif
f
ic
u
lt
n
o
r
t
o
o
easy
for
the gr
o
u
p
o
f
stu
de
nt
s to be
a
s
se
sse
d
) [28].
Th
e
a, b
an
d
c
p
arame
t
er
s
1.
The
a
par
am
et
er
O
n
e
feat
ure
of
a
goo
d
t
e
s
t
i
te
m
i
s
t
ha
t
h
i
gh-
abi
lit
y
s
t
u
d
e
n
t
s
w
ill
a
nsw
e
r
it
corre
ctl
y
m
ore
fre
que
n
t
l
y
tha
n
l
ow
er-
a
bili
ty
s
t
u
de
nts.
T
he
i
t
e
m
di
scr
i
m
i
na
t
i
o
n
‘
a’
p
a
r
am
e
te
r
expre
s
se
s
h
o
w
w
ell
an
i
te
m
c
a
n
di
ffe
re
nt
ia
te
a
mong
e
x
a
m
ine
e
s
w
ith
d
i
f
fere
nt
a
b
ili
ty.
A
test
i
t
e
m
h
a
s
p
osit
i
v
e
d
i
scrim
i
na
tio
n
w
h
e
n
l
ow
er
abi
l
i
t
y
s
t
ude
n
t
s
have
a
l
o
w
p
r
oba
b
i
l
i
t
y
o
f
a
n
s
w
e
r
i
ng
a
n
i
t
e
m
co
rr
ectly,
an
d
h
i
gh
e
r
a
bi
li
ty
s
t
ude
n
t
s
ha
ve a
h
ig
h
pro
b
ab
i
lit
y
of
g
et
t
i
n
g
t
he
ite
m
righ
t
.
A
t
es
t
item
has
neg
a
ti
ve
d
i
s
c
r
i
m
i
n
at
i
o
n
(a-v
a
l
u
e
s)
w
h
e
n
h
i
gh
a
b
ilit
y
ca
ndi
da
te
s
ha
v
e
a
l
ow
p
ro
ba
bil
i
t
y
o
f
a
n
sw
er
i
n
g
an
i
tem
corre
ctl
y
an
d
l
o
w
ab
il
i
t
y
ca
ndi
da
te
s
ha
ve
a
h
i
ghe
r
pro
b
ab
i
lit
y
of
a
nsw
e
rin
g
a
n
i
t
em
c
orr
ect
ly.
The
d
i
scrim
i
na
tio
n
va
lues
o
f
a
g
o
o
d
tes
t
i
te
m
range
s
be
tw
e
e
n
0.5
t
o
2
a
n
d
t
h
e
s
t
e
e
p
e
r
t
h
e
s
l
o
p
e
o
f
a
n
i
t
e
m
c
h
a
r
a
c
t
e
r
i
s
t
i
c
c
u
r
v
e
,
t
he
h
i
g
he
r
a
n
i
t
e
m’
s
dis
c
rimina
ti
o
n
v
a
l
ue
s
(a
-
val
u
es).
H
igh
di
sc
rimina
ti
on
l
e
v
e
l
i
n
d
i
ca
t
e
s
t
h
a
t
t
he
item
d
i
sc
rimi
na
tes
w
e
ll
be
tw
ee
n
l
o
w
an
d
h
i
gh
s
kil
l
e
d
in
div
i
dua
ls.
A
di
scrim
i
na
ti
on
pa
ram
e
te
r
is
a
m
ea
sur
e
t
h
a
t
can
b
e
g
r
a
p
h
i
c
a
ll
y
e
x
pressed
by
t
h
e
s
t
ee
pne
ss
o
f
the
it
e
m
c
har
a
cter
i
s
ti
cs
c
ur
v
e
(ICC).
The
Ite
m
d
isc
r
imina
t
i
on
v
al
ue
(
a-va
lue
s
)
abo
v
e
1
i
s
norm
a
ll
y
de
s
i
r
a
b
l
e
for
a
go
o
d
t
es
t
item
a
nd
a-
va
lue
s
a
b
ove
0
.
7
5
c
a
n
a
l
so
b
e
acc
ept
able.
Interpreting
disc
rim
i
nation
para
meter
val
u
es
a
re
pr
e
sen
t
ed
i
n
Tab
l
e
1.
T
a
b
l
e 1.
Inte
r
preti
ng D
i
scr
i
m
i
nat
i
o
n
P
a
ram
e
te
r V
a
l
u
e
s
[29]
D
i
sc
ri
m
i
na
t
i
on
Va
lue
Qua
lit
y
of a
n
Item
a
≥
1.
70
I
t
e
m
i
s
func
ti
on
i
ng
qu
i
t
e
s
a
tisf
a
c
t
orily
1.
35
≤
a
≤
1.
69
0.
65
≤
a
≤
1.
34
Good
ite
m
;
l
i
ttl
e
or
no
re
visi
on i
s
r
e
quire
d
Mo
de
r
a
t
e
: litt
l
e
or no r
e
vision is re
qui
r
e
0.
35
≤
a
≤
0.
64
I
te
m
is m
a
r
gi
n
a
l a
nd
nee
d
r
e
v
i
s
ion
a
≤
0.
34
Poor i
tem
;
s
hould
be
e
l
i
m
i
n
a
t
e
d
or
r
e
v
ise
d
2.
The
b
par
am
et
er
Item
d
i
f
fic
u
l
t
y
r
efers
t
o
t
he
b
p
ar
am
eter
,
is
t
he
p
o
i
nt
w
h
e
re
t
he
S
-sha
p
e
d
c
u
rve
has
t
h
e
s
t
eep
e
s
t
slope
.
E
x
a
m
in
ee
m
ust
ha
ve
h
ig
he
r
ab
i
lit
y
in
o
r
d
er
t
o
a
n
sw
er
a
d
if
ficult
i
t
e
m
c
o
rrec
tly.
Item
w
it
h
h
i
g
h
b
val
u
es
i
s
a
hard
o
r
d
i
ffic
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lt
i
t
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,
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ha
t
i
s
,
va
l
u
e
of
b
g
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e
a
t
er
t
han
1
in
d
i
ca
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a
di
ff
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ul
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it
em
a
nd
l
ow
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i
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ity
exa
m
ine
e
s
a
r
e
m
ore
li
kel
y
t
o
fa
i
l
b
e
c
a
u
se
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he
y
w
i
l
l
f
i
n
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it
d
i
f
fic
u
l
t
t
o
a
n
sw
er
c
or
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t
l
y
.
S
i
m
i
l
a
rl
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a
n
ite
m
w
ith
l
ow
b
v
a
l
ue
b
e
l
ow
-
1
i
n
dica
te
e
as
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m
,
w
h
ich
mos
t
o
f
t
h
e
e
xa
m
i
n
e
e
s
w
i
t
h
l
o
w
abi
lit
y
leve
l,
w
ill
hav
e
at
l
ea
s
t
a
m
o
d
e
ra
t
e
c
ha
nce
o
f
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nsw
e
rin
g
i
t
corre
ctl
y
.
W
h
e
n
a
n
i
te
m
ha
s
a
b-
va
l
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e
of
b
etw
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-
1.00
t
o
1.
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t
h
i
s
v
alu
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ndic
a
t
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s
a
n
i
t
e
m
wi
th
m
od
e
r
a
t
e
di
ff
i
c
ult
y
[
2
9
]
.
In
t
e
r
pre
tin
g
item
d
i
f
f
icu
l
t
y
v
al
ues
are
pr
e
s
en
ted
in Table 2.
Tab
l
e 2.
Inte
r
preti
ng I
t
e
m
D
iffic
u
lt
y V
a
l
u
e
s
[2
9
]
Diffic
u
lt
y v
a
lu
e
(b
)
I
nt
e
r
pr
e
t
a
t
i
o
n
-3.
00≤
-
2.
00
-2.
00
≤-
1.
00
-1.00 ≤1.00
V
e
r
y
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y
Ea
sy
Mode
r
a
t
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c
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lt
1
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Di
f
f
ic
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l
t
b
>
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.0
0
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if
fic
u
lt
Evaluation Warning : The document was created with Spire.PDF for Python.
IJERE
I
S
S
N
:
2252-
88
22
I
t
e
m
Re
sponse
T
h
e
o
ry:
An Int
r
od
uc
t
i
o
n
t
o
L
a
t
e
n
t
T
r
ai
t
Mo
del
s
to
T
e
st …
(Ado
Abd
u
Bic
h
i)
14
9
3.
The
c pa
ram
e
te
r
The 3P
LM i
n
c
lude
a
p
se
u
do-
g
u
e
ssin
g
p
a
r
am
ete
r
know
n
as c-pa
r
am
et
e
r
,
t
h
i
s
p
a
r
amet
er
e
xp
res
s
es
t
h
e
p
r
ob
a
b
il
i
t
y
t
h
at
a
n
e
x
amin
ee
wi
th
l
o
w
a
b
ili
t
y
c
an
b
e
ab
le
t
o
g
e
t
a
n
i
t
e
m
correctly
a
nd,
t
herefore,
has
a
grea
ter-
t
h
a
n
-ze
r
o
pro
b
a
b
i
lit
y
of
a
nsw
e
r
i
ng
a
n
item
corre
ct
l
y
i
n
a
t
es
t.
T
he
g
u
e
ssi
n
g
para
me
t
e
r
c
is
t
he
l
ow
e
s
t
v
a
lu
e
t
h
at
a
n
ICC
at
t
a
in
s.
F
or
e
x
a
mp
l
e
,
a
st
ud
en
t
,
w
ho
r
a
n
do
mly
s
e
l
e
c
t
s
r
e
spo
n
se
s
t
o
i
tem
s
t
ha
t
have
f
o
u
r
r
e
s
p
o
n
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e
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h
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e
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c
a
n
a
n
s
w
e
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t
h
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s
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t
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m
s
c
o
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t
l
y
a
b
o
u
t
1
o
u
t
o
f
4
t
i
m
e
s,
m
ea
ni
n
g
t
ha
t
the
pro
b
a
b
i
l
ity
o
f
gue
ssi
ng
corr
ectl
y
is a
b
o
u
t
0.
25.
2.5.
It
e
m S
e
l
e
ction
in
I
R
T
A
s
i
s
t
h
e
c
a
s
e
w
ith
c
lass
ica
l
t
es
t
the
o
r
y
,
it
em
r
esponse
the
o
r
y
a
l
s
o
bas
e
s
i
t
s
it
e
m
s
e
l
ec
tio
n
on
t
h
e
pur
pose
o
f
t
he
t
es
t.
T
h
e
f
i
n
a
l
s
e
l
e
c
t
i
on
of
t
e
s
t
item
s
w
i
ll
d
e
p
e
n
d
o
n
t
he
i
nfor
ma
t
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ac
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item
c
o
ntr
i
b
u
te
t
o
the
ove
ra
ll
in
form
ati
on
s
u
p
p
l
i
e
d
by
the
test.
A
n
e
spe
c
ia
l
l
y
use
ful
fea
t
ure
of
t
he
i
tem
in
for
m
a
t
i
o
n
f
unc
tio
ns
use
d
i
n IRT
tes
t
de
v
e
l
opm
en
t
i
s
t
ha
t
,
t
hey
pe
rm
it t
h
e t
e
st
d
e
v
e
lop
e
r
to
d
e
t
e
r
mi
n
e
t
h
e
c
ont
ri
but
ion
of
e
a
c
h i
t
em
to
t
he
t
es
t
inf
o
rm
ation
fu
nc
t
i
o
n
i
n
d
epe
n
de
nt
o
f
ot
her
ite
m
s
i
n
t
h
e
tes
t
.
A
s
o
u
t
l
i
ne
d
by
[1
5]
a
p
ro
c
e
dure
,
ori
g
ina
lly
c
onc
ept
u
a
l
i
zed
by
[30],
for
t
h
e
use
of
i
t
e
m
i
n
for
m
a
t
io
n
fu
nc
ti
ons
i
n
t
h
e
te
st
b
ui
ldi
n
g
pr
oce
ss.
T
hi
s
proce
dure
ent
a
il
s
tha
t
a
test d
e
ve
l
ope
r ta
ke t
he
f
o
l
low
i
n
g
f
our
s
t
e
p
s
:
First,
d
e
s
cri
b
e
the
sha
p
e
o
f
t
h
e
d
es
ired
t
e
s
t
i
n
form
ati
o
n
fu
n
c
t
i
o
n
over
t
h
e
des
i
r
e
d
ra
n
g
e
of
a
b
i
l
i
t
i
es.
[1
5]
c
a
l
l
s
t
h
i
s
t
h
e
tar
g
e
t
i
nfor
m
a
tio
n
fu
n
c
tio
n.
S
ec
ond,
s
e
l
e
c
t
i
t
e
m
s
w
i
th
item
in
for
m
a
t
i
on
func
t
i
o
n
s
t
h
at
w
i
l
l
f
ill
up
the
hard-to-fill
a
r
eas
u
nd
er
t
h
e
t
a
r
ge
t
i
n
fo
rma
t
i
on
fun
c
tio
ns.
Th
i
r
d,
a
fter
e
ac
h
ite
m
is
a
dd
e
d
,
the
n
ca
l
c
u
l
a
t
e
t
h
e
tes
t
i
n
f
or
ma
t
i
on
f
u
n
ct
i
on
for
t
h
e
selec
t
e
d
t
es
t
i
tem
s
.
F
ourt
h
,
sele
c
t
i
tem
s
u
n
t
il
t
h
e
te
st
in
form
ation
fu
nc
ti
on
a
p
pro
x
i
m
a
t
e
the
tar
g
e
t
i
n
f
orm
a
ti
o
n
f
u
n
c
tio
n
t
o
a
s
at
isfac
t
or
y
le
ve
l
.
L
as
t
l
y,
c
o
n
t
e
nt
val
i
d
at
io
n
cons
idera
t
i
o
ns a
re moni
t
o
red
d
u
ri
ng t
h
e i
t
em
se
l
e
c
ti
on process.
Th
is
p
roce
d
u
re
a
llow
s
t
he
t
es
t
de
vel
o
per
t
o
bui
ld
a
t
e
s
t
th
at
w
il
l
pre
c
i
se
l
y
f
u
l
fi
l
a
ny
se
t
of
d
es
ired
tes
t
s
pe
c
i
fica
ti
ons.
T
h
us,
i
t
i
s
p
o
ss
ib
l
e
t
o
b
u
i
l
d
a
t
est
th
a
t
"
d
iscr
imina
t
es"
w
e
l
l
a
t
a
n
;
pa
rt
i
c
u
l
a
r
r
e
g
ion
o
n
t
h
e
a
b
i
l
i
t
y
c
o
n
t
i
n
u
u
m
.
T
h
a
t
i
s
t
o
s
a
y
,
i
f
w
e
h
a
v
e
a
g
o
o
d
i
d
e
a
o
f
t
h
e
a
b
i
lit
y
o
f
a
g
roup
o
f
ex
ami
n
e
e
s
t
e
s
t
it
ems
ca
n
be se
l
ec
ted
so
as to m
ax
imise
t
e
s
t
i
n
f
orm
a
ti
on,
t
he re
g
io
n
of
a
bil
i
t
y
sp
an
ne
d
by
t
he e
xam
i
nees
b
e
i
n
g
tes
t
e
d, of
course
,
th
is
o
p
t
im
um
s
elec
t
i
o
n
o
f
t
e
s
t
item
s
w
ill
c
o
ntri
b
u
te
s
u
bs
t
a
ntia
l
l
y
to
t
he
p
r
ecis
i
on
w
i
th
w
h
i
c
h
a
bi
l
i
t
y
sc
o
r
es
a
re
e
sti
m
at
ed
.
Fu
rth
e
rmo
r
e,
w
ith
c
r
i
terion-referenced
t
e
st
s,
i
t
i
s
c
omm
on
t
o
obser
ve
l
o
w
e
r
t
e
s
t
perform
ance
on
a
pr
e-test
t
ha
n
on
a
po
st-t
est.
G
ive
n
t
hi
s
kn
ow
l
e
dge,
a
test
i
ns
truc
t
o
r
sho
u
l
d
se
lec
t
e
asi
e
r
it
e
m
s
for
the
pre-
t
e
st
a
nd
m
o
re
d
iffic
u
lt
item
s
f
or
t
he
pos
t-te
s
t
.
Th
en,
for
bo
th
t
est
i
n
g
a
dm
i
n
i
s
trat
io
ns,
me
asure
m
e
n
t
pr
ecis
i
o
n
w
i
ll
have
b
ee
n
m
a
xim
i
ze
d
in
t
he
a
bi
l
ity
r
eg
io
n
w
h
ere
t
h
e
ex
am
inee
s
w
oul
d
m
o
st
like
l
y
be
l
oca
t
ed.
M
o
reo
v
e
r
,
beca
u
s
e
item
s
o
n
bo
th
t
est
s
m
easur
e
t
h
e
sa
m
e
a
bi
lit
y,
a
n
d
ab
il
it
y
e
s
tim
a
t
es
a
r
e
in
de
pen
d
e
n
t
of
t
he
p
art
i
c
u
lar
cho
i
ce
o
f
t
e
s
t
item
s
,
the
i
n
str
u
c
t
o
r
c
a
n
m
e
a
s
ur
e
grow
t
h
by
sub
t
rac
t
i
ng
the
pr
e-
tes
t
abi
l
ity
e
st
im
ate
fr
o
m
t
he
pos
t-te
st ab
i
li
ty
e
st
i
m
ate
2.6.
Be
n
e
f
i
t
s of
I
t
e
m R
e
sp
o
n
se
T
h
e
or
y
Item
R
e
s
po
nse
The
o
r
y
m
ea
sur
e
m
e
nt
m
o
d
e
l
s,
w
he
n
c
o
m
p
ar
ed
t
o
c
l
ass
ica
l
m
ode
ls,
offer
se
v
e
ra
l
di
st
i
n
ct a
dva
n
t
ages.
The
s
e
i
n
clu
d
e the
f
o
ll
o
w
in
g:
(a
)
I
t
e
m sta
t
i
s
t
i
cs a
re in
d
epe
n
de
nt o
f t
h
e sam
p
le fr
o
m w
h
ich
the
y
w
e
r
e
esti
m
a
ted
(b)
Exa
m
inee
s
c
o
re
s
ar
e
in
depe
n
d
e
n
t
of
t
es
t
d
ifficulty
(c)
Item
a
naly
sis
acc
omm
oda
te
s
mat
c
h
i
ng
t
est
i
t
e
ms
t
o
ex
ami
n
e
e
k
no
wl
e
d
ge
l
e
v
el
(
d
)
T
est
an
a
l
ys
is
d
oe
sn’t
r
e
q
u
i
re
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trict
par
a
l
l
e
l
t
e
s
ts
f
o
r
assessi
n
g
re
l
ia
b
i
l
ity
(
E) Item
sta
tis
tic
s
an
d
exam
inee
a
b
i
l
i
t
y
a
r
e
bot
h re
por
t
e
d o
n
t
he
sam
e
scale
[31]
.
2.7.
Limitati
o
ns
o
f It
em
R
espon
s
e
T
h
eory
IRT
m
ode
ls
h
a
v
e
se
ver
a
l
tec
h
ni
c
a
l
a
nd
prac
tic
al
s
h
o
rtc
o
m
i
n
g
s.
A
ssu
mp
tio
n
s
und
e
r
l
y
in
g
t
h
e
u
s
e
of
IRT
model
s
a
re
m
ore
s
t
ringent
th
an
t
ho
s
e
r
e
q
ui
red
of
c
l
a
ssi
c
a
l
test
t
he
ory.
I
RT
m
o
d
e
l
s
als
o
t
en
d
t
o
b
e
mor
e
com
p
le
x
a
nd
t
h
e
mode
l
o
u
t
p
u
t
s
h
arde
r
t
o
u
n
d
e
r
s
t
an
d,
p
art
i
cu
lar
l
y
w
i
th
n
on-
tec
h
nic
a
l
l
y
orie
nte
d
a
u
d
i
e
n
c
e
s.
A
d
d
iti
ona
l
l
y
,
IRT
mod
e
ls
r
e
q
uire
l
a
r
ge
s
am
ple
s
t
o
o
b
ta
i
n
a
c
c
ura
t
e
a
n
d
s
t
a
b
l
e
p
a
r
a
m
e
t
e
r
e
s
tim
ate
s
,
alth
ou
g
h
Ra
sc
h
me
asure
m
e
n
t
m
ode
ls
a
re
u
se
fu
l
w
ith
s
ma
l
l
t
o
mo
der
a
te
s
am
p
le
s.
C
onse
q
ue
nt
ly,
t
h
e
ch
o
i
ce
o
f
a
m
ode
l
ma
y de
pen
d
o
n
t
h
e
sam
p
le a
v
a
ila
b
l
e,
par
t
i
c
u
lar
l
y
in t
he
f
iel
d
-t
es
tin
g
pha
se
o
f a
ce
rtifica
t
i
on e
x
am
.
3.
CONCL
U
S
ION
In
c
onc
l
u
s
i
o
n
,
there
a
r
e
many
l
i
m
ita
t
i
o
n
s
in
t
h
e
C
T
T
t
ha
t
c
once
rn
s
wi
t
h
c
a
lib
ra
ti
on
o
f
i
t
e
m
di
ffic
ul
t
y
,
sam
p
le
d
e
p
e
nde
nc
y
of
c
oe
ffic
ien
t
m
ea
sures,
a
n
d
e
s
t
i
ma
tes
of
m
ea
surem
e
nt
e
r
r
or
w
hic
h
i
n
turn
i
s
addre
s
se
d
b
y
t
he
I
R
T
.
exam
ple
I
R
T
mo
de
ls
r
epre
sen
t
t
he
a
b
i
l
i
t
y
of
t
he
e
xam
i
ne
es
a
nd
t
he
d
iffi
c
u
l
t
y
of
t
he
it
e
m
s
as
i
n
d
e
p
ende
nt
p
ar
am
eters.
I
t
c
a
n
se
par
a
te
t
hese
t
w
o
p
ara
me
ters
e
m
p
irica
l
ly
i
n
a
w
a
y
tha
t
n
o
othe
r
psy
c
h
o
me
tric
m
ode
ls
c
a
n
do.
S
im
i
l
a
r
ly,
In
c
on
trast
t
o
C
TT
t
he
e
s
t
i
m
a
t
i
on
f
r
a
me
wo
rk
o
f
IR
T
mod
e
ls
m
a
k
e
i
t
st
r
a
i
g
h
t
forw
ar
d
t
o
a
na
lyse
i
tem
s
t
ha
t
ha
v
e
r
a
ndom
m
issing
da
ta.
I
R
T
c
a
n
s
t
i
l
l
c
a
l
i
b
r
a
t
e
i
t
e
m
s
a
n
d
s
c
o
r
e
su
bj
ect
s
by
u
si
ng
a
l
l
t
h
e
av
ai
l
a
bl
e
in
fo
rmat
i
on
b
a
sed
on
t
h
e
l
i
ke
l
i
ho
od
;
the
like
lih
oo
d-
ba
se
d
me
tho
d
s
ar
e
implem
e
n
te
d
i
n
t
he
I
RT
p
r
o
cedur
e.
I
RT
d
iffers
c
o
n
si
der
a
bly
fr
om
t
he
l
i
n
ea
r
a
p
proa
c
h
t
o
te
st
a
nd
i
te
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2252-
8822
IJERE
V
ol
.
7,
N
o.
2,
June
2
0
18 :
1
4
2
– 1
51
15
0
ana
l
ys
is
(
CTT
)
a
nd
h
as
s
om
e
ide
n
t
i
f
ie
d
cr
ucia
l
the
o
r
e
t
i
c
a
l
an
d
em
pi
r
i
c
a
l
ga
ins
over
CTT
d
u
e
to
t
h
i
s,
it
is
expe
c
t
e
d
t
ha
t
t
h
ere
w
o
u
l
d
be
a
ppr
ec
iab
l
e
d
i
ffe
r
enc
e
s
be
t
w
ee
n
t
h
e
tw
o
a
nd
I
R
T
base
d
pa
ram
e
ter
e
s
tim
ates
sho
u
l
d
b
e
s
u
p
e
rior
a
n
d
r
el
i
a
ble
t
h
a
n
C
TT
b
a
s
e
d
p
ar
am
eter
e
st
ima
t
e
s.
W
i
t
h
t
h
es
e
f
e
a
t
u
r
e
s
,
IR
T
can
h
elp
resol
v
e
the pr
oble
m
s a
sso
c
i
a
t
ed w
i
t
h te
st
d
e
s
ign
base
d
o
n
CTT.
REFE
RENCES
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F
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“
Se
con
d
a
r
y T
e
ache
rs’ Assess
m
e
nt
a
n
d
G
r
adi
n
g
P
r
ac
t
i
ces
i
n
I
n
c
l
usiv
e
Class
r
oom
s”
.
A
Thes
is
S
ubm
i
tte
d
to
t
he
C
olle
ge
o
f
G
r
ad
uate
S
t
u
die
s
a
n
d
R
e
s
ear
c
h
i
n
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a
r
tia
l
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u
l
f
i
llme
n
t
of
t
h
e
Re
qu
i
r
em
ents for
the
D
egre
e o
f
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ter
of
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duca
t
io
n,
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ni
v
e
rsity
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aska
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u
al
ity
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c
ed
ure
s
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idi
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g
t
h
e De
vel
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t
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ssica
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t
Th
e
o
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a
nd
I
t
e
m
Re
sp
on
se
T
h
e
o
r
y
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the
i
r
A
p
p
lic
at
ion
s
t
o
Te
st
D
eve
l
o
p
m
e
n
t
”.
Educa
t
i
o
n
a
l
Measur
em
e
n
t
:
Issues and P
r
a
c
ti
c
e
,
1
2
(3)
,
38-
47,
1
9
9
3
[7]
B
o
c
k
,
R
.
D
.
"
A
B
r
i
e
f
H
i
s
t
o
r
y
o
f
I
t
e
m
R
e
s
p
o
n
s
e
T
h
e
o
r
y
.
"
Edu
c
ati
o
n
a
l
Me
a
s
u
r
eme
n
t
:
Issu
es
and
Prac
tice
,
16
(
4
)
, 21–3
3,
19
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[8]
Bine
t,
A
.,
&
S
im
on,
T
.
“
M
e
t
ho
ds
N
o
u
v
el
le
s
P
our
L
e
D
i
agn
o
s
t
ic
D
u
N
ie
vea
u
I
nte
l
le
c
t
ual
A
nor
mou
x
[N
e
w
m
ethods
f
or
the
d
i
a
gnos
is of le
ve
ls of i
n
te
l
l
e
c
t
ua
l a
b
n
o
rm
a
lity]”
.
AnneePsy
c
h
o
l
o
g
i
que
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[9]
Lord,
F.
M
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&
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M.
R
.
"
S
t
at
i
s
ti
ca
l Th
e
o
ri
e
s
of Ment
al
He
a
l
th
S
c
o
r
e
s
”
.
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s
o
n
W
e
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e
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ead
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r
etson, S. E.
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nd
boo
k
of
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e
rn
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e
m
Re
spo
n
s
e
Th
e
o
ry
.
Ne
w Yo
rk
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p
ri
ng
e
r
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r
l
a
g
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i
n
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a
ten
t
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l
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i
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a
te
n
t
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r
ait Mode
ls
”
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a
m
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let
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.
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a
t
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n,
H
.,
&
R
oger
s
,
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.
J
.
“
Fu
nd
am
en
t
a
l
s
o
f
Item
Re
sp
ons
e T
h
eo
ry”
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e
w
bur
y P
a
rk, CA
:
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a
g
e
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ubli
c
a
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ons,
19
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u
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i
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r
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.
S
.,
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83).
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r
es
po
nse
t
h
eor
y
:
A
p
p
l
ica
t
i
ons
t
o
psy
c
h
o
l
og
ic
al
m
easur
em
ent. H
ome
w
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I
L: Ir
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i
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.
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“
App
l
ica
t
i
o
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Item
Re
sp
on
se T
h
eory t
o
Pra
c
tic
a
l
T
e
sti
n
g
Prob
lem
s
”
.
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n
ce
E
r
l
ba
um
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c
huma
c
ker
,
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.
E.
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I
t
e
m
R
es
p
onse
The
o
ry”.
2
0
1
0
b
ht
tp:
/
/
a
p
p
lie
dm
ea
surem
e
ntass
o
cia
t
e
s
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m
a
/assets
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i
l
e/
ITEM_
R
E
S
P
ONS
E_THEORY.pd.
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etr
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ugus
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[17]
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a
m
b
let
on,
R
.
K
.,
&
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w
a
m
i
n
a
t
ha
n,
H
.
“
It
e
m
res
p
on
se the
o
ry
: Prin
ci
pl
es
an
d
ap
pli
c
at
io
n
s
(
V
o
l
.
7
)
”
:
S
p
r
i
n
g
e
r
,
1985
[18]
Ma
gis,
D
.
“
I
nfl
u
e
n
c
e
,
Informa
ti
o
n
a
n
d
I
te
m
Re
spo
n
se
T
heor
y
in
D
iscr
ete
D
a
ta
A
nal
y
s
i
s”
.
20
0
7
ht
tp:
/
/
b
i
c
t
e
l
.
u
lg
.
a
c
.
be/ETD
-
db/c
o
l
l
ec
t
i
o
n
/a
va
ila
b
l
e
/
U
L
ge
t
d
0
6
1
2
2
0
07-
10
0
1
4
7
/
.
A
c
cesse
d
o
n
20
June
,
2
014
.
[19]
Oj
e
r
i
n
d
e
,
D.
“
Cl
assi
ca
l
Te
st
T
h
e
o
r
y
(C
TT)
V
S
It
e
m
R
esp
o
n
s
e
Th
eo
ry
(
IRT):
A
n
E
val
u
a
t
i
o
n
of
t
he
C
o
m
p
a
r
a
b
i
l
i
t
y
o
f
I
t
e
m
A
n
a
l
y
s
i
s
R
e
s
u
l
t
s
”
.
A
g
u
e
s
t
l
e
c
t
u
r
e
p
r
e
s
e
n
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