latent rank theory: test theory for making can-do chart
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Latent Rank Theory:Test Theory for Making Can-Do Chart
SHOJIMA KojiroThe National Center for University Entrance Examinations, Japan
shojima@rd.dnc.ac.jp
1
Accuracy• Scale (Weighing machine)
– A1 weighs 73 kg
– fW(A1)=73
• fW (A1)≠74
• fW (A1)≠72
• Academic test– B1 scores 73 points
– fT(B1)=73
• fT(B1)≠74 ?
• fT(B1)≠72 ?
Discriminating Power• Scale (Weighing machine)
– A1 weighs 73 kg
– A2 weighs 75 kg
• fW(A1)<fW (A2)
• Academic test– B1 scores 73 points
– B2 scores 75 points
• fT(B1)<fT (B2) ?
Resolution• Scale (Weighing machine)
– A1 weighs 73 kg
– A2 weighs 75 kg
– A3 weighs ...
• Academic test– B1 scores 73 points
– B2 scores 75 points
– B3 scores ...kgT
Test Limitations• Precise measurement is almost impossible
– CTT reliabilities: 10% measurement error
• A test is at best capable of classifying academic ability into 5–20 levels
• Why continuous scale?– Classical Test Theory: Continuous Scale
– Item Response Theory: Continuous Scale
• Common European Framework of Reference for Languages (CEFR)– 6 levels: A1, A2, B1, B2, C1, C2
Graded evaluation↓
Accountability↓
Qualification testOrdinal academic ability evaluation scale based on Neural Test Theory
Ordinal academic ability evaluation scale based on Neural Test Theory
Continuous academic ability evaluation scale based on IRT or CTT
Continuous academic ability evaluation scale based on IRT or CTT
It is difficult to explain the relationship between scores and abilities because individual abilities also change continuously
It is difficult to explain the relationship between scores and abilities because individual abilities also change continuously
Because the individual abilities also change in stages, it is easy to explain the relationship between scores and abilities. This increases the test’s accountability.
Because the individual abilities also change in stages, it is easy to explain the relationship between scores and abilities. This increases the test’s accountability.
Latent Rank Theory(or Neural Test Theory)
• A test theory– Ordinal scale (not continuous scale)
– Self-organizing map (SOM) or generative topographic mapping (GTM) mechanism
• Shojima, K. (2009) Neural test theory. K. Shigemasu et al. (Eds.) New Trends in Psychometrics, Universal Academy Press, Inc., pp. 417-426.
• Shojima, K. (2011) Local dependence model in latent rank theory. Jpn J of Applied Statistics, 40, 141-156.
Statistical Learning Framework in LRT
・ For (t=1; t ≤ T; t = t + 1)
・ U(t)←Randomly sort row vectors of U
・ For (h=1; h ≤ N; h = h + 1)
・ Obtain zh(t) from uh
(t)
・ Select winner rank for uh(t)
・ Obtain V(t,h) by updating V(t,h−1)
・ V(t,N)←V(t+1,0)
Point 1
Point 2
LRT Mechanism (SOM)
0
0
0
1
0
0
0
1
0
0
0
1
0
1
1
1
1
0
1
0
1
0
0
1
Latent Rank Scale
Nu
mb
er
of
Item
s
InputPoint 1Point 2 Point 1Point 2
Point 1: Winner Rank Selection
Bayes
ML
)1,()()1,()(
1
)()1,()( 1ln1ln)|(
htqj
thj
htqj
thj
n
j
thj
htth vuvuzp Vu
Likelihood
)|(lnmaxarg: )1,()()(
htt
hQq
MLw pwR Vu
)(ln)|(lnmaxarg: )1,()()(q
htth
MAPw fppwR
Vu
Point 2: Update the Reference Vectors
• The nodes of the ranks nearer to the winner are updated to become closer to the input data
• h: tension
• α: size of tension
• σ: region size of learning propagation
)1,(')(')()()1,(),( )()'( htQ
thQ
th
tn
htht V1u1zh1VV
1
)1()(1
)1()(
2
)(exp
)1(}{
1
1
22
2)(
)()(
T
ttTT
ttT
Q
wq
N
Qh
nh
Tt
Tt
t
ttqw
tqw
t
h
Example• A geography test of the NCT
N 5000n 35Median 17Max 35Min 2Range 33Mean 16.911Sd 4.976Skew 0.313Kurt -0.074Alpha 0.704
0 5 10 15 20 25 30 35SCORE
0
100
200
300
400
500
YCNEUQERF
Fit Indices and N of RanksN of ranks is
10N of Ranks is 5
Item Reference Profile (IRP)
Monotonic increasing constraint can be imposed.
Test Reference Profile (TRP)
• Strongly ordinal alignment condition (SOAC)– All IRPs increase monotonically TRP also increases monotonically
• Weakly ordinal alignment condition (WOAC)– TRP increases monotonically, but not all IRPs increase monotonically
• For the scale to be ordinal, at least the WOAC must be satisfied.
• (Weighted) sum of IRPs• Expected score at each
latent rank
Rank Membership Profile (RMP)
• Posterior distribution of the latent rank to which each examinee belongs
Q
q qqi
qqiiq
fpp
fppp
1' '' )()|(
)()|(
vu
vuRMP
Examples of RMP
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 11
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 12
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 13
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 14
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 15
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 6
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 7
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 8
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 9
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 10
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 1
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 2
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 3
2 4 6 8 10LATENT RANK
0
0.2
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0.6
0.8
1
YTILIBABORP
Examinee 4
2 4 6 8 10LATENT RANK
0
0.2
0.4
0.6
0.8
1
YTILIBABORP
Examinee 5
17
Extended Models• Graded LRT Model (RN07-03)
– LRT model for ordinal polytomous data
• Nominal LRT Model (RN07-21)– LRT model for nominal polytomous data
• Continuous LRT Model
• Multidimensional LRT Model
Graded LRT ModelBoundary Category Reference Profiles
0 0 01
1 1
2
2
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
0 0 01 1 1
2
2
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N KPR
OB
AB
ILIT
Y
0 0 01 1 1
22
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
0 0 0
11
1
2
2
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
0 0 0
11 1
2
2
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
0 0 0
11
1
2
2
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N KPR
OB
AB
ILIT
Y
Graded LRT ModelItem Category Reference Profile
00 0
11
1
22
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
0 0 0
11
1
22
2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N KPR
OB
AB
ILIT
Y
0 0 0
11
1
2 2 23
33
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
00
0
11
12
2 2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
00 0
11
1
22 2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
00
0
1 1
122 2
33
3
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N KPR
OB
AB
ILIT
Y
Nominal LRT ModelItem Category Reference Profile
*Correct selection, x Combined categories selected less than 10% of the time
2 22
3 33
4 4 4x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
1 11
33
3
44
4
x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
2 2 23 3
3
4 44
x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
2 22
33
3
4 4 4x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
22 2
3 3 3
4
44
x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
11
12 2 23 3 34
4
4
x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
3
33
44 4
x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
22
2
3 3
3x x x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
44 4
xx x
2 4 6 8 1 00 .0
0 .2
0 .4
0 .6
0 .8
1 .0
L A T E N T R A N K
PRO
BA
BIL
ITY
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