Estimation. The Model Probability The Model for N Items — 1 The vector probability takes this form...
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![Page 1: Estimation. The Model Probability The Model for N Items — 1 The vector probability takes this form if we assume independence.](https://reader036.fdocuments.in/reader036/viewer/2022082713/5697bfd11a28abf838cab3d8/html5/thumbnails/1.jpg)
Estimation
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The Model
expPr ; ,
1 expni n i
ni ni n in i
xX x
0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3 4
n i
Pro
babi
lity
expPr 1; ,
1 expn i
ni n in i
X
![Page 3: Estimation. The Model Probability The Model for N Items — 1 The vector probability takes this form if we assume independence.](https://reader036.fdocuments.in/reader036/viewer/2022082713/5697bfd11a28abf838cab3d8/html5/thumbnails/3.jpg)
The Model for N Items — 1
1
1
Pr ; , Pr ; ,
exp
1 exp
I
n n n ni ni n ii
Ini n i
i n i
X x
x
X x δ
1 2Let , , ,
denotea response vector for person
T
n n n nIX X X
n
X
The vector probability takes this form if we assume independence
![Page 4: Estimation. The Model Probability The Model for N Items — 1 The vector probability takes this form if we assume independence.](https://reader036.fdocuments.in/reader036/viewer/2022082713/5697bfd11a28abf838cab3d8/html5/thumbnails/4.jpg)
The Model for N Items — 2
1
1
1
1
1
expPr ; ,
1 exp
exp
1 exp
exp
1 exp
Ini n i
n n ni n i
I
ni n ii
I
n ii
I
n n ni ii
I
n ii
x
x
r x
X x δ
1
the raw scoreof person
I
n nii
r x
n
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Probabilities of scoring 2 for different response patterns
-0.01
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
n
Pro
babi
lity
Mode is ability that makes this pattern
most likely
Likelihood principle: Which ability maximises the probability of what was obtained?
All modes at same location
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Graphical Display of the likelihoods for a five item test
0
0.01
0.02
0.03
0.04
0.05
0.06
Pr 1; ,n nR δ
Pr 0; ,n nR δ Pr 5; ,n nR δ Pr 3; ,n nR δ Pr 2; ,n nR δ
Pr 4; ,n nR δ
n
Pro
babi
lity
All items with difficulty parameter zero
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Estimation Methods
• Approximate Methods– PROX– UFORM
• Pair-wise• Minimum Chi-square• Maximum likelihood
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Maximum Likelihood Methods–1
• Joint maximum likelihood– also called unconditional maximum likelihood
(UCON)– method used in Quest, WinSteps, Facets,
ConQuest, TAM– Ability and difficulty estimates
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Maximum Likelihood Methods–2
• Marginal Maximum Likelihood– really a new model that invokes a population
distribution assumption– Works well with more general models– ConQuest, TAM, default method
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Joint Maximum Likelihood
1
1 1
1 1
1 1
1 1
1 1
Pr ; ,
exp
1 exp
exp
1 exp
exp
1 exp
N
n n nn
N Ini n i
n i n i
N I
ni n in i
N I
n in i
N I
n n i in i
N I
n in i
L
x
x
r s
X X x δ
1
1
scoreof person
scoreof item
I
n nii
N
i nin
r x n
s x i
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Maximising the Joint Likelihood — 1
1 1
1 1
1 1
1 1
log
exp
log1 exp
log 1 exp
N I
n n i in i
N I
n in i
N I
n n i in i
N I
n in i
L
r s
r s
X X
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Maximising the Joint Likelihood — 2
1 1
1 1
1
log 1 exp
exp
1 exp
0
N I
n n i in it t
N I
n in it
It i
ti t i
r s
r
X
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Maximising the Joint Likelihood — 3
1 1
1 1
1
log 1 exp
exp
1 exp
0
N I
n n i in iu u
N I
n in iu
Nn u
un n u
r s
s
X
A total of N+I equations to be solved
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At the solution — 1
First derivatives are zero (called scores)
1
1
exp
1 exp
exp
1 exp
It i
ti t i
Nn u
un n u
r
s
What the student scored
What the model predicts
It happens to be the expected score
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We only need the marginals!
• Many sets of patterns will satisfy a given set of marginals• Estimates, errors, reliability do not depend on the
patterns• Parameter estimates do not depend upon fit• Implications for the order debate
Item 1 Item 2 Item 3 … Student ScoreStudent 1 0 1 1 … 23Student 2 1 1 1 … 34Student 3 0 0 1 … 15… … .. … … …Item Score 56 49 89 …
We do not need to use this
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Further Implications
– Student and item scores are sufficient statistics for Rasch estimation.
– Students with the same score will have the same ability estimate.
– One-to-one match between raw score and Rasch ability estimate (when no missing data).
– Use of score equivalence table.– So why (and when) do we need Rasch scores?
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At the solution: What must the distribution of estimates look like?
First derivatives are zero (called scores)
1
1
exp
1 exp
exp
1 exp
It i
ti t i
Nn u
un n u
r
s
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The Resulting Ability Distribution
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale