Estimation

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

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

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

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

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

Estimation Methods

• Approximate Methods– PROX– UFORM

• Pair-wise• Minimum Chi-square• Maximum likelihood

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

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

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

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

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r s

r s

X X

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

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

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

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s

What the student scored

What the model predicts

It happens to be the expected score

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

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?

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

The Resulting Ability Distribution

Score 0

Score 1

Score 2

Score 3Score 4

Score 5

Score 6

Proficiency on Logit Scale