Introduction to Item Response Theory

Introduction to Item Response Theory

Wim Krijnen

Lector Analyse Technieken voor Praktijkonderzoek

Lectoraat Healthy Ageing, Allied Health Care and Nursing

Hanze University of Applied Sciences

July 1, 2015

Outline

1

Definitions

2

_{Two Examples}

3

_{Some Further Considerations}

Odds of accomplishing a task

P = the probability of successfully accomplishing a task.

Many applications:

1

_{intelligence (item)}

_{stadia of development}

_{health care questionnaires}

Models for P:

1

_{Model P directly}

_{Model the Odds ratio:}

P

1 − P

3

_{Model the log of Odds ratio:}

log



P

1 − P



Remark: Last appears to be great idea (large variation,

computationally convenient, very close to normal)!!

Rasch model for log odds of getting item i correct

logarithm of odds of getting item i correct:

log



P

i

(θ)

1 − P

i

(θ)



= α(θ − β) =

Z ⇔ P(θ) =

e

Z

1 + e

Z

P

_i

(θ) =

P(X

_i

=

1|θ) probability of getting item correct

value increases with increasing ability

item characteristic curve

β

i

item difficulty parameter

P(x

im

=

1|β

i

) =

0.50 ability at which probability of correct

response is 0.5

parameter locating the function horizontally

α

i

discrimination parameter of item i

determines steepness (slope) of ICC

Some examples of Item Characteristic Curves = Plots of

P

i

(θ) =

P(X

i

=

1|θ) as function of Ability θ

Outline

1

Definitions

2

_{Two Examples}

3

_{Some Further Considerations}

Opinion on Abortion

Should the law allow abortion under the circumstance presented in (0:

No; 1: Yes):

Item 1: The woman decides on her own that she does not.

Item 2: The couple agree that they do not wish to have a child.

Item 3: The woman is not married and does not wish to marry the

man.

Item 4: The couple cannot afford any more children.

Dffclt

Dscrmn

Item 1

0.17

4.45

Item 2

-0.24

4.32

Item 3

-0.34

5.66

Item 4

-0.32

3.63

Latent variable: Tendency to alow for abortion

Predicted Ability Scores

I1

I2

I3

I4

Obs

Exp

z1

se.z1

1

0

0

0

0

103

103.41

-0.90

0.46

2

0

0

0

1

13

15.52

-0.50

0.26

3

0

0

1

0

10

11.15

-0.38

0.24

4

0

0

1

1

21

14.32

-0.18

0.24

5

0

1

0

0

9

8.91

-0.46

0.25

6

0

1

0

1

6

6.81

-0.25

0.23

7

0

1

1

0

7

12.05

-0.14

0.24

8

0

1

1

1

44

41.82

0.09

0.28

9

1

0

0

0

1

1

-0.45

0.25

10

1

0

1

1

6

7.19

0.10

0.28

11

1

1

0

0

3

0.90

-0.21

0.23

12

1

1

0

1

3

2.56

0.01

0.26

13

1

1

1

0

12

7.11

0.16

0.29

14

1

1

1

1

141

142.71

0.66

0.52

Workplace Industrial Relation Survey (WIRS)

Before change in workplace there were

Item 1: Informal discussion with individual workers.

Item 2: Meeting with groups of workers.

Item 3: Discussions in established joint consultative committee.

Item 4: Discussions in specially constituted committee to consider

the change.

Item 5: Discussions with the union representatives at the

establishment.

Item 6: Discussions with paid union officials from outside.

Dffclt

Dscrmn

Item 1

3.40

0.15

Item 2

-0.94

0.37

Item 3

0.81

1.72

Item 4

1.37

1.01

Item 5

0.48

2.03

Item 6

1.68

1.37

Outline

1

Definitions

2

_{Two Examples}

3

_{Some Further Considerations}

Testing for unidimensionality

Testing for equal discrimination parameters between items

“Factor Scores” with corresponding S.E. and fit

Amount of information in range of ability scores

Generalization of Rasch model to Likert scales (sufficient number

of data points needed)

Computerized Adaptive Testing