eRm Extended Rasch Modeling Item response theory Rasch - - PowerPoint PPT Presentation

SLIDE 1

eRm – Extended Rasch Modeling

An R Package for the Analysis of Extended Rasch Models Hatzinger, R., Mair, P., useR Conference, June 2006

Contents

• Introduction to Rasch measurement
• Item response theory
• Rasch measurement scale
• The Rasch model and extensions
• RM and LLTM
• RSM and LRSM
• PCM and LPCM
• Model hierarchy
• A unified CML approach
• Organization of the eRm routine

Rasch Measurement

• Item Response Theory (IRT)
• Analysis of response patterns in tests and questionnaires.
• Functional relationship between the probability to solve an item

and some item and person parameter.

• A simple model is the Rasch model with one parameter for

each item and one parameter for each subject.

• Rasch model as a seal of approval of a test (fairness, scaling).
• Rasch measurement scale
• Example: A temperature scale in physics is clearly defined. What

about a scale for mathematical ability?

• The Rasch model generates a scale for a latent trait .

i

β

v

θ Ψ

The Rasch Model

• Rasch model equation for dichotomous items
• Assumptions
• Unidimensionality
• Sufficiency of the raw score
• Parallel item characteristic curves
• Local independence

I1 I2 I3 I4 I5 Rv P1 1 1 1 1 1 5 P2 1 1 1 1 4 P3 1 1 1 3 P4 1 1 P5 …

X

( ) ( ) ( )

exp 1| , 1 exp

v i vi i v v i

P X θ β β θ θ β − = = + − item parameter

i

β … person parameter

v

θ …

SLIDE 2

Linear Logistic Test Model

• LLTM (Linear Logistic Test Model)
• Linear reparameterization of the

Rasch model.

• LLTM as a more parsimonious model.
• LLTM as a more general model in

terms of repeated measurements and certain effects.

• Concept of virtual items.

1 p i ij j j

w β η

=

=∑

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥

• 1

2 K

β β β τ δ ν ρ …

* 1 * 2 * * 1 * 2 * 2 * 2 1 * 2 2 * 3 * 3 1 * 3 2 * 4 * 4 1 * 4 2 * 5 K K K K K K K K K K K K K

β β β β β β β β β β β β β β β

+ + + + + + + +

• B1

B2 B3 B4 B5

Rating Scale Models

• RSM (Rating Scale Model)
• Item response categories are rating scales (polytomous)
• LRSM (Linear Rating Scale Model)
• Linear decomposition of the item parameter

( ) ( )

( )

( )

( )

exp | , , , , exp

v i k vi v i m m v i h h

k P X k h θ β ω θ β ω ω θ β ω

=

+ + = = + +

∑

… category parameter

h

ω … , categories; 0, , h k h m = … …

1 p i ij j j

w β η

=

=∑

Partial Credit Models

• PCM (Partial Credit Model)
• Each item category gets a partial credit (item-category parameter)
• Different number of categories per item allowed
• LPCM (Linear Partial Credit Model)
• Linear decomposition of the item-category parameter

1 p ih ihj j j

w β η

=

= ∑

( ) ( ) ( )

exp 1| , exp

i

v ik vik v ik m v ih h

k P X h θ β θ β θ β

=

+ = = +

∑

item-category parameter

ih

β …

Model Hierarchy

• Model Nesting
• LPCM is the most general model
• All other models can be viewed as special cases of LPCM.
• Parameterization through appropriate choice of W.
• Unified CML procedure which is able to estimate these models.

⊂

Rasch LLTM

⊂ ⊂ ⊂

LPCM LRSM PCM RSM

⊂ ⊂

SLIDE 3

A Unified CML Approach

• Linear item parameter decomposition
• CML approach
• Estimation of
• Likelihood conditioned on the raw score

vanishes

= β Wη

design matrix W… parameter vector η…

• η

θ ( )

log log

l l l

C lh lh r r l h r

L x n β γ ε

+

= −

∑∑ ∑

( ) ( )

( )

log

l l l l l

l r h C lh a lh lh r l h r a r

L w x n γ ε ε η γ ε

− +

⎛ ⎞ ∂ = − ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠

∑∑ ∑

Organization of the eRm Routine

function RM

X

function LLTM

W X,Times,W,G W X W X W X,Times,W,G W X,Times,W,G W

function RSM function PCM function LRSM function LPCM

Unified CML

Likelihood, parameter estimators, standard errors

LR model test

LR, plot

References

• Fischer, G., & Molenaar, I. (1995). Rasch Models – Foundations,

Recent Developments, and Applications. Springer.

• Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-

573.

• Fischer, G.H., & Parzer, P. (1991). An extension of the rating scale model with an application to the

measurement of change. Psychometrika, 56, 637-651.

• Fischer, G.H., & Ponocny, (1994). An extension of the partial credit model with an application to the

measurement of change. Psychometrika, 59, 177-192.

• Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
• Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: The

Danish Institute of Educational Research.

• Scheiblechner, H. (1972). Das Lernen und Lösen komplexer Denkaufgaben. [The learning and

solving of complex reasoning items.] Zeitschrift für Experimentelle und Angewandte Psychologie, 3, 456-506.