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 itemand 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. Whatabout a scale for mathematical ability?

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

iβvθ

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 RvP1 1 1 1 1 1 5

P2 1 1 1 0 1 4

P3 0 0 1 1 1 3

P4 1 0 0 0 0 1

P5 0 0 0 0 0 0

…

X( ) ( )

( )exp

1| ,1 exp

v ivi i v

v i

P Xθ β

β θθ β−

= =+ −

item parameteriβ …

person parametervθ …

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 jj

wβ η=

=∑

11

11 1

1 1

1 11 1 1

1 1 1

1 1 11 1 1

1 1 1

1 1 11 1 1 1 1

1 1 1 1 1

1 1 1 1 1

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

1 2 Kβ 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

ββ

βββ

βββ

βββ

βββ

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

( ) ( )( )( )( )

0

0

exp| , , , ,

exp

v i kvi v i m m

v i hh

kP X k

h

θ β ωθ β ω ω

θ β ω=

+ += =

+ +∑…

category parameterhω …, categories; 0, ,h k h m=… …

1

p

i ij jj

wβ 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 jj

wβ η=

=∑

( ) ( )

( )0

exp1| ,

expi

v ikvik v ik m

v ihh

kP X

h

θ βθ β

θ β=

+= =

+∑

item-category parameterihβ …

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

PCMRSM⊂⊂

A Unified CML Approach

• Linear item parameter decomposition

• CML approach

• Estimation of

• Likelihood conditioned on the raw score vanishes

=β Wη design matrixW…parameter vectorη…

η

θ

( )log logl l

l

C lh lh r rl h r

L x nβ γ ε+= −∑∑ ∑

( ) ( )( )

log l

l l l

l

lr hC

lh a lh lh rl h ra r

L w x nγ ε

εη 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 CMLLikelihood, 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.