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

Contents • Introduction to Rasch measurement eRm – Extended Rasch Modeling • Item response theory • Rasch measurement scale • The Rasch model and extensions An R Package for the Analysis of Extended Rasch Models • RM and LLTM • RSM and LRSM • PCM and LPCM • Model hierarchy Hatzinger, R., Mair, P., • A unified CML approach useR Conference, June 2006 • Organization of the eRm routine Rasch Measurement The Rasch Model • Rasch model equation for dichotomous items • Item Response Theory (IRT) ( ) X θ − β exp ( ) • Analysis of response patterns in tests and questionnaires. = β θ = + v i 1| , P X ( ) I 1 I 2 I 3 I 4 I 5 R v θ − β vi i v 1 exp v i • Functional relationship between the probability to solve an item P 1 1 1 1 1 1 5 P 2 1 1 1 0 1 4 β … and some item and person parameter. item parameter i P 3 0 0 1 1 1 3 β • A simple model is the Rasch model with one parameter for θ … P 4 1 0 0 0 0 1 person parameter i v θ each item and one parameter for each subject. P 5 0 0 0 0 0 0 v … • Rasch model as a seal of approval of a test (fairness, scaling). • Assumptions • Rasch measurement scale • Unidimensionality • Example: A temperature scale in physics is clearly defined. What • Sufficiency of the raw score about a scale for mathematical ability? • Parallel item characteristic curves Ψ • The Rasch model generates a scale for a latent trait . • Local independence

Linear Logistic Test Model Rating Scale Models β β β τ δ ν ρ … 1 2 K • LLTM (Linear Logistic Test Model) • RSM (Rating Scale Model) β * ⎡ ⎤ 1 1 ⎢ ⎥ β * 1 ⎢ ⎥ 2 B 1 • Linear reparameterization of the • Item response categories are rating scales (polytomous) ⎢ ⎥ � � ⎢ ⎥ β * Rasch model. 1 ⎢ ⎥ ( ) K ( ) ⎢ ⎥ θ + β + ω β * exp 1 1 k + ⎢ ⎥ ( ) K 1 = θ β ω ω = v i k p … β | , , , , = ∑ * P X k ⎢ 1 1 ⎥ β η + K 2 B 2 vi v i 0 m w ⎢ ⎥ m ( ) ∑ ( ) � � � θ + β + ω i ij j ⎢ ⎥ exp h = β j 1 * ⎢ 1 1 ⎥ v i h 2 K ⎢ ⎥ = h 0 β * 1 1 1 ⎢ ⎥ + 2 K 1 = β ⎢ ⎥ * 1 1 1 … … • LLTM as a more parsimonious model. , categories; 0, , h k h m + 2 2 K ⎢ ⎥ B 3 � � � � ⎢ ⎥ ω … ⎢ ⎥ category parameter β • LLTM as a more general model in * 1 1 1 3 ⎢ ⎥ K h β * ⎢ 1 1 1 ⎥ + 3 K 1 terms of repeated measurements and ⎢ ⎥ β * 1 1 1 + ⎢ ⎥ 3 K 2 B 4 • LRSM (Linear Rating Scale Model) certain effects. � ⎢ � � � ⎥ ⎢ ⎥ β * 1 1 1 ⎢ ⎥ 4 K • Linear decomposition of the item parameter β ⎢ ⎥ • Concept of virtual items. * 1 1 1 1 1 + 4 K 1 ⎢ ⎥ β * 1 1 1 1 1 ⎢ ⎥ + 4 K 2 B 5 p ⎢ ⎥ = ∑ � � � � � � β η ⎢ ⎥ w β ⎢ ⎥ * ⎣ ⎦ ⎢ 1 1 1 1 1 ⎥ i ij j 5 K = 1 j Partial Credit Models Model Hierarchy • PCM (Partial Credit Model) • Model Nesting ⊂ ⊂ • Each item category gets a partial credit (item-category parameter) Rasch RSM PCM ⊂ ⊂ ⊂ • Different number of categories per item allowed LPCM ⊂ ( ) LLTM LRSM θ + β exp k ( ) = θ β = v ik 1| , P X vik v ik m ∑ i ( ) θ + β exp h v ih = h 0 • LPCM is the most general model β … item-category parameter • All other models can be viewed as special cases of LPCM. ih • LPCM (Linear Partial Credit Model) • Parameterization through appropriate choice of W . • Linear decomposition of the item-category parameter • Unified CML procedure which is able to estimate these models. = ∑ p β η w ih ihj j = 1 j

A Unified CML Approach Organization of the eRm Routine • Linear item parameter decomposition function RM function LLTM function RSM function PCM function LRSM function LPCM X X,Times,W,G X X X,Times,W,G X,Times,W,G W … design matrix = β W η η … parameter vector W W W W W W • CML approach � η • Estimation of Unified CML • Likelihood conditioned on the raw score � θ vanishes Likelihood, parameter estimators, standard errors ∑∑ ∑ ( ) = β − γ ε log log L x n + C lh lh r r l l l h r l LR model test ( ) ( ) ⎛ ⎞ n γ ε l ∂ log LR, plot L ∑∑ ∑ − = ⎜ − ε ⎟ r h C w x l ( ) ⎜ ⎟ + ∂ η γ ε lh a lh lh r l l l ⎝ ⎠ l h r a r l 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.