Noise and Random Error

Question: In Rasch analysis, how does noise differ from random error?

Answer: Every observation is conceptualized to consist of three components:

1. Its expected value. This is the amount predicted from the Rasch model and the parameter estimates (ability, difficulty and rating scale structure).

2. Model randomness or modeled random error. This is the randomness in the data predicted by the Rasch model, which is a probabilistic model. It is the Bernoulli binomial variance or multinomial variance, "the model variance of the observation around its expectation". The Rasch model uses this for estimating the distance between the parameter estimates, the Rasch measures.

3. Unmodeled randomness. This is the part of each observation that contradicts the Rasch model. It makes the mean-square statistics depart from 1.0. We don't want this randomness because it degrades measurement. From the perspective of the Rasch model, this component is random, i.e., unpredictable, but it may be highly predictable from other perspectives, e.g., "Robin has a response set."

Statistically, "noise" is "2.+3.", but often we use "noise" to mean "3." or even "2.". If there is obvious ambiguity, we use terms like "modeled randomness" for "2.", and "unmodeled noise" for "3.".

There is the paradoxical situation that some of the "3. Unmodeled randomness" can cancel out some of the "2. Model randomness" This happens when the data overfit the model, and the mean-squares are less than 1.0. So sometimes, "noise" only refers to the part of "3. Unmodeled randomness" that adds to the model randomness in the observations.

Noise and Random Error … Rasch Measurement Transactions, 2007, 21:2 p. 1103

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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