# 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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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