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
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