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Measurement Error Models
Measurement Error Models (ISBN: 9780470316665) was written by Wayne A. Fuller in 1987.
A measurement Xt is decomposed into the true value xt and the measurement error ut.
When a linear model is constructed as Yt = β0 + β1xt + et but estimated as Yt = βˆ0 + βˆ1Xt = βˆ0 + βˆ1(xt + ut), then the expected values are characterized by:
And the covariance matrix is specified as:
We assume that xt, et, and ut are all independent.
The regression coefficient that has been attenuated (biased towards 0) by measurement error is represented as γ.
The relationship between γ and β1 is characterized by κ.
In more plain terms, this is:
κ is the reliability ratio.