Question: I was taught that all raw scores on a test have the same raw score standard error, SEM, and this is:
SEM = raw score S.D. * sqrt (1-Reliability).
Why do standard errors for person measures differ?
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Answer: The raw score "test" reliability is based on an average raw score standard error for the sample. But each raw score has a different standard error. The raw score standard errors are biggest at the center of the test and smallest at the extremes. In contrast, the standard error of a Rasch measure is smallest in the center of the test and biggest at the extremes. Zero and perfect raw scores have standard errors of zero, but the corresponding Rasch measures have infinite standard errors.
The plot is an idealization plot of their relationship for a 30 item dichotomous test. But, like the raw score reliability, the Rasch reliability is also based on the average standard error of the sample.
Classical Test Theory (CTT) computes a "test" reliability = R. This is based on the average SEM (standard error of measurement) of the raw scores. Average raw-score SEM = sqrt (1-R) * (observed raw score S.D.).
Rasch computes an S.E. = SEM for each measure
SEM(Raw Score) ≈ 1 / ( SEM(Rasch Measure in logits) )
and
SEM(Rasch Measure in logits) ≈ 1 / ( SEM(Raw Score) )
Standard Errors and Reliabilities: Rasch and Raw Score, Rasch Measurement Transactions, 2007, 20:4 p. 1086
The URL of this page is www.rasch.org/rmt/rmt204f.htm
Website: www.rasch.org/rmt/contents.htm