Using Residuals to Estimate the Size and Statistical Significance of Local Bias

Let us use the notation of Rating Scale Analysis (Wright & Masters, 1982) p. 100

Xni = the observation generated by person n on item i

Eni = its expectation = sum k*Pnik, for k = 0,m

Yni = its residual = Xni - Eni

Wni = its variance = sum (k*k*Pnik) - Eni*Eni, for k = 0,m

Then the logit bias for subset S of persons on item i is

Bias(S) = sum (Yni) / sum (Wni) where sum is across all n in S

Standard error of Bias is

SE (S) = sqrt ( sum (Wni) ) where sum is across all n in S

Statistical significance of Logit Bias departure from its expectation of 0.0 is

t = Bias(S)/SE(S)


Alternatively, the standardized residual of an observation is

Zni = Yni / sqrt(Wni)

The mean standardized residual for Subset S comprised of s persons responding to item i is

M(S) = Sum(Zni) / s

It has standard error = sqrt(s)

The significance of this mean departure from its expectation of 0.0 is

t = M(S) / sqrt (s)

MESA Research Note #1 by John Michael Linacre, April 1997

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April 12, 2019, Fri. On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch
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May 22 - 30, 2019, Wed.-Thu. Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/
June 17-19, 2019, Mon.-Wed. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement
June 20-21, 2019, Thurs.-Fri. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 2-5, 2019, Tue.-Fri. 2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org
July 11-12 & 15-19, 2019, Thu.-Fri. A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses
Aug 5 - 10, 2019, Mon.-Sat. 6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html
Aug. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
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Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Nov. 3 - Nov. 4, 2019, Sun.-Mon. International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm
Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
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