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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Coming Rasch-related Events
Jan. 18 - 19, 2019, Fri.-Sat. In-person workshop, Munich, Germany: Introduction to Rasch Measurement With Winsteps (William Boone, Winsteps), boonewjd@gmail.com
Jan. 25 - Feb. 22, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 28, 2019, Mon. On-line course: Understanding Rasch Measurement Theory (ACER), https://www.acer.org/professional-learning/postgraduate/Rasch
Feb. 4 - 7, 2019, Mon.-Thur. RUMM-based Rasch Workshop (in Italian), Bologna, Italy,https://mailinglist.acer.edu.au/pipermail/rasch/attachments/20190114/de6886f8/attachment.pdf
March 21, 2019, Thur. 13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019
April 4 - 8, 2019, Thur.-Mon. NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual
April 5 - 9, 2019, Fri.-Tue. AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting
May 24 - June 21, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
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. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - June 19, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 26 - July 24, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 7 - Sept. 4, 2020, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com

 

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