Log-linear (or Logistic) Regression vs. Logit-linear Rasch
Estimation	Log-linear Rasch (CMLE)	Logit-linear Rasch (JMLE)
Data matrix	Contingency table: one cell per response string and demographic combination: 4 dichotomies + 2 genders: 2x2x2x2x2 = 32 cells (see TGK) 3 4-category items: 4x4x4 = 64 (Agresti)	Response strings for all subjects. Persons coded with demographic variables.
Missing data	Must be imputed or subject omitted	Merely lessens precision
Basic element	Frequency of persons in cell: e.g., (TGK) F{X_1010M} for response string "1010", Male	Observation: X_ni
Model	Log_e(F{X₁₀₁₀}) = 1E₁ + 0E₂ + 1E₃ + 0E₄ + š_(1+0+1+0) (see TGK)	log_e(P_ni1/P_ni0) = B_n + E_i
Interaction terms	Yes, but no longer Rasch model	Yes, post-hoc to explain residuals
Constraints	To eliminate terms, and establish local origin.	To establish local origin
Estimation bias	Negligible - equivalent to Conditional Maximum Likelihood (CMLE) Rasch	Up to 2, corrected by (L-1)/L
Global fit	Decisive as to acceptability of model.	Uninformative
Items
Maximum items	13, i.e., 2¹³ cells	>3,000
Item calibrations	Yes, but relative to the anchored item	Yes, with mean calibration of zero or anchor item(s).
Item S.E.	Test-dependent, because relative to anchored item. Anchored item has S.E.=0	As test-independent as possible. S.E.s reported for all items.
Item fit diagnosis	Unexpected cell frequencies, summarized by tests of local independence (see TGK)	Unexpected response patterns, summarized by sums of residuals
Persons
Maximum persons	Unlimited, because accumulated in cells	>20,000
Person measures	Only obtained by secondary analysis	Yes, modeled
Person S.E.	Obtained by secondary analysis	Yes, modeled
Person fit diagnosis	Unexpected cell frequencies: Agresti: 8 strings of "322", but 2.9 expected	Unexpected response patterns: in Agresti data: pattern "122".
Unexpected responses	No	Yes, by residual size
Best for
Item calibration	<=13 items with local S.E.s	>=5 items with general S.E.s
Person measurement	No	Yes
Misfit diagnosis	No	Yes
Software	Standard statistical: SAS, SPSS	Custom: BIGSTEPS, QUEST

Log-linear (or Logistic) Regression vs. Logit-linear Rasch

Estimation

Log-linear Rasch (CMLE)

Logit-linear Rasch (JMLE)

Data matrix

Contingency table: one cell per response string and demographic combination:
4 dichotomies + 2 genders: 2x2x2x2x2 = 32 cells (see TGK)
3 4-category items: 4x4x4 = 64 (Agresti)

Response strings for all subjects. Persons coded with demographic variables.

Missing data

Must be imputed or subject omitted

Merely lessens precision

Basic element

Frequency of persons in cell: e.g., (TGK)
F{X_1010M} for response string "1010", Male

Observation: X_ni

Model

Log_e(F{X₁₀₁₀}) =
1*E₁ + 0*E₂ + 1*E₃ + 0*E₄ + š_(1+0+1+0) (see TGK)

log_e(P_ni1/P_ni0) = B_n + E_i

Interaction terms

Yes, but no longer Rasch model

Yes, post-hoc to explain residuals

Constraints

To eliminate terms, and establish local origin.

To establish local origin

Estimation bias

Negligible - equivalent to Conditional Maximum Likelihood (CMLE) Rasch

Up to 2, corrected by (L-1)/L

Global fit

Decisive as to acceptability of model.

Uninformative

Items

Maximum items

13, i.e., 2¹³ cells

>3,000

Item calibrations

Yes, but relative to the anchored item

Yes, with mean calibration of zero or anchor item(s).

Item S.E.

Test-dependent, because relative to anchored item. Anchored item has S.E.=0

As test-independent as possible. S.E.s reported for all items.

Item fit diagnosis

Unexpected cell frequencies, summarized by tests of local independence (see TGK)

Unexpected response patterns, summarized by sums of residuals

Persons

Maximum persons

Unlimited, because accumulated in cells

>20,000

Person measures

Only obtained by secondary analysis

Yes, modeled

Person S.E.

Obtained by secondary analysis

Yes, modeled

Person fit diagnosis

Unexpected cell frequencies:
Agresti: 8 strings of "322", but 2.9 expected

Unexpected response patterns:
in Agresti data: pattern "122".

Unexpected responses

Yes, by residual size

Best for

Item calibration

<=13 items with local S.E.s

>=5 items with general S.E.s

Person measurement

Yes

Misfit diagnosis

Yes

Software

Standard statistical: SAS, SPSS

Custom: BIGSTEPS, QUEST

Agresti: Agresti A (1993) Computing conditional maximum likelihood estimates for generalized Rasch models using simple log-linear models with diagonals parameters. Scandinavian Journal of Statistics 20(1) 63-71.

TGK: TenVergert E, Gillespie M, & Kingma J (1993) Testing the assumptions and interpreting the results of the Rasch model using log-linear procedures in SPSS. Behavior Research Methods, Instruments & Computers 25(3) 350-359.

Log-linear (logistic) regression vs. Logit-linear Rasch. Linacre J.M. … Rasch Measurement Transactions, 1997, 11:3 p. 586.

Rasch Publications
Rasch Measurement Transactions (free, online)	Rasch Measurement research papers (free, online)	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Applying the Rasch Model 3rd. Ed., Bond & Fox	Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters	Introduction to Rasch Measurement, E. Smith & R. Smith	Introduction to Many-Facet Rasch Measurement, Thomas Eckes	Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.	Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Journal of Applied Measurement	Rasch models for measurement, David Andrich	Constructing Measures, Mark Wilson	Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish:	Análisis de Rasch para todos, Agustín Tristán	Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

Rasch Publications

Rasch Measurement Transactions (free, online)

Rasch Measurement research papers (free, online)

Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch

Applying the Rasch Model 3rd. Ed., Bond & Fox

Best Test Design, Wright & Stone

Rating Scale Analysis, Wright & Masters

Introduction to Rasch Measurement, E. Smith & R. Smith

Introduction to Many-Facet Rasch Measurement, Thomas Eckes

Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.

Statistical Analyses for Language Testers, Rita Green

Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar

Journal of Applied Measurement

Rasch models for measurement, David Andrich

Constructing Measures, Mark Wilson

Rasch Analysis in the Human Sciences, Boone, Stave, Yale

in Spanish:

Análisis de Rasch para todos, Agustín Tristán

Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
Jan. 25 - March 8, 2023, Wed..-Wed.	On-line course: Introductory Rasch Analysis (M. Horton, RUMM2030), medicinehealth.leeds.ac.uk
Apr. 11-12, 2023, Tue.-Wed.	International Objective Measurement Workshop (IOMW) 2023, Chicago, IL. iomw.net
June 23 - July 21, 2023, Fri.-Fri.	On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 11 - Sept. 8, 2023, Fri.-Fri.	On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com