# 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{X1010M} for response string "1010", Male
Observation: Xni
Model Loge(F{X1010}) =
1*E1 + 0*E2 + 1*E3 + 0*E4 + š(1+0+1+0) (see TGK)
loge(Pni1/Pni0) = Bn + Ei
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

John Michael Linacre

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

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June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com