Anderson, C.J, & Yu, H.T. (2007, in press). Log-multiplicative association models as item response theory models. Psychometrika.

Input/output files used to perform analyses:

• Table 1. Probabilities generated from a LMA2 model for 7 items and estimated parameters of LMA2 modele when item with the largest and smallest nu is dropped from the data set.
  • SAS program used to generate the probabilities. The probabilities were multiplied by 10000 to retain extra digits.
  • LMA2 fit to all 7 items (Lem).
  • LMA2 fit to all 6 items with largest nu dropped (Lem).
  • LMA2 fit to all 6 items with smallest nu dropped (Lem).
  • 2Pl fit to all 6 items with largest nu dropped (Lem).
  • 2Pl fit to all 6 items with smallest nu dropped (Lem).



• Table 3. Dissimilarity indices for the LMA1 and Rasch models fit to probabilities generated by the Rasch model and the indices for the LMA2 and 2PL models fit to probabilities generated from 2PL. The probabilities were generated using theta ~ normal (0,1) and theta ~ exp(1).
  • Examples of SAS input used to generate probabilities via numerical integration:
  • Rasch model for 12 items.
  • 2PL model for 12 items.
  • LEM input/output files for 4 items:
    • Rasch normal(0,1) theta.
    • LMA1 normal(0,1) theta.
    • MLLE 2PL normal(0,1) theta.
    • LMA2 normal(0,1) theta.
    • Rasch exp(1) theta.
    • LMA1 exp(1) theta.
    • MLLE 2PL exp(1) theta.
    • LMA2 exp(1) theta.
  • LEM input/output files for 6 items:
    • Rasch normal(0,1) theta.
    • LMA1 normal(0,1) theta.
    • MLLE 2PL normal(0,1) theta.
    • LMA2 normal(0,1) theta.
    • Rasch exp(1) theta.
    • LMA1 exp(1) theta.
    • MLLE 2PL exp(1) theta.
    • LMA2 exp(1) theta.
  • LEM input/output files for 8 items:
    • Rasch normal(0,1) theta.
    • LMA1 normal(0,1) theta.
    • MLLE 2PL normal(0,1) theta.
    • LMA2 normal(0,1) theta.
    • Rasch exp(1) theta.
    • LMA1 exp(1) theta.
    • MLLE 2PL exp(1) theta.
    • LMA2 exp(1) theta.
  • LEM input/output files for 10 items:
    • Rasch normal(0,1) theta.
    • LMA1 normal(0,1) theta.
    • MLLE 2PL normal(0,1) theta.
    • LMA2 normal(0,1) theta.
    • Rasch exp(1) theta.
    • LMA1 exp(1) theta.
    • MLLE 2PL exp(1) theta.
    • LMA2 exp(1) theta.
  • LEM input/output files for 12 items:
    • Rasch normal(0,1) theta.
    • LMA1 normal(0,1) theta.
    • MLLE 2PL normal(0,1) theta.
    • LMA2 normal(0,1) theta.
    • Rasch exp(1) theta.
    • LMA1 exp(1) theta.
    • MLLE 2PL exp(1) theta.
    • LMA2 exp(1) theta.



• Table 4. LSAT7 analyses (Lem) and a little extra:
  • LSAT7 data
  • Complete independence log-linear model.
  • MMLE of Rasch model.
  • LMA1 Rasch as a log-multiplicative model.
  • MMLE of 2 PL model. (Rasch)
  • LMA2 2PL as a log-multiplicative model.
  • All 2-way interaction log-linear model. .
  • Two dimensional log-multiplicative association model. (LMA2)



• Figure 3. Simulation of response to 100 items and model fitting:
  • SAS: randomly selects alpha and betas and simulates responses to 2PL.
  • Simulated 2PL data.
  • SAS: Simulated data according to Rasch model. (uses same betas as used for 2PL data.
  • Simulated Rasch data.
  • SAS: Pseudo-likelihood estimation of LMA1 (fit to Rasch data). Code fits all 100 items, as well as subsets of 50 and 20 items.
  • BILOG output for Rasch model.
  • BILOG output for 2PL model.
  • FORTRAN program for fitting LMA2 is available upon request.
  • SAS: compares results. Note that this code shows the transformations, computes correlations, generates the figure in the paper, and does some other things. The notes in the figures will only be in the correct place if your monitor is the same size as mine.

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Last revised June 20, 2008