Bayesian Analysis of Mixed Rasch Model

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Bayesian Analysis of Mixed Rasch Model

• Ru Lu Robert J. Mislevy
• 13 November 2006

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Rasch Model

• Let ?i be the location of ith individual on the
  latent continuum
• Let ßj be the location of jth item on the latent
  continuum
• The Rasch Model is given by

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Rasch Model

• The Rasch model for 0/1 items

?less able q more able?
Person A
Person B
Person D
Item 1
Item 4
Item 5
Item 3
Item 6
Item 2
?easier b harder?

• Same item ordering for all people.

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Mixed Rasch Model (MRM)

• Incorporates an unobserved student covariate
  (latent variable) f into the probability model

• Where Fi is an index variable that indicates
  which latent group student i belongs to.

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MRM
?less able q more able?
Person A
Person B
Person D
Item 1
Item 4
Item 5
Item 3
Item 6
Item 2
?easier b harder?

?less able q more able?
Person F
Person G
Person H
Item 4
Item 2
Item 6
Item 1
Item 5
Item 3
?easier b harder?
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Application of Mixture IRT Models

• Mixture IRT models including MRM have been used
  for a variety of purposes (Fieuws, Spiessens,
  Draney, 2004)
• Classification into psychological diagnostic
  groups (Waller Mechl, 1998)
• Personality assessment (Reise Gomel, 1995)
• Analysis of strategy use in problem solving
  (Mislevy Verhelst, 1990)
• The use of cognitive strategies (Rijmen De
  Boeck, 2003)
• Speedness effects in time-limit tests (Bolt,
  Cohen, Wollack, 2002)

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Assumptions of MRM

• Like the Rasch Model, it has the usual
  assumptions of
• Local Independence
• Examinee Response Independence
• Different Assumption
• The RM does not hold for the entire population,
  but does so within (latent) subpopulations of
  individuals.

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Some Potential Issues

• Use of priors
• Selection of the number of latent classes
• Label-switching

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Prior Settings in MRM

• The Mixed Rasch Model for 0/1 items

• T normal distribution
• b normal distribution
• F multinomial distribution
• pF dirichlet distribution

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Selection of Number of LCs in MRM

• As other finite mixture models, use information
  criteria, such as AIC, BIC. CAIC. Some simulation
  study suggested that BIC works better (Li, Cohen,
  Kim, Cho, 2006)
• However, DIC is not available for mixture models
  in WinBugs build-in yet. We have to apply the
  options individually (Celeux, Forbes, Robert,
  Titterington, 2006)

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Label Switching Problem

• Why it happens?
• How to detect it?
• How to solve it?
• A simulation study would be used to demonstrate
  the whole process.

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Generating Data

• True values in the MRM
• number of LCs 2
• number of Items 20
• sample size 1000
• proportion of students being in LC1.4
• four sets of the following item parameters

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WinBugs Code for MRM

• model Mixed Rasch Model
• Person parameters
• for (j in 1N)
• classj dcat(pi)
• thetaj dnorm(0,tautheta)
• Item Parameters
• for (k in 1I)
• for (c in 1G)
• bk,c dnorm(0, .25)

• Response Model
• for (j in 1N)
• for (k in 1I)
• pj,klt- (exp(thetaj-bk,clas
  sj)/(1exp(thetaj-bk, classj)))
• rj,k dbern(pj,k)
• Other prior
• pi1G ddirch(alpha)
• tautheta dgamma(.5,1)
• vartheta lt- 1/tautheta

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WinBugs Results (I) Trace
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WinBugs Results (I) Density
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WinBugs Result (I) Stat
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WinBugs Results (II) Trace
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WinBugs Results (II) Density
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WinBugs Result (II) Stat
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Control of Label Switching Problem

• There is a couple of ways to control label
  switching problem
• Control of proportions, e.g., pii duni(0,.5)
• Pre-assigning one observation to each component
  as prior (Chung, Loken, Schafer (2004)

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Example of assigning membership in Winbugs

• data
• list(N1000, I20, G2, alphac(5, 15),
• classc(NA, 1, 1, NA, 1, 1, 1, NA, 1, 1,
• NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
• NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
• NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
• NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
• 2, 2, 2, 2, 2, 2, 2, 2, 2, 2),

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Extensions (I)

• The MRM can easily be extended to more complex
  IRT models, such as
• Other dichotomous IRT models, like 2-PL,3-PL,
  LLTM,...
• Polytomous IRT Models (i.e.,Von Davier Rost,
  1995)
• Multidimensional IRT Models (i.e.,Anderson, 1971)

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Extensions (II)

• Borrowing information about mixtures (Von Davier
  Carstenson, 2006)
• Covariates of mixture components (i.e., Smit et
  al., 1999)
• Partial knowledge of class membership (i.e., Von
  Davier Yamanoto, 2004)
• Mixtures of diagnostic Rasch models (i.e., Von
  Davier, 2005)

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Main References

• Rost, J. (1990). Rasch Models in latent classes.
  An integration of two approaches to item
  analysis. Applied Psychological Measurement, 14,
  271-282.
• Rost, J. (1991). A logistic mixture distribution
  model for polychotomous item responses. British
  Journal of Mathematical and Statistical
  Psychology, 44,75-92.
• Mislevy, R. J., Verhelst, N. (1990). Modeling
  item responses when different subjects employ
  different solution strategies. Psychometrika, 55,
  195-215.
• Fischer, G. (1995). Rasch models foundations,
  recent development and applications. New York
  Springter.
• Von Davier, M., Carstensen,C. H. (2006).
  Multivariate and Mixture Distribution Rasch
  Models Extensions and Applications. New York
  Springter.