A Simple Graded Response Item Response Theory Model

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A Simple Graded Response Item Response Theory
Model

• Robert J. Mislevy
• University of Maryland
• November 10, 2006

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Topics

• Review of the Rasch model for dichotomous items
• A Samejima-style model for graded responses
• A full Bayesian model (same structure as for
  dichotomous model)

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What is IRT?

• A single latent variable measures students
  overall proficiency in some domain of tasks.
• The structure of the probability model
  conditional independence among observations given
  q and item parameters b.

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What is IRT?

• For Item j, the IRT model expresses the
  probability of a given response xj as a function
  of q and parameters bj that characterize Item j
  (such as its difficulty)
• f(xjq,bj).

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The Rasch model for dichotomous (right/wrong)
items

• Prob(Xij1qi,bj) f(1qi,bj) Y(qi - bj),
  where
• Xij is response of Student i to Item j, 1
  right, 0 wrong
• qi is the proficiency parameter of Student i
• bj is the difficulty parameter of Item j
• Y(x) is the logistic function, Y(x)
  exp(x)/1exp(x).
• The probability of an incorrect response is then
• Prob(Xij0qi,bj) f(0qi,bj) 1-Y(qi - bj).

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The Rasch model for dichotomous (right/wrong)
items

• Two Rasch model curves, with b1-1 and b22.

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A model for items with responses in K ordered
categories (k1,2,,K)

• Build a model in two stages
• Rasch model for above category k for categories
  1 to K-1. These are the cumulative response
  probabilities
  P(Xgtkq,bjk) exp(q-bjk)/1exp(q-bjk).
• Get category response probabilities for each
  category by subtraction
• P(X1q,bj) 1 - P(Xgt1q,bj1)
• P(Xkq,bj) P(Xgtk-1q,bjk) - P(Xgtkq,bjk-1)
• P(XKq,bj) P(XgtK-1q,bjK-1).

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A model for items with responses in K ordered
categories (k1,2,,K)

• Approach due to Dr. Fumiko Samejima (1969)
• Item difficulty parameters are ordered from lower
  to higher categories bj1ltbj2ltltbjK.
• Usually see version with slopes that is,
  P(Xgtkq,aj,bjk)
  expaj(q-bjk)/1expaj (q-bjk).
• This is not a Rasch model for graded response
  data.

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A three category model

• Two cumulative probability curves, with bj1-1
  and bj22.

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A three category model

• Three response category probability curves, with
  bj1-1 and bj22.

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A full Bayesian model A generic measurement
model

• Xij Response of Person i to Item j
• qi Parameter(s) of Person i
• bj Parameter(s) of Item j
• h Parameter(s) for distribution of qs
• t Parameter(s) for distribution of bs
• Note Exchangeability assumed here for qs and
  for bs--i.e., modeling all with the same prior.
  Later well incorporate additional info, about
  people and/or items.

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A full Bayesian model The recursive expression
of the model
The measurement model Item response given
person item parameters Distributions for person
parameters Distributions for item
parameters Distribution for parameter(s) of
distributions for item parameters Distribution
for parameter(s) of distributions for person
parameters
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A full Bayesian model A BUGS diagram
bj
pij
qi
t
h
Xij
Items j
Persons i

• Plates for people and items
• Item parameters explicit
• q population distribution structure explicit
• In dichotomous IRT, item person parameters give
  probability parameter in a binomial distribution
  for the observed response.