Differences between IRT and Rasch models

1
Differences between IRT and Rasch models

• Robert W. Massof
• Lions Vision Research and Rehabilitation Center
• Wilmer Ophthalmological Institute
• The Johns Hopkins University School of Medicine
• Baltimore, Maryland

2
Disclaimers

• Not history Does not purport to explain how
  models were developed
• Not new Uses basic Thurstonian and
  psychophysical principles
• Not threatening Does not challenge accepted
  beliefs
• Not generalizable At current stage of
  development, applies to rating scales and other
  surrogate measures

3
General measurement theory

• For a given trial with a given person, we wish to
  measure the magnitude of the latent stochastic
  decision variable d
• d is distributed across trials within a person
  according to density f(d) and has an expected
  value m and variance sd2

m
sd
4
Conjoint Variable
Rasch
IRT
5
General measurement theory

• Person constructs ordered response thresholds,
  tx,to partition the latent decision variable into
  concatenated intervals
• Intervals need not be equal

t2
t1
t3
6
General measurement theory

• Within a person, response thresholds are
  stochastic with density across trials of g(tx)
• Expected value of response threshold for category
  x is tx and the variance is sx2

7
Measurement decision rule
Person assigns category x to d if
t1, t2, , tx lt d lt tx1, , tm-1, tm
8
Express variables on each trial as expected
values plus residuals

• tx tx rx

• d m rd

9
Measurement decision rule
Assign category x to d if
t1, t2, , tx lt d lt tx1, , tm-1, tm
t1r1, , txrx lt m rdlt tx1rx1, ,tmrm
tx tx rx
d µ rd
10
Measurement decision rule
Assign category x to d if
t1, t2, , tx lt d lt tx1, , tm-1, tm
t1r1, , txrx lt m rdlt tx1rx1, ,tmrm
t1-r(1), , tx-r(x) lt m lt tx1-r(x1), ,tm-r(m)
r(x) rd - rx
11
Measurement decision rule
Assign category x to d if
t1, t2, , tx lt d lt tx1, , tm-1, tm
t1r1, , txrx lt m rdlt tx1rx1, ,tmrm
t1-r(1),.., tx-r(x) lt m lt tx1-r(x1),..,tm-r(m)
t1-r(1)-m,..,tx-r(x)-m lt 0lt tx1-r(x1)-m,..,tm-r(
m)-m
12

• The only random variables in the decision rule
  are the combined residuals r(1),,r(x),r(x1),,r(
  m)
• The joint density function is
• This function has a covariance matrix that
  describes the variance of each of the combined
  residuals and the correlations between each pair

13

• The probability of satisfying the decision rule
  for category x is the cumulative joint
  probability
• The probability of responding with category x
  must be conditioned on the limits of the decision
  rule

14
Comparison of models
tx1 tx (interval x) sx12 sx2 2?x sx1
sx sx2 sx12 2 sx2(1 ?x)

• Rasch rating scale model (Andrich, 1978)
• rx 0
• Vx (interval) 2sx2

• IRT rating scale model (Samejima, 1969)
• rx 1
• Vx (interval) 0

15

• For the Samejima IRT model,
• s1s2sxsmsr r12r13r1xrmx1.0,
  therefore
• r(1)r(2)r(x)r(m)r , and
• so

16
Samejima model
17

• For Andrichs version of the Rasch model
  r12r13r1xrmx0, therefore
• and

18
(No Transcript)
19
Andrich model
20
Two classes of rating scale model

• Samejima (IRT)
• Uncertain trait
• Fixed thresholds
• Interval sizes fixed

• Andrich (Rasch)
• Fixed trait
• Uncertain thresholds
• Interval sizes variable

21
Dichotomous responses

• Single threshold
• Cannot distinguish between uncertain trait and
  uncertain threshold
• Rasch and IRT models identical

22
But what about the discrimination index?

• That is another story.