Presentation conducted in "ReMa-IDA: Psychometrics: Measurement (400929-M-6)" class at Tilburg University. The presentation is based on the literature: Masters, G.N. A rasch model for partial credit scoring. Psychometrika 47, 149–174 (1982). https://doi.org/10.1007/BF02296272

1. Partial Credit Model
December 9, 2021
Kento Okuyama

2. Latent trait model for items with ordered categories will be discussed.
• The Rasch family has the potential to measure ability irrespective of the
difficulties of the chosen items and item difficulty irrespective of the
abilities of the persons in our sample.
• The Graded Response Model (Samejima 1969) is not a member of the
Rasch family, i.e., the person and item parameters are not separable.
• Now the question arises: can we develop a model for items with
ordered categories whose person and item parameters are separable
(i.e., a member of the Rasch family)?
Motivation

3. δi corresponds to the level of ability
resulting in 50% probability of a
correct response.
The Rasch family
The Rasch model for dichotomous responses:
In the dichotomous Rasch model,
• person parameters can be eliminated from the estimation of item
parameters given the sum score.
• item parameters can be eliminated from the estimation of person
parameters given the counts.
(δi: item difficulty)

4. λik represents “category boundary”: it is the level of ability resulting in the
50% probability of choosing the category or above the category.
The probability of choosing the k’th category or above the category:
Samejima’s Graded Response Model (GRM)
(λik: difficulty at the k’th category)
In the GRM, "category boundaries" (item parameters) cannot be separated
from person parameters.

5. The probability of responding in the k’th category “rather than” the (k−1)’th
category (i.e., each “step” is treated dichotomously):
The Partial Credit Model (PCM)
(δik: “step” difficulty at the k’th category)
δik corresponds to the level of ability resulting in the 50% “step” probability.
In the Partial Credit Model (Masters 1982), "step" difficulties (item
parameters) can be separated from person parameters!

6. • In the PCM, person and item parameters are separable via sum score
and counts. This is the advantage of using the PCM over the GRM.
• For the parameter estimation, conditional maximum likelihood will be
used. The procedure is based on the estimation of the parameters
used for the Rasch’s dichotomous model.
• Some limitations of the PCM:
• The PCM assumes unidimensionality. The model cannot be used where
multiple latent traits are observed.
• The PCM does not allow varying slope coefficients for each step.
However, an extended model called Generalized Partial Credit Model
(Muraki 1992) exists.
Some Details about the PCM

7. The Partial Credit Model...
• is a latent trait model for items with ordered categories.
• uses the “step” difficulty instead of the “boundary”.
• achieves separability of the person and item parameters.
• will be estimated using conditional maximum likelihood.
• assumes unidimensionality and equal slope coefficients.
Summary