Polytomous Rasch Model | Real Statistics Using Excel

Basic Concepts

The UCON algorithm for building a Rasch model with dichotomous scores can be extended to more than scores of 0 or 1, i.e. to polytomous scores. These types of models are appropriate for Likert scores and for exams with partial credit.

Notation

Suppose that we have test data with values x_si as defined in Basic Concepts of Rasch Analysis, except that now we will allow scores of 0, 1, 2, …, m where m is any positive integer including values larger than 1. For any ability parameter β and difficulty parameter δ, we define

where

Here, the Rasch-Andrich threshold parameter τ_h is defined as the difficulty of obtaining scoring category h relative to category h −1. Note that the same threshold parameters are used for any subject s and item i. For any item i, the difficulty δ_i parameter can be viewed as the average of the δ_i + τ_h. Let

Then for m = 2

and so

For m = 3, and P(x = 0) and P(x = 1) are as above and

Note too that –τ₁ – τ₂ can be replaced by +τ₃.

Iteration

As in the dichotomous case, the ability, difficulty, and threshold parameters are estimated using an iterative approach. In particular, the threshold value τ_k,t at step t is calculated by