University of Illinois IRT Laboratory

Selecting a Polytomous IRT Model

Numerous IRT models are available for examining polytomous data. As always, the choice of a model should be based on both theoretical and empirical considerations, i.e., model-data fit.

Two commonly used models among applied psychologists are:

Samejima's Grade Response model (SGR) - where options are ordered along a continuum, as with Likert scales.
Bock's Nominal Model (BNM) - where the options have no explicit ordering, as is often the case with measures of biographical data and some multiple choice tests.

Description of Samejima's Graded Response model

For dichotomously scored items, it is common practice to discuss only the item response function for the positive response to an item, although a response function also exists for the negative category.

For polytomous items, each category is respresented by an option response function (ORF).

According to the SGR model, the probability of selecting option k on item i is

where:

v denotes the person's response to the polytomously scored item i;

k is the particular option selected by the respondent
(k = 1 to s, where "s" refers to the number of options for that item);

a is the item discrimination parameter, assumed to be the same for each option within a particular item;

b is the extremity parameter that varies from option to option

theta represents the value of the latent trait (e.g., conscientiousness or cognitive ability),

P(theta) represents the probability of a positive response

For a more in-depth description of the SGR model, please refer to Samejima (1969) or Hambleton and Swaminathan (1991). In estimating these parameters with the MULTILOG program (described next), one should note that MULTILOG's estimates for a_i contain a scaling factor of 1.7. Therefore, one must divide a_i by 1.7 before further analyses are conducted.

Representative SGR option response functions for an item with 5 options are shown below:

Graph

Samejima's Graded Response
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Estimating Parameters (SGR)

Description of Bock's Nominal model

Bock's Nominal model (1972) is an alternative model for polytomous data where the options for each item are not ordered along some continuum. According to the model, the probability of endorsing option k on item i is:

where:

or alternatively,

z represents the propensity to select option k of item i,

a_ik and b_ik are the option discrimination and location parameters,

c_ik is commonly called the option extremity parameter, where the 1.7a term is absorbed.

For a more in-depth description of the Nominal model please refer to Bock (1972) or Hambleton and Swaminathan (1985). In estimating these parameters with the MULTILOG program (described next), one should note that MULTILOG's estimates for a_ik and c_ik contain a scaling factor of 1.7. Thus, the user must divide the a_ik and c_ik parameters by 1.7, then compute b_ik by dividing the new c_ik by the new a_ik and multiplying by -1.

Representative BNM option response functions for an item with 5 options are shown below:

Estimating Parameters (BNM)