Item Response Theory

Item response theory (IRT) is a framework for predicting a response given some latent factor.

Description

The framework is used to predict a particular response to a given item. Often there is a predetermined 'correct' response, so the framework is used to predict an individual's ability to correctly response. This is the case of the Rasch model introduced below.

In general the framework is applicable to predicting any particular response to an item.

Formulation

Let i index individuals and j index items. Individuals also have a 1-dimensional latent ability, usually notated as θ_i but sometimes as η_i instead.

The Rasch model is a logistic model that predicts correct responses (x_ij = 1) based on latent ability and item difficulty α_j. This model is variably considered an example of an item characteristic curve (ICC) or a 1-parameter logistic (1PL) model.

An evolution of the Rasch model, sometimes called a 2-parameter logistic (2PL) model, introduces a new term to allow for varying slopes. λ_j reflects the discrimination of an item. High discrimination suggests that responses are strongly related to latent ability, whereas low discrimination suggests that responses are very similar across individuals with high or low ability.

Further evolutions add guessing and inattention (ceiling) parameters. (3PL, 4PL, etc.) In these cases, the parameters are more commonly notated a, b, c, d, and so on. See this live demo of a 3PL model.

Generalizations of the model include:

• there is a category of polytomous models for categorical responses

  • graded response models (GRM) for ordinal responses

• multidimensional models for multiple latent factors

Normal Ogive Models

Normal ogive models trade the logit link function for probit. Note that the parameters in this specification are almost always notated as:

• β_j for item difficulty

• α_j for item discrimination

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