IRT 1

• Classical Test Theory (CTT) - analyses are the easiest and most widely used form of analyses

  • Most of tests are constructed based on CTT

• Classical Analyses are performed on the test rather than on the item and although item statistics can be generated - Test reliability, test validity

• IRT/latent trait models aim to look beyond that at the underlying traits which are producing the test performance. They are measured at item level

  • Basically, a logistic regression of P(1) on the latent trait

• IRT refers to a family of latent trait models used to establish psychometric properties of items and scales

1. Item response function (IRF) – Mathematical function relating the latent trait to the probability of endorsing an item

2. Invariance – position on the latent trait can be estimated by any items with known IRFs. Item characteristics are population independent

3. Item Information Function – an indication of item quality; an item’s ability to differentiate among respondents

• Item Response Function (IRF) - characterizes the relation between a latent variable (i.e., individual differences on a construct) and the probability of endorsing an item.

• The IRF models the relationship between examinee trait level, item properties and the probability of endorsing the item.

• Examinee trait level is signified by the Greek letter theta (0-)

Dichotomous case

• The problem: Dependent variable is binary, with a yes or a no answer

• Can be coded, 1 for yes and 0 for no

• There are no other valid responses

Solution: Use a Different Functional Form

The properties we need

• The model should be bounded by 0 and 1

• The model should estimate a value for the dependent variable in terms of the probability of being in one category or the other

Solution: Use a Different Functional Form cont.

• We want to know the probability, p, that a particular case falls in the 0 or the 1 category

• We want to derive a model which gives good estimates of 0 and 1, or put another way, that a particular case is likely to be a 0 or a 1

  - Thus, modelling the probability of correctly answering a dichotomously scored item means predicting the probability of getting it right conditional on the latent trait theta (q)

Item difficulty

• An item’s location is defined as the amount of the latent trait needed to have a .5 probability of endorsing/solving the item

• The higher the “delta” parameter the higher on the trait level a respondent needs to be in order to endorse/solve the item

Item discrimination

• Indicates the steepness of the IRF at the item’s location

• An items discrimination indicates how strongly related the item is

  to the latent trait like loadings in a factor analysis

• Items with high discriminations are better at differentiating

  respondents around the location point; small changes in the

  latent trait lead to large changes in probability

• Vice versa for items with low discriminations