1. Item Response Theory (Cohen)

Item Response Theory (IRT), also known as latent-trait theory, is a family of theories and methods that provides a way to model the probability that a person with X ability (e.g., a particular personality trait) is able to perform (e.g., on a personality test) at a level of Y.

IRT is not a single method or theory. Instead, IRT is a family of theories and methods, comprising well over a hundred different models. Each model as its own assumptions and data characteristics to handle data. There are, for instance, IRT models designed specifically for tests with dichotomous test items (yes/no, true/false). Other models are specifically designed to for tests polytomous items (test items with three or more answer categories). Another important group of IRT models is developed by the Danish mathematician George Rash. He developed the so-called Rasch model in which each item on the test is assumed to have an equivalent relationship with the ability, or whatever construct is being measured by the test.

Two very important characteristics in IRT are: (1) the difficulty level of an item; (2) the discrimination of an item's level. Difficulty refers to the attribute of not being easily accomplished, solved, or comprehended. Discrimination refers to the degree to which an item differentiaties among people with higher of lower levels of the trait, ability, or whatever construct is being measured. 

IRT differs in important ways from classical test theory (CTT). First of all, in CTT, no assumptions are made about the frequency distribution of test scores. In contrast, such assumptions are inherent in IRT. More specifically, for most applications in educational and psychological testing, there are three assumptions made regarding the data to be analyzed within an IRT framework. Those three assumptions are: (1) unidimensionality; (2) local independence; (3) monotonicity.

1. The unidimensionality assumption states that the set of items measures a single latent construct, often denoted by the Greek symbol theta (Θ).
2. The local independence assumption states that (a) there is a systematic relationship between all of the test items, and; (b) that relationship has to do with the ability level of the test taker.
3. The monotonicity assumption states that the probability of endorsing or selecting an item response indicative of higher levels of the ability (Θ) should increase as the underlying level of Θ increases.

In practice, IRT models tend to be robust, which implies that they can handle minor violations of these three assumptions. Still, the better the data meets the assumptions, the better the IRT model will fit the data and provide insight into the construct of interest.

Finally, two more definitions are discussed. First, the probabilistic relationship between a test taker's response to an item of the test and the testtaker's level of the latent construct being measures can be expressed in a graphic form by an Item Characteristics Curve (ICC). For each response category a unique curve will be plotted. Next, in such a plot, the vertical axis indicates the probability bounded between 0 and 1 that a person will select one of the item response categories. The horizontal axis indicates the ability level (Θ). Second, another useful curve is the information curve (IC) which provides insight into what items work best with test takers at a particular Θ level as compared to other items in the test.