Psychometrics

Psychometrics refers to the measurement of abilities, traits, and attitudes with questionnaires and tests. While such techniques as factor analysis and structural equation modeling can be thought to fall under the umbrella of psychometrics, this article focuses on item response theory (IRT), which studies the relationship between traits of individuals and their responses to items on a test. Although the roots of IRT are in educational testing, it has applications in many other disciplines. Much of the pioneering work in the field is attributed to Frederick Lord.

The trait studied in IRT is most often a latent trait, one that cannot be measured directly. IRT describes how a person's observable behavior, or his or her performance on a test, is related to this latent trait or ability. Characteristics of each item on the test determine the likelihood of a particular response at a given level of ability. The purpose of IRT modeling is usually to estimate respondents’ abilities, although the item characteristics may also be of interest.

Test items need to discriminate well in order to inform researchers about respondents’ abilities. That is, people with lower values of the trait should tend to respond one way to an item while people with higher values of the trait respond differently. The relationship between values of the trait under consideration and the probability of a particular response is modeled with the item characteristic curve (ICC), sometimes called an item response function. The most basic IRT models are presented first, and then more sophisticated models are discussed.

Logistic and Ogive Models

Logistic and ogive curves are natural choices for modeling ICCs when items have only two response options (or are coded to be binary) because they are monotonically increasing in the latent trait and they allow the probability of a correct response, conditional on the trait, to vary from 0 to 1. Ogive curves are defined by the cumulative normal distribution and logistic curves by the cumulative logistic distribution.

Before stating the models, the notation used to refer to the test is given. θ is the ability parameter or level of the trait, and the true ability of individual i is denoted θi. Items are indexed with the letter j, and Yij is the response of individual i to item j. The number 1 denotes a correct answer and 0 an incorrect answer. Pj(θ) represents the probability that an individual with ability θ answers item j correctly; it is the height of the ICC curve for item j at θ. For notational convenience in working with logistic curves, define

. This is the height of the cumulative distribution function (c.d.f.) of the logistic distribution with location parameter b and scale parameter a at θ. [Page 1139]Also, define Φ(θ, a, b) to be the height of the normal cdf with mean b and standard deviation a at θ. The Rasch model, or one-parameter logistic model, is the most basic of the logistic models for IRT. The single-item parameter bj is a location parameter that is interpreted as the difficulty of item j. Under the Rasch

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