2PL3PL4PL

2PL3PL4PL is an informal label used in item response theory (IRT) to refer to the family of logistic models that extend the basic two-parameter logistic model by incorporating additional asymptotes or guessing parameters. It is not a single formal model, but a shorthand used in discussions of model choice, comparative analyses, or software documentation to encompass 2PL, 3PL, and 4PL formulations.

Two-parameter logistic model: In the 2PL, each item is characterized by a discrimination parameter a and a difficulty parameter b. The probability of a correct response for a person with ability theta is typically written as P(theta) = 1 / (1 + exp(-D a (theta - b))) where D is a scaling constant often set to 1.7. This model assumes a perfect upper and lower bound and does not include guessing or asymptote flexibility.

Three-parameter logistic model: The 3PL adds a lower asymptote c, representing the probability of a correct response due to guessing. The probability becomes P(theta) = c + (1 - c) * [1 / (1 + exp(-D a (theta - b)))], introducing a minimum probability of success regardless of ability.

Four-parameter logistic model: The 4PL adds both a lower asymptote c and an upper asymptote d, allowing the upper bound to be less than 1. The form is P(theta) = c + (d - c) * [1 / (1 + exp(-D a (theta - b)))], with 0 <= c < d <= 1. This provides greater flexibility to model ceiling effects and non-perfect performance.

Applications and considerations: The choice among 2PL, 3PL, and 4PL depends on test design and data characteristics. 3PL is common for multiple-choice assessments due to guessing, while 4PL is used when both floor and ceiling effects are pronounced. Estimation typically uses maximum likelihood or Bayesian methods, and model fit is assessed via information criteria or likelihood-based tests.