# Item Response Theory (IRT)

**Item Response Theory (IRT)** is a paradigm in psychometrics and education that deals with the design, analysis, and scoring of questionnaires, tests, and other instruments for measuring abilities, attitudes, and other variables. Unlike Classical Test Theory, which focuses on test-level measures, IRT concentrates on individual items in a test to examine how different items function across varying levels of the attribute being measured.

## History

IRT began to gain prominence in the mid-20th century as computer technology advanced, allowing for more complex statistical models. Early contributions were made by psychologists and statisticians like Frederic M. Lord and Georg Rasch.

## Key Concepts

- Parameters: In IRT, items are described by one or more parameters that explain different aspects like difficulty, discrimination, and guessing. These parameters are used to construct the Item Characteristic Curve (ICC).
- Item Characteristic Curve (ICC): The ICC is a graphical representation showing the probability of a correct response to an item across different levels of the latent trait being measured. The curve is usually S-shaped (sigmoidal).

## Models

- 1PL Model: The One-Parameter Logistic (1PL) model, also known as the Rasch model, is the simplest IRT model, considering only the difficulty parameter.
- 2PL Model: The Two-Parameter Logistic (2PL) model considers both difficulty and discrimination parameters. It is more flexible than the 1PL model but requires more data.
- 3PL Model: The Three-Parameter Logistic (3PL) model adds a guessing parameter to the 2PL model, making it suitable for multiple-choice tests where guessing is a factor.

## Applications

- Educational Testing
- Personality Assessment
- Clinical Diagnosis
- Job Performance Measurement
- Adaptive Testing

## Software

- R packages like ltm and mirt
- Commercial software like Winsteps and IRTPro

## Limitations

- Requires large sample sizes
- Complex to understand and implement
- Assumes unidimensionality, which might not always be realistic