Item Response Theory

Item response theory (IRT) is a family of psychometric models in which the probability of a particular response to an item is expressed as a function of the examinee's latent ability and the item's parameters. Where classical test theory works at the level of total scores and assumes a single error term, IRT models the response to each item directly and yields ability and item parameters on a common scale. The canonical treatment for educational measurement is Lord's 1980 Applications of Item Response Theory to Practical Testing Problems.

Logistic models

For dichotomously scored items, three models dominate. The one-parameter logistic (1PL) model, equivalent to the Rasch model, allows items to vary only in difficulty. The two-parameter logistic (2PL) model adds an item-specific discrimination parameter, so items can differ in how sharply they distinguish examinees of nearby ability. The three-parameter logistic (3PL) model adds a lower asymptote — the pseudo-guessing parameter — which models the probability that a low-ability examinee answers correctly by chance. Polytomous extensions include the graded response model and the generalized partial credit model for ordered category responses.

Item characteristic curve

Each item's behaviour is summarised by an item characteristic curve (ICC), the S-shaped function relating ability to the probability of a correct response. The point of inflexion locates difficulty; the slope at that point indexes discrimination; the lower asymptote indexes guessing. Test information functions, the sum of item information across the bank, identify ability ranges where measurement is precise.

Use in language testing

IRT underwrites computer-adaptive testing (used by TOEFL iBT and the Pearson Test of English), test equating across forms, item banking, and DIF analysis. The conditional standard error of measurement derived from IRT replaces the single classical SEM with one that varies along the ability scale, enabling targeted item development for borderline cut-score candidates.

References

• Lord, F. M. (1980). Applications of Item Response Theory to Practical Testing Problems. Lawrence Erlbaum Associates.
• Hambleton, R. K., Swaminathan, H., & Rogers, H. J. (1991). Fundamentals of Item Response Theory. Sage.
• AERA, APA, & NCME (2014). Standards for Educational and Psychological Testing. American Educational Research Association.