Approximate Functional Relationship between IRT and CTT Item Discrimination Indices: A Simulation, Validation, and Practical Extension of Lord's (1980) Formula

John T. Kulas, Jeffrey A. Smith, Hui Xu

Research output: Contribution to journalArticle


Lord (1980) presented a purely conceptual equation to approximate the nonlinear functional relationship between classical test theory (CTT; aka true score theory) and item response theory (IRT) item discrimination indices. The current project proposes a modification to his equation that makes it useful in practice. The suggested modification acknowledges the more common contemporary CTT discrimination index of a corrected item-total correlation and incorporates item difficulty. We simulated slightly over 768 trillion individual item responses to uncover a best-fitting empirical function relating the IRT and CTT discrimination indices. To evaluate the effectiveness of the function, we applied it to real-world test data from 16 workforce and educational tests. Our modification results in shifted functional asymptotes, slopes, and points of inflection across item difficulties. Validation with the workforce and educational tests suggests good prediction under common assumption testing conditions (approximately normal distribution of abilities and moderate item difficulties) and greater precision than Lord's (1980) formula.

Original languageEnglish
Pages (from-to)393-407
Number of pages15
JournalJournal of applied measurement
Issue number4
Publication statusPublished - 1 Jan 2017


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