Lord’s Equity Theorem Revisited

Wim J. van der Linden*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


Lord’s (1980) equity theorem claims observed-score equating to be possible only when two test forms are perfectly reliable or strictly parallel. An analysis of its proof reveals use of an incorrect statistical assumption. The assumption does not invalidate the theorem itself though, which can be shown to follow directly from the discrete nature of the equating problem it addresses. But, surprisingly, an obvious relaxation of the problem is enough to obtain exactly the opposite result: As long as two test forms measure the same ability, they can always be equated, no matter their reliability, degree of parallelness, or even difference in length. Also, in spite of its lack of validity, the original proof of Lord’s theorem has an important interim result directly applicable to the problem of assembling a new test form pre-equated to an old form.

Original languageEnglish
Pages (from-to)415-430
Number of pages16
JournalJournal of educational and behavioral statistics
Issue number4
Early online date24 Mar 2019
Publication statusPublished - 1 Aug 2019


  • UT-Hybrid-D
  • equity
  • local equating
  • observed-score equating
  • Q-Q transformation
  • compound binomial distribution


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