### Abstract

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 language | English |
---|---|

Pages (from-to) | 415-430 |

Number of pages | 16 |

Journal | Journal of educational and behavioral statistics |

Volume | 44 |

Issue number | 4 |

Early online date | 24 Mar 2019 |

DOIs | |

Publication status | Published - 1 Aug 2019 |

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### Keywords

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

### Cite this

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*Journal of educational and behavioral statistics*, vol. 44, no. 4, pp. 415-430. https://doi.org/10.3102/1076998619837627

**Lord’s Equity Theorem Revisited.** / van der Linden, Wim J.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Lord’s Equity Theorem Revisited

AU - van der Linden, Wim J.

N1 - Sage deal

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

KW - UT-Hybrid-D

KW - equity

KW - local equating

KW - observed-score equating

KW - Q-Q transformation

KW - compound binomial distribution

UR - http://www.scopus.com/inward/record.url?scp=85063572950&partnerID=8YFLogxK

U2 - 10.3102/1076998619837627

DO - 10.3102/1076998619837627

M3 - Article

AN - SCOPUS:85063572950

VL - 44

SP - 415

EP - 430

JO - Journal of educational and behavioral statistics

JF - Journal of educational and behavioral statistics

SN - 1076-9986

IS - 4

ER -