A bivariate lognormal response-time model for the detection of collusion between test takers

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A bivariate lognormal model for the distribution of the response times on a test by a pair of test takers is presented. As the model has parameters for the item effects on the response times, its correlation parameter automatically corrects for the spuriousness in the observed correlation between the response times of different test takers because of variation in the time intensities of the items. This feature suggests using the model in a routine check of response-time patterns for possible collusion between test takers using an estimate of the correlation parameter or a statistical test of a hypothesis about it. Closed-form expressions for the maximum-likelihood estimations of the model parameters and a Lagrange multiplier test for the correlation parameter are presented. As in any type of statistical decision making, results from such procedures should be corroborated by evidence from other sources, for example, results from a response-based analysis or observations during the test session. The effectiveness of the model in removing the spuriousness from correlated response times is illustrated using empirical response-time data.
Original languageEnglish
Pages (from-to)378-394
Number of pages17
JournalJournal of educational and behavioral statistics
Issue number3
Publication statusPublished - 2009


  • Cheating
  • Lagrange multiplier test
  • Lognormal model
  • Maximum-likelihood estimation
  • Response time


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