A Paradox by Another Name Is Good Estimation

  • Mark D. ReckaseEmail author
  • Xin Luo
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 89)


This chapter describes the property of estimates of points in a multidimensional space that is labeled by some as paradoxical, shows when this property of the estimates is present, and also shows that the paradoxical result is not flaw in estimation because estimates improve with additional information even when the paradox occurs. The paradox is that when a correct response to a test item is added to the string of responses for an examinee to previous items, at least one of the coordinates of the new estimated θ-point decreases compared to the estimate based on the initial string of responses. The information presented in the chapter shows that this can occur whenever the likelihood function for the estimates has a particular form. This form is present in many cases when the item responses for a test can not be described by simple structure. Results are presented to show that the additional response improves the estimate of the θ-point even though the paradoxical result occurs.


Multidimensional item response theory Estimation Likelihood function Compensatory model 


  1. Finkelman M, Hooker G, Wang J (2010) Prevalence and magnitude of paradoxical results in multidimensional item response theory. J Educ Behav Stat 35:744–761CrossRefGoogle Scholar
  2. Flexner SB, Hauck LC (eds) (1987) The Random House dictionary of the English language, 2, unabridgedth edn. Random House, New YorkGoogle Scholar
  3. Hooker G (2010) On separable tests, correlated priors, and paradoxical results in multidimensional item response theory. Psychometrika 75:694–707CrossRefzbMATHMathSciNetGoogle Scholar
  4. Hooker G, Finkelman M (2010) Paradoxical results and item bundles. Psychometrika 75:249–271CrossRefzbMATHMathSciNetGoogle Scholar
  5. Hooker G, Finkelman M, Schwartzman A (2009) Paradoxical results in multidimensional item response theory. Psychometrika 74:419–442CrossRefzbMATHMathSciNetGoogle Scholar
  6. Jordan P, Spiess M (2012) Generalizations of paradoxical results in multidimensional item response theory. Psychometrika 77:127–152CrossRefzbMATHMathSciNetGoogle Scholar
  7. Reckase M (2009) Multidimensional item response theory. Springer, New YorkCrossRefGoogle Scholar
  8. Van der Linden WJ (2012) On compensation in multidimensional response modeling. Psychometrika 77:21–30CrossRefzbMATHMathSciNetGoogle Scholar
  9. van Rijn PW, Rijmen F (2012) A note on explaining away and paradoxical results in multidimensional item response theory (Research Report RR-12-13). Educational Testing Service, Princeton, NJGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.CEPSEMichigan State UniversityEast LansingUSA
  2. 2.Department of Counseling Educational Psychology and Spec Ed, MQM, CEPSECollege of Education, Michigan State UniversityEast LansingUSA

Personalised recommendations