Computation of the Response Similarity Index M4 in R under the Dichotomous and Nominal Item Response Models

Abstract

Unusual response similarity among test takers may occur in testing data and be an indicator of potential test fraud (e.g., examinees copy responses from other examinees, send text messages or pre-arranged signals among themselves for the correct response, item pre-knowledge). One index to measure the degree of similarity between two response vectors is M4 proposed by Maynes (2014). M4 index is based on a generalized trinomial distribution and it is computationally very demanding. There is currently no accessible tool for practitioners who may want to use M4 in their research and practice. The current paper introduces the M4 index and its computational details for the dichotomous and nominal item response models, provides an R function to compute the probability distribution for the generalized trinomial distribution, and then demonstrates the computation of the M4 index under the dichotomous and nominal item response models using R.

Keywords

• response similarity
• M4
• test fraud
• item response theory
• test security

Computation of the Response Similarity Index M4 in R under the Dichotomous and Nominal Item Response Models

Abstract

Unusual response similarity among test takers may occur in testing data and be an indicator of potential test fraud (e.g., examinees copy responses from other examinees, send text messages or pre-arranged signals among themselves for the correct response, item pre-knowledge). One index to measure the degree of similarity between two response vectors is M4 proposed by Maynes (2014). M4 index is based on a generalized trinomial distribution and it is computationally very demanding. There is currently no accessible tool for practitioners who may want to use M4 in their research and practice. The current paper introduces the M4 index and its computational details for the dichotomous and nominal item response models, provides an R function to compute the probability distribution for the generalized trinomial distribution, and then demonstrates the computation of the M4 index under the dichotomous and nominal item response models using R.

Keywords

• response similarity
• M4
• test fraud
• item response theory
• test security

References

1. Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37(1), 29-51.
2. Chalmers, R.P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. URL http://www.jstatsoft.org/v48/i06/.
3. Charalambides, C. A. (2005). Combinatorial methods in discrete distributions (Vol. 600). John Wiley & Sons.
4. Gabriel, T. (2010, December 27). Cheaters find an Adversary in Technology. The New York Times. Retrieved from https://www.nytimes.com/2010/12/28/education/28cheat.html
5. Maynes, D.D. (2014). Detection of non-independent test taking by similarity analysis. In N. M. Kingston and A. K. Clark (Eds.) Test fraud: Statistical detection and methodology (pp. 53-82). Routledge: New York, NY. Maynes, D. D. (2017). Detecting potential collusion among individual examinees using similarity analysis. In GJ Cizek and JA Wollack (eds.), Handbook of quantitative methods for detecting cheating on tests, Chapter 3, 47-69. Routledge, New York, NY.
6. Penfield, R. D., de la Torre, J., & Penfield, R. (2008). A new response model for multiple-choice items. Presented at the annual meeting of the National Council on Measurement in Education, New York.
7. Thissen, D., & Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49(4), 501-519.
8. van der Linden, W. J., & Sotaridona, L. (2006). Detecting answer copying when the regular response process follows a known response model. Journal of Educational and Behavioral Statistics, 31(3), 283-304. Wollack, J. A. (1997). A nominal response model approach for detecting answer copying. Applied Psychological Measurement, 21(4), 307-320.