Abstract
A cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described. Statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method/degree of smoothing. These include: (a) asymptotic significance tests that compare no equating and linear equating to equipercentile equating; (b) a scheme for estimating total equating error and for dividing total estimated error into systematic and random components. The smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test.
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References
Angoff, W. H. (1971). Scales, norms, and equivalent scores. In R. L. Thorndike (Ed.),Educational measurement (2nd ed.). Washington, DC: American Council on Education.
Angoff, W. H. (1982). Summary and derivation of equating methods used at ETS. In P. W. Holland & D. B. Rubin (Eds.),Test equating. New York: Academic Press.
Beran, R. (1984). Bootstrap methods in statistics.Jahresbericht der Deutschen Mathematiker-Vereiningung, 86, 14–30.
Bickel, P. J., & Freedman, D. A. (1981). Some asymptotic theory for the bootstrap.The Annals of Statistics, 9, 1196–1217.
Blommers, P. J., & Forsyth, R. A. (1977).Elementary statistical methods in psychology and education. Boston, MA: Houghton Mifflin.
Braun, H. I., & Holland, P. W. (1982). Observed score test equating: A mathematical analysis of some ETS equating procedures. In P. W. Holland & D. B. Rubin (Eds.),Test equating. New York: Academic Press.
Efron, B. (1981). Nonparametric estimates of the standard error: The jackknife, the bootstrap, and other methods.Biometrika, 68, 589–599.
Efron, B. (1982).The jackknife, the bootstrap, and other resampling plans. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Freedman, D. A. (1981). Bootstrapping regression models.The Annals of Statistics, 9, 1218–1228.
Jarjoura, D., & Kolen, M. J. (1985). Standard errors of equipercentile equating for the common item nonequivalent populations design.Journal of Educational Statistics, 10, 143–160.
Klein, L. W., & Jarjoura, D. (1985). The importance of content representation for common-item equating with nonrandom groups.Journal of Educational Measurement, 22, 197–206.
Kolen, M. J. (1984). Effectiveness of analytic smoothing in equipercentile equating.Journal of Educational Statistics, 9, 25–44.
Kolen, M. J. (1985). Standard errors of Tucker equating.Applied Psychological Measurement, 9, 209–223.
Lord, F. M. (1980).Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
Morrison, D. F. (1976).Multivariate statistical methods (2nd ed.). New York: McGraw Hill.
Petersen, N. S., Marco, G. L., & Stewart, E. E. (1982). A test of the adequacy of linear score equating models. In P. W. Holland & D. B. Rubin (Eds.),Test equating. New York: Academic Press.
Rao, C. R. (1973).Linear statistical inference and its applications (2nd ed). New York: Wiley.
Reinsch, C. H. (1967). Smoothing by spline functions.Numerische Mathematik, 10, 177–183.
Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap.The Annals of Statistics, 9, 1187–1195.
Tapia, R. A., & Thompson, J. R. (1978).Nonparametric probability density estimation. Baltimore: The Johns Hopkins University Press.
Yang, S. (1985). A smooth nonparametric estimator of a quantile function.Journal of the American Statistical Association, 80, 1004–1011.
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The authors thank Robert L. Brennan for reviewing an earlier draft of this manuscript. Most of the work was completed while the second author was at The American College Testing Program.
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Kolen, M.J., Jarjoura, D. Analytic smoothing for equipercentile equating under the common item nonequivalent populations design. Psychometrika 52, 43–59 (1987). https://doi.org/10.1007/BF02293955
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DOI: https://doi.org/10.1007/BF02293955