Occasionally mixtures of two binomials with one known success parameter are met. An example in educational testing is the mastery or random guessing model, in which an examinee is supposed either to master the items or not to master them and to guess blindly. This paper gives moment estimators for such mixtures and presents results from a Monte Carlo investigation into their statistical properties. The results suggest excellent estimators which can safely be used in most instances. It is also indicated how the properties of these estimators relate to those of moment estimators in case both success parameters are unknown. Finally, it is pointed out that in situations in which errors in specifying the true value of the known parameter may occur, it might be prudent to consider this parameter as unknown and to estimate accordingly.
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References
1.
Bergan, J. R. , Cancelli, A. A., and Luiten, J. W.Mastery assessment with latent class and quasi-independence models representing homogeneous item domains. Journal of Educational Statistics, 1980, 5, 65-82.
2.
Besel, R.Using group performance to interpret individual responses to criterion-referenced tests . Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, Louisiana, February-March 1973. (EDRS No. ED 076 658).
3.
Blischke, W. R.Moment estimators for the parameters of a mixture of two binomial distributions . Annals of Mathematical Statistics, 1962, 33, 444-454.
4.
Blischke, W. R.Mixtures of discrete distributions . In Proceedings of the International Symposium on Discrete Distributions, Montreal, 1963.
5.
Dayton, C. M. and Macready, G. B.A probabilistic model for validation of behavioral hierarchies. Psychometrika, 1976, 41, 189-204.
6.
Emrick, J. A.An evaluation model for mastery testing . Journal of Educational Measurement, 1971, 8, 321-326.
7.
Emrick, J. A. and Adams, E. N.An evaluation model for individualized instruction (Report RC 2674). Yorktown Hts., N.Y.: IBM Thomans J. Watson Research Center, October 1969.
8.
Harris, C. W. and Pearlman, A. P.An index for a domain of completion or short answer items. Journal of Educational Statistics, 1978, 3, 285-303.
9.
Keats, J. A. and Lord, F. M.A theoretical distribution for mental test scores. Psychometrika, 1962, 27, 59-72.
10.
Knapp, T. R.The reliability of a dichotomous test item: A `correlationless' approach . Journal of Educational Measurement, 1977, 14, 237-252.
11.
Macready, G. B. and Dayton, C. M.The use of probabilistic models in the assessment of mastery. Journal of Educational Statistics, 1977, 2, 99-120.
12.
Macready, G. B. and Dayton, C. M.The nature and use of state mastery models. Applied Psychological Measurement, 1980, 4, 493-516.
13.
Meskauskas, J. A.Evaluation models for criterion-referenced testing: Views regarding mastery and standard-setting . Review of Educational Research, 1976, 46, 133-158.
14.
Muench, H.Probability distributions of protection test results . Journal of the American Statistical Association, 1936, 31, 677-689.
15.
Muench, H.Discrete frequency distributions arising from mixtures of several single probability values . Journal of the American Statistical Association, 1938, 33, 390-398.
Powers, L.Gene analysis by the partitioning method when interaction of genes are involved . Botanical Gazette, 1951, 113, 1-23.
18.
Rao, C. R.The utilization of multiple measurements in problems of biological classification . Journal of the Royal Statistical Society, 1948, 10B, 159-203.
19.
Rao, C. R.Linear statistical inference and its applications. New York: Wiley, 1973.
20.
Reulecke, W. A.Ein Modell für die kriteriumorientierte Testauswertung . Zeitschrift für Empirische Pädagogik, 1977, 1, 49-72. (a)
21.
Reulecke, W. A.A statistical analysis of deterministic theories. In H. Spada and W. F. Kempf (Eds.), Structural models of thinking and learning. Bern, Switzerland: Hans Huber Publishers, 1977. (b)
22.
van der Linden, W. J.Decision models for use with criterion-referenced measurements . Applied Psychological Measurement, 1980, 4, 469-492. (a)
23.
van der Linden, W. J.Simple estimators for the simple latent class mastery testing model . Paper presented at the Fourth International Symposium on Educational Testing, Antwerp, Belgium, June 24-27, 1980. (b)
24.
van der Linden, W. J.Estimating the parameters of Emrick's mastery testing model . Applied Psychological Measurement, 1981, 5, 517-530.
25.
van der Linden, W. J.On the estimation of the proportion of masters in criterion-referenced testing . Zeitschrift für Empirische Paedagogik, 1982, 6, 195-201.
26.
Walsh, J. E.Approximate probability values for observed number of successes from statistically independent binomial events with unequal probabilities . Sanhkyà, Series A, 1953, 15, 281-290.
27.
Walsh, J. E.Definition and use of generalized percentage points . Sankhyà, Series A, 1959, 21, 281-288.
28.
Walsh, J. E.Corrections to two papers concerned with binomial events . Sankhyà, Series A, 1963, 25, 247.
29.
Wilcox, R. R.New methods for studying equivalence. In C. W. Harris, A. P. Pearlman, and R. R. Wilcox, Achievement test items: Methods for study (CSE Monograph No. 6). Los Angeles: University of California, Center for the Study of Evaluation, 1977. (a)
30.
Wilcox, R. R.New methods for studying stability. In C. W. Harris, A. P. Pearlman, and R. R. Wilcox, Achievement test items: Methods for study (CSE Monograph No. 6). Los Angeles: University of California, Center for the Study of Evaluation, 1977. (b)
