The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework

R. A. Weitzman

doi:doi:10.11648/j.ajtas.20221101.12

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The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework

R. A. Weitzman

Published in American Journal of Theoretical and Applied Statistics (Volume 11, Issue 1)

Received: 28 December 2021 Accepted: 17 January 2022 Published: 21 January 2022

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Abstract

Principally to reduce the cost by reducing the sample size required to conduct survey research, this article presents and illustrates the use of a method to determine the sample sizes required to obtain Bayesian estimates of population proportions with specified margins of error. The development proceeds within a regression framework derived from mental test theory. Specifically, building on prior work, the development presented here enables a researcher to conduct a survey to obtain pure Bayes estimates of the proportion of all members of a defined population choosing each one of a number of mutually exclusive and exhaustive options or falling into each one of a number of mutually exclusive and exhaustive categories, including two. The regression framework not only provides useful insight into Bayesian and classical statistics but also enables the development to proceed without explicit reference to the differing parent distributions of the sample and population proportions, both being asymptotically normal. In addition to the sample-size advantage, which is substantial, this article identifies other practical advantages that Bayesian has over classical estimation of population proportions and, in a somewhat in-depth comparison of the two, discusses other reasons a Bayesian method may be a powerful substitute for the classical method of estimating population proportions via independent random sampling.

Published in	American Journal of Theoretical and Applied Statistics (Volume 11, Issue 1)
DOI	10.11648/j.ajtas.20221101.12
Page(s)	13-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Population Proportion, Survey Research, Pure Bayes Estimate, Regression Model, Standard Error of Estimate, Sample Size

References

[1]	Alvarez, E., Arcos, A., Gonzalez, S., Munoz, J. F., & Rueda, M. (2013). Estimating population proportions in the presence of missing data, Journal of Computational and Applied Mathematics, 237 (1), 470-476.
[2]	Charter, Richard A. (1992). Revisiting the Standard Errors of Measurement, Estimate, and Prediction and Their Application to Test Scores, Perception and Motor Skills, 82 (3c), 1139-1144.
[3]	Efron, B., & Morris, C. (1973). Stein's Estimation Rule and its Competitors—an Empirical Bayes Approach. Journal of the American Statistical Association, 68, 117-130.
[4]	Fienberg, S. E., & Holland, P. W. (1973). Simultaneous Estimation of Multinomial Cell Probabilities. Journal of the American Statistical Association, 68, 683-691.
[5]	Gulliksen, H. (1950). Theory of mental tests. New York: John Wiley.
[6]	Healy, J. D. (1981). The Effects of Misclassification Error on the Estimation of Several Population Proportions, The Bell System Technical Journal, 60 (5), 697-705.
[7]	James, W, & Stein, C. M. (1961). Estimation with Quadratic Loss Function. Proceedings of the 4th Berkeley Symposium, 1, 361-379.
[8]	Kelley, T. L. (1947). Fundamental Statistics. Cambridge, MA: Harvard University Press.
[9]	Morris, C. N. (1983). Parametric Empirical Bayes Inference: Theory and Applications. Journal of the American Statistical Association, 78, 47-59.
[10]	Novick, M. R., Lewis, C, & Jackson, P. H. (1973). The Estimation of Proportions in m Groups, Psychometrika, 38, 19–46.
[11]	Pfeffermann, Danny (1993). The Role of Sampling Weights When Modeling Survey Data, International Statistical Review / Revue nternationale de Statistique,. 317-337.
[12]	Stigler, S. M. (1983). Comment on C. Morris, Parametric Empirical Bayes Inference: Theory and Applications. Journal of the American Statistical Association, 78, 62-63.
[13]	Stigler, S. M. (1990). The 1988 Neyman Memorial Lecture: a Galtonian Perspective on Shrinkage Estimators. Statistical Science, 5, 147-155.
[14]	Weitzman, R. A. (2006). Population-Sample Regression in the Estimation of Population Proportions. Journal of Educational and Behavioral Statistics, 31, 413-43.
[15]	Weitzman, R. A. (2009). Fitting the Rasch Model to Account for Variation in Item Discrimination, Educational and Psychological Measurement, 69, 216-231.

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APA Style

R. A. Weitzman. (2022). The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. American Journal of Theoretical and Applied Statistics, 11(1), 13-18. https://doi.org/10.11648/j.ajtas.20221101.12

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ACS Style

R. A. Weitzman. The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. Am. J. Theor. Appl. Stat. 2022, 11(1), 13-18. doi: 10.11648/j.ajtas.20221101.12

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AMA Style

R. A. Weitzman. The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. Am J Theor Appl Stat. 2022;11(1):13-18. doi: 10.11648/j.ajtas.20221101.12

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@article{10.11648/j.ajtas.20221101.12,
  author = {R. A. Weitzman},
  title = {The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {1},
  pages = {13-18},
  doi = {10.11648/j.ajtas.20221101.12},
  url = {https://doi.org/10.11648/j.ajtas.20221101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221101.12},
  abstract = {Principally to reduce the cost by reducing the sample size required to conduct survey research, this article presents and illustrates the use of a method to determine the sample sizes required to obtain Bayesian estimates of population proportions with specified margins of error. The development proceeds within a regression framework derived from mental test theory. Specifically, building on prior work, the development presented here enables a researcher to conduct a survey to obtain pure Bayes estimates of the proportion of all members of a defined population choosing each one of a number of mutually exclusive and exhaustive options or falling into each one of a number of mutually exclusive and exhaustive categories, including two. The regression framework not only provides useful insight into Bayesian and classical statistics but also enables the development to proceed without explicit reference to the differing parent distributions of the sample and population proportions, both being asymptotically normal. In addition to the sample-size advantage, which is substantial, this article identifies other practical advantages that Bayesian has over classical estimation of population proportions and, in a somewhat in-depth comparison of the two, discusses other reasons a Bayesian method may be a powerful substitute for the classical method of estimating population proportions via independent random sampling.},
 year = {2022}
}

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TY  - JOUR
T1  - The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework
AU  - R. A. Weitzman
Y1  - 2022/01/21
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T2  - American Journal of Theoretical and Applied Statistics
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VL  - 11
IS  - 1
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Author Information

R. A. Weitzman

Naval Postgraduate School, Monterey, USA

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APA Style

R. A. Weitzman. (2022). The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. American Journal of Theoretical and Applied Statistics, 11(1), 13-18. https://doi.org/10.11648/j.ajtas.20221101.12

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ACS Style

R. A. Weitzman. The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. Am. J. Theor. Appl. Stat. 2022, 11(1), 13-18. doi: 10.11648/j.ajtas.20221101.12

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AMA Style

R. A. Weitzman. The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework. Am J Theor Appl Stat. 2022;11(1):13-18. doi: 10.11648/j.ajtas.20221101.12

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@article{10.11648/j.ajtas.20221101.12,
  author = {R. A. Weitzman},
  title = {The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {1},
  pages = {13-18},
  doi = {10.11648/j.ajtas.20221101.12},
  url = {https://doi.org/10.11648/j.ajtas.20221101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221101.12},
  abstract = {Principally to reduce the cost by reducing the sample size required to conduct survey research, this article presents and illustrates the use of a method to determine the sample sizes required to obtain Bayesian estimates of population proportions with specified margins of error. The development proceeds within a regression framework derived from mental test theory. Specifically, building on prior work, the development presented here enables a researcher to conduct a survey to obtain pure Bayes estimates of the proportion of all members of a defined population choosing each one of a number of mutually exclusive and exhaustive options or falling into each one of a number of mutually exclusive and exhaustive categories, including two. The regression framework not only provides useful insight into Bayesian and classical statistics but also enables the development to proceed without explicit reference to the differing parent distributions of the sample and population proportions, both being asymptotically normal. In addition to the sample-size advantage, which is substantial, this article identifies other practical advantages that Bayesian has over classical estimation of population proportions and, in a somewhat in-depth comparison of the two, discusses other reasons a Bayesian method may be a powerful substitute for the classical method of estimating population proportions via independent random sampling.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - The Determination of Sample Size in a Bayesian Estimation of Population Proportions: How and Why to Do It in a Regression Framework
AU  - R. A. Weitzman
Y1  - 2022/01/21
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221101.12
DO  - 10.11648/j.ajtas.20221101.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 13
EP  - 18
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221101.12
AB  - Principally to reduce the cost by reducing the sample size required to conduct survey research, this article presents and illustrates the use of a method to determine the sample sizes required to obtain Bayesian estimates of population proportions with specified margins of error. The development proceeds within a regression framework derived from mental test theory. Specifically, building on prior work, the development presented here enables a researcher to conduct a survey to obtain pure Bayes estimates of the proportion of all members of a defined population choosing each one of a number of mutually exclusive and exhaustive options or falling into each one of a number of mutually exclusive and exhaustive categories, including two. The regression framework not only provides useful insight into Bayesian and classical statistics but also enables the development to proceed without explicit reference to the differing parent distributions of the sample and population proportions, both being asymptotically normal. In addition to the sample-size advantage, which is substantial, this article identifies other practical advantages that Bayesian has over classical estimation of population proportions and, in a somewhat in-depth comparison of the two, discusses other reasons a Bayesian method may be a powerful substitute for the classical method of estimating population proportions via independent random sampling.
VL  - 11
IS  - 1
ER  -

Copy | Download