Health Services and Outcomes Research Methodology

, Volume 18, Issue 3, pp 155–174 | Cite as

Statistical testing when the populations from which samples are drawn are uncertain

  • Gary C. McDonald


The topic of this article is hypothesis testing when the populations from which the data are drawn are known only with a given probability distribution. Some important areas of application for which such a situation arises is reviewed briefly. The specific cases herein considered are testing a one-sided hypothesis involving two populations. An illustrative small data set, involving six observations, is used to demonstrate relevant approaches and calculations for such testing. Both a frequentist approach and a Bayesian approach are developed. In both of these approaches, use is made of all possible data configurations along with their corresponding probabilities. Various measures of goodness are developed for each of the two approaches. A simulation approach is developed for larger data sets.


Bayesian improved surname and geocoding Most likely strategy Averaging strategy Bayes factor Posterior distribution 


Compliance with ethical standards

Ethical approval

This article does not contain any studies with human participants or animals performed by the author.


  1. Adjaye-Gbewonyo, D., Bednarczyk, R.A., Davis, R.L., Omer, S.B.: Using the Bayesian improved surname geocoding method (BISG) to create a working classification of race and ethnicity in a diverse managed care population: a validation study. Health Serv. Res. 49(1), 268–283 (2014). CrossRefPubMedGoogle Scholar
  2. Albert, J.: Bayesian Computation with R, 2nd edn. Springer, New York (2009)CrossRefGoogle Scholar
  3. Brown, D.P., Knapp, C., Baker, K., Kaufmann, M.: Using Bayesian imputation to assess racial and ethnic disparities in pediatric performance measures. Health Serv. Res. 51(3 Pt. 1), 1095–1108 (2016). CrossRefPubMedGoogle Scholar
  4. Consumer Finance Protection Bureau: Using publicly available information to proxy for unidentified race and ethnicity (2014 Summer).
  5. Consumer Finance Protection Bureau: Fair lending report of the Consumer Financial Protection Bureau (2016 April).
  6. Elkadry, A.: Statistical analyses of randomly sourced data. Ph.D. Dissertation, Oakland University, Rochester, MI, Dept. of Mathematics and Statistics (2017)Google Scholar
  7. Elliott, M.N., Morrison, P.A., Fremont, A., McCaffrey, D.F., Pantoja, P., Lurie, N.: Using the census bureau’s surname list to improve estimates of race/ethnicity and associated disparities. Health Serv. Outcomes Res. Methodol. 9, 69–83 (2009).; Erratum (2009), 9, 252–253.
  8. Fiscella, K., Fremont, A.M.: Use of geocoding and surname analysis to estimate race and ethnicity. Health Serv. Res. 41(4 pt 1), 1482–1500 (2006). PubMedPubMedCentralGoogle Scholar
  9. Gill, J.: Bayesian Methods: A Social and Behavioral Sciences Approach, 3rd edn. CRC Press, Boca Raton (2015)Google Scholar
  10. Jeffreys, H.: The Theory of Probability, 3rd edn. Oxford University Press, Oxford (1961)Google Scholar
  11. McDonald, K.M., Rojc, K.J.: Automotive finance regulation: warning lights flashing. Bus Lawyer 70, 617–624 (2015)Google Scholar
  12. Navidi, W.: Statistics for Engineers and Scientists, 4th edn. McGraw Hill Education, New York (2015)Google Scholar
  13. Parmigiani, G.: Modeling in Medical Decision Making: A Bayesian Approach. Wiley, New York (2002)Google Scholar
  14. Republican Staff of the Committee on Financial Services, U. S. House of Representatives: Unsafe at any bureaucracy: CFPB junk science and indirect auto lending. In: 114th Congress, First Session, November 24, 2015Google Scholar
  15. Republican Staff of the Committee on Financial Services, U. S. House of Representatives: Unsafe at any bureaucracy, Part II: how the bureau of consumer financial protection removed anti-fraud safeguards to achieve political goals. In: 114th Congress, Second Session, January 20, 2016Google Scholar
  16. Republican Staff of the Committee on Financial Services, U. S. House of Representatives: Unsafe at any bureaucracy, Part III: the CFPB’s vitiated legal case against auto-lenders. In: 115th Congress, First Session, January 18, 2017Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Oakland UniversityRochesterUSA

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