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Testing a Precise Null Hypothesis: The Case of Lindley’s Paradox

Published online by Cambridge University Press:  01 January 2022

Jan Sprenger
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Abstract

Testing a point null hypothesis is a classical but controversial issue in statistical methodology. A prominent illustration is Lindley’s Paradox, which emerges in hypothesis tests with large sample size and exposes a salient divergence between Bayesian and frequentist inference. A close analysis of the paradox reveals that both Bayesians and frequentists fail to satisfactorily resolve it. As an alternative, I suggest Bernardo’s Bayesian Reference Criterion: (i) it targets the predictive performance of the null hypothesis in future experiments; (ii) it provides a proper decision-theoretic model for testing a point null hypothesis; (iii) it convincingly addresses Lindley’s Paradox.

Type
General Philosophy of Science
Information
Philosophy of Science , Volume 80 , Issue 5 , December 2013 , pp. 733 - 744
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

†

The author wishes to thank the Netherlands Organisation for Scientific Research (NWO) for support of his research through Veni grant 016.104.079, as well as José Bernardo, Cecilia Nardini, and the audience at PSA 2012, San Diego, for providing helpful input and feedback.

References

Berger, James O., and Delampady, Mohan. 1987. “Testing Precise Hypotheses.” Statistical Science 2:317–52.Google Scholar
Berger, James O., and Sellke, Thomas. 1987. “Testing a Point Null Hypothesis: The Irreconcilability of p-Values and Evidence.” Journal of the American Statistical Association 82:112–39.Google Scholar
Bernardo, José M. 1979. “Reference Posterior Distributions for Bayesian Inference.” Journal of the Royal Statistical Society B 41:113–47.Google Scholar
Bernardo, José M. 1999. “Nested Hypothesis Testing: The Bayesian Reference Criterion.” In Bayesian Statistics, Vol. 6, Proceedings of the Sixth Valencia Meeting, ed. Bernardo, J. M. et al., 101–30. Oxford: Oxford University Press.Google Scholar
Bernardo, J. M. 2012. “Integrated objective Bayesian estimation and hypothesis testing.” In Bayesian Statistics, Vol. 9, Proceedings of the Ninth Valencia Meeting, ed. Bernardo, J. M. et al., 1–68. Oxford: Oxford University Press.Google Scholar
Cohen, Jacob. 1994. “The Earth Is Round ().” American Psychologist 49:997–1001.CrossRefGoogle Scholar
Earman, John. 1992. Bayes or Bust? Cambridge, MA: MIT Press.Google Scholar
Good, I. J. 1952. “Rational Decisions.” Journal of the Royal Statistical Society B 14:107–14.Google Scholar
Goodman, S. N. 1999. “Towards Evidence-Based Medical Statistics.” Pt. 1, “The P Value Fallacy.” Annals of Internal Medicine 130:1005–13.Google Scholar
Jahn, R. G., Dunne, B. J., and Nelson, R. D.. 1987. “Engineering Anomalies Research.” Journal of Scientific Exploration 1:21–50.Google Scholar
Jefferys, William H. 1990. “Bayesian Analysis of Random Event Generator Data.” Journal of Scientific Exploration 4:153–69.Google Scholar
Lindley, Dennis V. 1957. “A Statistical Paradox.” Biometrika 44:187–92.CrossRefGoogle Scholar
Mayo, Deborah G. 1996. Error and the Growth of Experimental Knowledge. Chicago: University of Chicago Press.CrossRefGoogle Scholar
Popper, Karl R. 1934/1934. Logik der Forschung. Berlin: Akademie. English trans. The Logic of Scientific Discovery (New York: Basic, 1959).Google Scholar
Popper, Karl R. 1963. Conjectures and Refutations: The Growth of Scientific Knowledge. New York: Harper.Google Scholar
Royall, Richard. 1997. Scientific Evidence: A Likelihood Paradigm. London: Chapman & Hall.Google Scholar
Schmidt, Frank L., and Hunter, John E.. 1997. “Eight Common but False Objections to the Discontinuation of Significance Testing in the Analysis of Research Data.” In What If There Were No Significance Tests? ed. Harlow, Lisa L. et al., 37–64. Mahwah, NJ: Erlbaum.Google Scholar
Seidenfeld, Teddy. 1981. “On After-Trial Properties of Best Neyman-Pearson Confidence Intervals.” Philosophy of Science 48:281–91.CrossRefGoogle Scholar
Sprenger, Jan. 2013. “Bayesianism vs. Frequentism in Statistical Inference.” In Oxford Handbook of Probability and Philosophy, ed. Hájek, A. and Hitchcock, C.. Oxford: Oxford University Press, forthcoming.Google Scholar