Inferences on the Behrens–Fisher Problem Based on the Inferential Model

Abstract

The Behrens–Fisher problem is a fundamental and important problem in applied statistics. Among the existing approaches, the commonly used Welch's approximate t test has the greatest powers in most cases; however, Welch's t test may be liberal for small sample sizes. As three modified approaches, the GPQ, CC and MIM tests are conservative in controlling Type I risk. In this paper, a new test based on the inferential model (IM) framework is proposed to deal with Behrens–Fisher problem. Simulation studies show that the IM test is very easy to operate and does improve the GPQ, CC and MIM tests in terms of both Type I error rates and statistical powers. Moreover, the powers of our IM test are close to those of Welch's t test. In this sense, our IM test can be a competitive alternative method to the t test. More importantly, unlike the classical solution, the IM's output is posterior-probabilistic in nature and, therefore, has a meaningful interpretation within and not just across experiments. Finally, a real example is used to demonstrate the application of the proposed method.

Keywords

Behrens–Fisher problem / Welch's approximate t test / Predictive random set / Inferential model / Type I error rate / Statistical power / 62F03 / 62F25 / 62P30

References

[1] Ahmet S, Evren O, Berna Y. Comparison of confidence intervals for the Behrens–Fisher problem. Commun. Stat. Simul. Comput., 2017, 46: 3242-3266

[2] Aspin AA. An examination and further development of a formula arising in the problem of comparing two mean values. Biometrika, 1948, 35: 88-96

[3] Aspin AA. Tables for use in comparisons whose accuracy involves two variances, separately estimated. Biometrika, 1949, 36: 290-293

[4] Bather J. A conversation with Herman Chernoff. Stat. Sci., 1996, 11: 335-350

[5] Behrens BV. Ein Beitrag zur Fehlerberechnung bei wenige Beobachtungen. Landwirtch Jb, 1929, 6: 807-837

[6] Bellon A, Didier G. On the Behrens–Fisher problem: a globally convergent algorithm and a finite- sample study of the Wald, LR and LM tests. Ann. Stat., 2008, 36: 2377-2408

[7] Chapman DG. Some two sample tests. Ann. Math. Stat., 1950, 21: 601-606

[8] Cochran WG, Cox GM. Experiment Designs, 1950, New York, Wiley

[9] Dannenberg O, Dette H, Munk A. An extension of Welch’s approximate t-solution to comparative bioequivalence trials. Biometrika, 1994, 1: 91-101

[10] Dudewicz EJ, Ahmed SU. New exact and asymptotically optimal solution to the Behrens–Fisher problem, with tables. Am. J. Math. Manag. Sci., 1998, 18: 359-426

[11] Dudewicz EJ, Ma Y, Mai SE, Su H. Exact solutions to the Behrens–Fisher problem: asymptotically optimal and finite sample efficient choice among. J. Stat. Plan. Inference, 2007, 137: 1584-1605

[12] Fisher RA. The fiducial argument in statistical inference. Ann. Eugen., 1935, 6: 141-172

[13] Fraser DAS, Wong A, Sun Y. Three enigmatic examples and inference from likelihood. Can. J. Stat., 2009, 37: 161-181

[14] Ghosh M, Kim YH. The Behrens–Fisher problem revisited: a Bayes-frequentist synthesis. Can. J. Stat., 2001, 29: 5-17

[15] Hsu PL. Contribution to the Theory of 'Student's' t Test as Applied to the Problem of Two Samples, 1938, London, Statistical Research Memoirs, Department of Statistics, University College London

[16] Jeffreys H. Note on the Behrens–Fisher formula. Ann. Eugen., 1940, 10: 48-51

[17] Lehmann EL. Nonparametrics: Statistical Methods Based on Ranks, 1975, San Francisco, Holden-Day

[18] Lehmann EL. The Fisher, Neyman–Pearson theories of testing hypotheses: one theory or two. J. Am. Stat. Assoc., 1993, 88: 1242-1249

[19] Linnik JV. Statistical Problems with Nuisance Parameters (Scripta technica, Trans.), 1968, Providence, RI, American Mathematical Society

[20] Lu HZ, Jin H. A new prediction interval for binomial random variable based on inferential models. J. Stat. Plan. Inference, 2020, 205: 156-174

[21] Lu HZ, Jin H, Wang ZN, Chen C, Lu Y. Prior-free probabilistic interval estimation for binomial proportion. TEST, 2019, 28(2): 522-542

[22] Martin R. On an inferential model construction using generalized associations. J. Stat. Plan. Inference, 2018, 195: 105-115

[23] Martin R, Lingham RT. Prior-free probabilistic prediction of future observations. Technometrics, 2016, 58: 225-235

[24] Martin R, Liu CH. Inferential models: a framework for prior-free posterior probabilistic inference. J. Am. Stat. Assoc., 2013, 108: 301-313

[25] Martin R, Liu CH. Marginal inferential models: prior-free probabilistic inference on interest parameters. J. Am. Stat. Assoc., 2015, 110: 1621-1631

[26] Moserb K, Stevensg R. Homogenity of variance in the two-sample means test. Am. Stat., 1992, 1: 19-21

[27] Prokofyev VN, Shishkin AD. Successive classification of normal sets with unknown variances. Radio Engng. Electron. Phys., 1974, 19: 141-143

[28] Scheffe H. Practical solutions of the Behrens–Fisher problem. J. Am. Stat. Assoc., 1970, 12: 1501-1508

[29] Singh P, Saxena KK, Srivastava OP. Power comparisons of solutions to the Behrens–Fisher problem. Am. J. Math. Manag. Sci., 2002, 22: 233-250

[30] Tsui KH, Weerahandi S. Generalized p values in significance testing of hypotheses in the presence of nuisance parameters. J. Am. Stat. Assoc., 1989, 84: 602-607

[31] Wang YY. Probabilities of the type I errors of Welch tests for the Behrens–Fisher problem. J. Am. Stat. Assoc., 1971, 3: 605-608

[32] Wang ZN, Jin H, Lu HZ, Jin YL. An efficient test based on the Inferential Model for the non-inferiority of odds ratio in matched-pairs design. Stat. Methods Med. Res., 2018, 27: 2831-2841

[33] Welch BL. The significance of the difference between means when the population variances are unequal. Biometrika, 1938, 29: 350-362

[34] Welch BL. The generalization of Student’s problem when several populations are involved. Biometrika, 1947, 34: 28-35

[35] Welch BL. Appendix to Mrs. Aspin’s tables. Biometrika, 1949, 36: 293-296

Funding

China Postdoctoral Science Foundation(2021M690774)

National Natural Science Foundation of China(11731015)

Natural Science Foundation of Guangdong Province(2019A1515011717)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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