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BMC Medical Research Methodology

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Open Access 01-12-2021 | Research article

Bayesian Hodges-Lehmann tests for statistical equivalence in the two-sample setting: Power analysis, type I error rates and equivalence boundary selection in biomedical research

Author: Riko Kelter

Published in: BMC Medical Research Methodology | Issue 1/2021

Abstract

Background

Null hypothesis significance testing (NHST) is among the most frequently employed methods in the biomedical sciences. However, the problems of NHST and p-values have been discussed widely and various Bayesian alternatives have been proposed. Some proposals focus on equivalence testing, which aims at testing an interval hypothesis instead of a precise hypothesis. An interval hypothesis includes a small range of parameter values instead of a single null value and the idea goes back to Hodges and Lehmann. As researchers can always expect to observe some (although often negligibly small) effect size, interval hypotheses are more realistic for biomedical research. However, the selection of an equivalence region (the interval boundaries) often seems arbitrary and several Bayesian approaches to equivalence testing coexist.

Methods

A new proposal is made how to determine the equivalence region for Bayesian equivalence tests based on objective criteria like type I error rate and power. Existing approaches to Bayesian equivalence testing in the two-sample setting are discussed with a focus on the Bayes factor and the region of practical equivalence (ROPE). A simulation study derives the necessary results to make use of the new method in the two-sample setting, which is among the most frequently carried out procedures in biomedical research.

Results

Bayesian Hodges-Lehmann tests for statistical equivalence differ in their sensitivity to the prior modeling, power, and the associated type I error rates. The relationship between type I error rates, power and sample sizes for existing Bayesian equivalence tests is identified in the two-sample setting. Results allow to determine the equivalence region based on the new method by incorporating such objective criteria. Importantly, results show that not only can prior selection influence the type I error rate and power, but the relationship is even reverse for the Bayes factor and ROPE based equivalence tests.

Conclusion

Based on the results, researchers can select between the existing Bayesian Hodges-Lehmann tests for statistical equivalence and determine the equivalence region based on objective criteria, thus improving the reproducibility of biomedical research.

See Lehmann [15] for a balanced perspective which focusses on the appropriate frame of reference of the statistical inference.

In this paper Bayesian equivalence tests are investigated. Frequentist equivalence tests, superiority tests or non-inferiority tests are not studied, although the Bayesian versions of the latter two can be identified as slight modifications of Bayesian equivalence tests, which is clarified in the main text later.

However, the Cauchy distribution has fat tails so it could also be reasonable to use distributions with lighter tails as an alternative (for example, a normal distribution).

Van Ravenzwaaij et al. [44] also present examples of how to apply the Bayes factor for non-inferiority and superiority testing, for details see the original paper.

Results show that this is not the case.

Note that in this case, the support interval should not be given a Bayes factor interpretation: The interval includes parameter values which have been corroborated by the data, that is p(θ|x)/p(θ)>k. While the Savage-Dickey Bayes factor representation allows to interpret values inside the support interval as yielding BF₀₁>k, to reject the null based on Bayes factors one would logically require parameter values which yield a BF₁₀>k. Thus, in general, the support interval draws its legitimation from including values which have been corroborated by the data, and not by the fact that sometimes a Bayes factor interpretation can be given to them.

An exception is given by the OH model of Morey et al. [41], where only the widths r₀ and r₁ need to be specified. However, r₀ can be interpreted as the width of the equivalence region in the OH model.

In the frequentist paradigm a widespread equivalence testing procedure is the two one-sided tests (TOST) procedure described in Lakens et al. [38], see Appendix A.

Similar proposals for default ROPEs as β=0.05 for regression coefficients have been made for logistic and linear regression models. For a mathematical derivation see Kruschke ([36], p. 277).

Lakens et al. [38] underline that this is the weakest possible justification.

Other options next to the JZS model of Rouder et al. [28] which is used to compute the NOH model of Morey and Rouder [41] would be the Bayesian t-test models of Kruschke [58] or Kelter [32, 71].

The Bayes factor of Van Ravenzwaaij et al. [44] is not reported here because it is identical to the NOH model of Morey et al. [41], but it can be computed using the baymedr R package [45], see the provided replication script.

An exception is the OH model of Morey et al. [41], in which the associated Bayes factor is not consistent as discussed above.

The NOH Bayes factor \(BF_{01}^{NOH}\) for R=[−0.05,0.05] yields 9.27 under the wide C(0,1) prior in this case, indicating also moderate evidence for the interval null hypothesis, compare Jeffreys [60].

Altman DG. Statistics in medical journals: Some recent trends. Stat Med. 2000; 19(23):3275–89.PubMedCrossRef

Ioannidis JPA. Why Most Clinical Research Is Not Useful. PLoS Med. 2016; 13(6). https://doi.org/10.1371/journal.pmed.1002049.

Wasserstein RL, Schirm AL, Lazar NA. Moving to a World Beyond “p<0.05”. Am Stat. 2019; 73(sup1):1–19. https://doi.org/10.1080/00031305.2019.1583913.CrossRef

Wasserstein RL, Lazar NA. The ASA’s Statement on p-Values: Context, Process, and Purpose. Am Stat. 2016; 70(2):129–33. https://doi.org/10.1080/00031305.2016.1154108.CrossRef

Colquhoun D. An investigation of the false discovery rate and the misinterpretation of p-values. R Soc Open Sci. 2014; 1(3):140216. https://doi.org/10.1098/rsos.140216.PubMedPubMedCentralCrossRef

Colquhoun D. The problem with p-values. Aeon. 2016. https://doi.org/10.1016/S1369-7021(08)70254-2.

Edwards W, Lindman H, Savage LJ. Bayesian statistical inference for psychological research. Psychol Rev. 1963; 70(3):193–242. https://doi.org/10.1037/h0044139.CrossRef

Berger JO, Wolpert RL. The Likelihood Principle. Hayward: Institute of Mathematical Statistics; 1988, p. 208.

Kruschke JK, Liddell TM. The Bayesian New Statistics : Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective. Psychon Bull Rev. 2018; 25:178–206. https://doi.org/10.3758/s13423-016-1221-4.PubMedCrossRef

10.

Birnbaum A. On the Foundations of Statistical Inference (with discussion). J Am Stat Assoc. 1962; 57(298):269–306. https://doi.org/10.2307/2281640.CrossRef

11.

Pratt JW. Bayesian Interpretation of Standard Inference Statements. J R Stat Soc Ser B (Methodol). 1965; 27(2):169–92. https://doi.org/10.1111/j.2517-6161.1965.tb01486.x.

12.

Basu D. Statistical Information and Likelihood (with discussion). Sankhya Indian J Stat Ser A. 1975; 37(1):1–71. https://doi.org/10.1007/978-1-4612-3894-2.

13.

Wagenmakers E-J, Morey RD, Lee MD. Bayesian Benefits for the Pragmatic Researcher. Curr Dir Psychol Sci. 2016; 25(3):169–76. https://doi.org/10.1177/0963721416643289.CrossRef

14.

Morey RD, Hoekstra R, Rouder JN, Lee MD, Wagenmakers E-J. The fallacy of placing confidence in confidence intervals. Psychon Bull Rev. 2016; 23(1):103–23. https://doi.org/10.3758/s13423-015-0947-8.PubMedCrossRef

15.

Lehmann EL. The Fisher, Neyman-Pearson Theories of Testign Hypotheses: One Theory or Two?J Am Stat Assoc. 1993; 88(424):1242–9.CrossRef

16.

Morey RD, Romeijn JW, Rouder JN. The philosophy of Bayes factors and the quantification of statistical evidence. J Math Psychol. 2016; 72:6–18. https://doi.org/10.1016/j.jmp.2015.11.001.CrossRef

17.

Hendriksen A, de Heide R, Grünwald P. Optional stopping with bayes factors: A categorization and extension of folklore results, with an application to invariant situations. Bayesian Anal. 2020. https://doi.org/10.1214/20-ba1234.

18.

Rouder JN. Optional stopping: no problem for Bayesians. Psychon Bull Rev. 2014; 21(2):301–8. https://doi.org/10.3758/s13423-014-0595-4.PubMedCrossRef

19.

Ioannidis JPA. What Have We (Not) Learnt from Millions of Scientific Papers with p-Values?,. Am Stat. 2019; 73:20–5. https://doi.org/10.1080/00031305.2018.1447512.CrossRef

20.

Pratt JW. On the Foundations of Statistical Inference: Discussion. J Am Stat Assoc. 1962; 57(298):307–26.

21.

Dawid AP. Recent Developments in Statistics. In: Proceedings of the European Meeting of Statisticians. Grenoble: North-Holland Pub. Co.: 1977.

22.

Kruschke JK, Liddell TM. Bayesian data analysis for newcomers. Psychon Bull Rev. 2018; 25(1):155–77. https://doi.org/10.3758/s13423-017-1272-1.PubMedCrossRef

23.

Nuijten MB, Hartgerink CHJ, van Assen MALM, Epskamp S, Wicherts JM. The prevalence of statistical reporting errors in psychology (1985-2013). Behav Res Methods. 2016; 48(4):1205–26. https://doi.org/10.3758/s13428-015-0664-2.PubMedCrossRef

24.

Wetzels R, Matzke D, Lee MD, Rouder JN, Iverson GJ, Wagenmakers E-J. Statistical evidence in experimental psychology: An empirical comparison using 855 t tests. Perspect Psychol Sci. 2011; 6(3):291–8. https://doi.org/10.1177/1745691611406923.PubMedCrossRef

25.

Chen Z, Hu J, Zhang Z, Jiang S, Han S, Yan D, Zhuang R, Hu B, Zhang Z. Efficacy of hydroxychloroquine in patients with COVID-19: results of a randomized clinical trial. medRxiv. 2020; 7. https://doi.org/10.1101/2020.03.22.20040758.

26.

Gönen M, Johnson WO, Lu Y, Westfall PH. The Bayesian Two-Sample t Test. Am Stat. 2005; 59(3):252–7. https://doi.org/10.1198/000313005X55233.CrossRef

27.

Jeffreys H. Scientific Inference. Cambridge: Cambridge University Press; 1931.

28.

Rouder JN, Speckman PL, Sun D, Morey RD, Iverson G. Bayesian t tests for accepting and rejecting the null hypothesis. Psychon Bull Rev. 2009; 16(2):225–37. https://doi.org/10.3758/PBR.16.2.225.PubMedCrossRef

29.

Wetzels R, Raaijmakers JGW, Jakab E, Wagenmakers E-J. How to quantify support for and against the null hypothesis: A flexible WinBUGS implementation of a default Bayesian t test. Psychonomic Bulletin and Review. 2009; 16(4):752–60. https://doi.org/10.3758/PBR.16.4.752.PubMedCrossRef

30.

Wang M, Liu G. A Simple Two-Sample Bayesian t-Test for Hypothesis Testing. Am Stat. 2016; 70(2):195–201. https://doi.org/10.1080/00031305.2015.1093027.CrossRef

31.

Gronau QF, Ly A, Wagenmakers E-J. Informed Bayesian t -Tests. Am Stat. 2019; 00(0):1–7. https://doi.org/10.1080/00031305.2018.1562983.

32.

Kelter R. Bayest: An R Package for effect-size targeted Bayesian two-sample t-tests. J Open Res Softw. 2020; 8(14). https://doi.org/10.5334/jors.290.

33.

Kelter R. Bayesian and frequentist testing for differences between two groups with parametric and nonparametric two-sample tests. WIREs Comput Stat. 2020; 7. https://doi.org/10.1002/wics.1523.

34.

Cohen J. Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Hillsdale: Routledge; 1988.

35.

Berger JO, Brown LD, Wolpert RL. A Unified Conditional Frequentist and Bayesian Test for fixed and sequential Hypothesis Testing. Ann Stat. 1994; 22(4):1787–807. https://doi.org/10.1214/aos/1176348654.CrossRef

36.

Kruschke JK. Rejecting or Accepting Parameter Values in Bayesian Estimation. Adv Methods Pract Psychol Sci. 2018; 1(2):270–80. https://doi.org/10.1177/2515245918771304.CrossRef

37.

Lakens D. Equivalence Tests: A Practical Primer for t Tests, Correlations, and Meta-Analyses. Soc Psychol Personal Sci. 2017; 8(4):355–62. https://doi.org/10.1177/1948550617697177.PubMedPubMedCentralCrossRef

38.

Lakens D, Scheel AM, Isager PM. Equivalence Testing for Psychological Research: A Tutorial. Adv Methods Pract Psychol Sci. 2018; 1(2):259–69. https://doi.org/10.1177/2515245918770963.CrossRef

39.

Berger JO, Boukai B, Wang Y. Unified Frequentist and Bayesian Testing of a Precise Hypothesis. Stat Sci. 1997; 12(3):133–60.CrossRef

40.

Kelter R. Analysis of Bayesian posterior significance and effect size indices for the two-sample t-test to support reproducible medical research. BMC Med Res Methodol. 2020; 20(88). https://doi.org/10.1186/s12874-020-00968-2.

41.

Morey RD, Rouder JN. Bayes Factor Approaches for Testing Interval Null Hypotheses. Psychol Methods. 2011; 16(4):406–19. https://doi.org/10.1037/a0024377.PubMedCrossRef

42.

Hodges JL, Lehmann EL. Testing the Approximate Validity of Statistical Hypotheses. J R Stat Soc Ser B (Methodol). 1954; 16(2):261–8. https://doi.org/10.1111/j.2517-6161.1954.tb00169.x.

43.

Lindley DV. Decision Analysis and Bioequivalence Trials. Stat Sci. 1998; 13(2):136–41.CrossRef

44.

Van Ravenzwaaij D, Monden R, Tendeiro JN, Ioannidis JPA. Bayes factors for superiority, non-inferiority, and equivalence designs. BMC Med Res Methodol. 2019; 19(1):1–12. https://doi.org/10.1186/s12874-019-0699-7.CrossRef

45.

Linde M, van Ravenzwaaij D. baymedr: An R Package for the Calculation of Bayes Factors for Equivalence, Non-Inferiority, and Superiority Designs. arXiv preprint: arXiv:1910.11616v1. 2020.

46.

Makowski D, Ben-Shachar MS, Chen SHA, Lüdecke D. Indices of Effect Existence and Significance in the Bayesian Framework. Front Psychol. 2019; 10:2767. https://doi.org/10.3389/fpsyg.2019.02767.PubMedPubMedCentralCrossRef

47.

Makowski D, Ben-Shachar M, Lüdecke D. bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. J Open Source Softw. 2019; 4(40):1541. https://doi.org/10.21105/joss.01541.CrossRef

48.

Haaf JM, Ly A, Wagenmakers EJ. Retire significance, but still test hypotheses. Nature. 2019; 567(7749):461. https://doi.org/10.1038/d41586-019-00972-7.PubMedCrossRef

49.

Tendeiro JN, Kiers HAL. A Review of Issues About Null Hypothesis Bayesian Testing. Psychol Methods. 2019; 24(6):774–95. https://doi.org/10.1037/met0000221.PubMedCrossRef

50.

Robert CP. The expected demise of the Bayes factor. J Math Psychol. 2016; 72(2009):33–7. https://doi.org/10.1016/j.jmp.2015.08.002.CrossRef

51.

Stern JM. Significance tests, Belief Calculi, and Burden of Proof in legal and Scientific Discourse. Front Artif Intell Appl. 2003; 101:139–47.

52.

Wagenmakers E-J, Lodewyckx T, Kuriyal H, Grasman R. Bayesian hypothesis testing for psychologists: A tutorial on the Savage-Dickey method. Cogn Psychol. 2010; 60(3):158–89. https://doi.org/10.1016/j.cogpsych.2009.12.001.PubMedCrossRef

53.

Dickey JM, Lientz BP. The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain. Ann Math Stat. 1970; 41(1):214–26. https://doi.org/10.1214/AOMS/1177697203.CrossRef

54.

Verdinelli I, Wasserman L. Computing Bayes factors using a generalization of the Savage-Dickey density ratio. J Am Stat Assoc. 1995; 90(430):614–8. https://doi.org/10.1080/01621459.1995.10476554.CrossRef

55.

Gronau QF, Sarafoglou A, Matzke D, Ly A, Boehm U, Marsman M, Leslie DS, Forster JJ, Wagenmakers E-J, Steingroever H. A tutorial on bridge sampling. J Math Psychol. 2017; 81:80–97. https://doi.org/10.1016/j.jmp.2017.09.005.PubMedPubMedCentralCrossRef

56.

Gronau QF, Wagenmakers E-J, Heck DW, Matzke D. A Simple Method for Comparing Complex Models: Bayesian Model Comparison for Hierarchical Multinomial Processing Tree Models Using Warp-III Bridge Sampling. Psychometrika. 2019; 84(1):261–84. https://doi.org/10.1007/s11336-018-9648-3.PubMedCrossRef

57.

Liao JG, Midya V, Berg A. Connecting and Contrasting the Bayes Factor and a Modified ROPE Procedure for Testing Interval Null Hypotheses. Am Stat. 2020. https://doi.org/10.1080/00031305.2019.1701550.

58.

Kruschke JK. Bayesian estimation supersedes the t-test,. J Exp Psychol Gen. 2013; 142(2):573–603. https://doi.org/10.1037/a0029146.PubMedCrossRef

59.

Kelter R. Bayesian alternatives to null hypothesis significance testing in biomedical research: a non-technical introduction to Bayesian inference with JASP. BMC Med Res Methodol. 2020; 20(1). https://doi.org/10.1186/s12874-020-00980-6.

60.

Jeffreys H. Theory of Probability, 3rd ed. Oxford: Oxford University Press; 1961.

61.

Kass RE, Raftery AE. Bayes factors. J Am Stat Assoc. 1995; 90(430):773–95.CrossRef

62.

Goodman SN. Toward Evidence-Based Medical Statistics. 2: The Bayes Factor. Ann Intern Med. 1999; 130(12):1005. https://doi.org/10.7326/0003-4819-130-12-199906150-00019.PubMedCrossRef

63.

Lee MD, Wagenmakers E-J. Bayesian Cognitive Modeling : a Practical Course. Amsterdam: Cambridge University Press; 2013, p. 264.

64.

Held L, Ott M. On p-Values and Bayes Factors. Ann Rev Stat Appl. 2018; 5(1):393–419. https://doi.org/10.1146/annurev-statistics-031017-100307.CrossRef

65.

van Doorn J, van den Bergh D, Bohm U, Dablander F, Derks K, Draws T, Evans NJ, Gronau QF, Hinne M, Kucharský S, Ly A, Marsman M, Matzke D, Raj A, Sarafoglou A, Stefan A, Voelkel JG, Wagenmakers E-J. The JASP Guidelines for Conducting and Reporting a Bayesian Analysis. psyarxiv preprint. 2019. https://doi.org/10.31234/osf.io/yqxfr. https://psyarxiv.com/yqxfr.

66.

Westlake WJ. Symmetrical confidence intervals for bioequivalence trials. Biometrics. 1976; 32(4):741–4.PubMedCrossRef

67.

Kirkwood TBL. Bioequivalence Testing - A Need to Rethink. Biometrics. 1981; 37(3):589–94. https://doi.org/10.2307/2530573.CrossRef

68.

Carlin BP, Louis TA. Bayesian Methods for Data Analysis. Boca Raton: Chapman & Hall, CRC Press; 2009.

69.

Hobbs BP, Carlin BP. Practical Bayesian design and analysis for drug and device clinical trials. J Biopharm Stat. 2007; 18(1):54–80.CrossRef

70.

Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. J Pharmacokinet Biopharm. 1987; 15(6):657–80.PubMedCrossRef

71.

Kelter R. Bayest - Effect Size Targeted Bayesian Two-Sample t-Tests via Markov Chain Monte Carlo in Gaussian Mixture Models. Comprehensive R Archive Network. 2019. https://cran.r-project.org/web/packages/bayest/index.html.

72.

Kruschke JK. Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan, 2nd ed. Oxford: Academic Press; 2015, pp. 1–759. https://doi.org/10.1016/B978-0-12-405888-0.09999-2.

73.

Wagenmakers E-J, Gronau QF, Dablander F, Etz A. The Support Interval. Erkenntnis. 2020; 0123456789. https://doi.org/10.1007/s10670-019-00209-z.

74.

Zieba M, Tomczak JM, Lubicz M, Światek J. Boosted SVM for extracting rules from imbalanced data in application to prediction of the post-operative life expectancy in the lung cancer patients. Appl Soft Comput J. 2014; 14(PART A):99–108. https://doi.org/10.1016/j.asoc.2013.07.016.CrossRef

75.

U.S. Food and Drug Administration Center for Drug Evaluation and Research. Guidance for industry: Statistical approaches to establishing bioequivalence. 2001. Web archive: https://www.fda.gov/regulatory-information/search-fda-guidance-documents/statistical-approaches-establishing-bioequivalence. Accessed 01 Mar 2021.

76.

Senn S. Statistical issues in bioequivalance. Stat Med. 2001; 20(17-18):2785–99. https://doi.org/10.1002/sim.743.PubMedCrossRef

77.

Cook JA, Hislop JA, Adewuyi TE, Harrild KA, Altman DG, Ramsay DG, Fraser C, Buckley B, Fayers P, Harvey I, Briggs AH, Norrie JD, Fergusson D, Ford I, Vale LD. Assessing methods to specify the target difference for a randomised controlled trial: DELTA (Difference ELicitation in TriAls) review. Health Technol Assess. 2014; 18(28):1–172. https://doi.org/10.3310/hta18280.PubMedPubMedCentralCrossRef

78.

Cook JA, Julious SA, Sones W, Hampson LV, Hewitt C, Berlin JA, Ashby D, Emsley R, Fergusson DA, Walters SJ, Wilson ECF, MacLennan G, Stallard N, Rothwell JC, Bland M, Brown L, Ramsay CR, Cook A, Armstrong D, Altman D, Vale LD. DELTA 2 guidance on choosing the target difference and undertaking and reporting the sample size calculation for a randomised controlled trial. Trials. 2018; 19(1):1–6. https://doi.org/10.1136/bmj.k3750.CrossRef

79.

Jaeschke R, Singer J, Guyatt GH. Measurement of health status: Ascertaining the minimal clinically important difference. Control Clin Trials. 1989; 10(4):407–15. https://doi.org/10.1016/0197-2456(89)90005-6.PubMedCrossRef

80.

Weber R, Popova L. Testing equivalence in communication research: theory and application. Commun Methods Measures. 2012; 6(3):190–213. https://doi.org/10.1080/19312458.2012.703834.CrossRef

81.

Simonsohn U. Small Telescopes: Detectability and the Evaluation of Replication Results. Psychol Sci. 2015; 26(5):559–69. https://doi.org/10.1177/0956797614567341.PubMedCrossRef

82.

Ferguson CJ. An effect size primer: A guide for clinicians and researchers. Prof Psychol Res Pract. 2009; 40(5):532–8. https://doi.org/10.1037/a0015808.CrossRef

83.

Beribisky N, Davidson H, Cribbie RA. Exploring perceptions of meaningfulness in visual representations of bivariate relationships. PeerJ. 2019; 2019(5):6853. https://doi.org/10.7717/peerj.6853.CrossRef

84.

Rusticus SA, Eva KW. Defining equivalence in medical education evaluation and research: does a distribution-based approach work?Pract Assess Res Eval. 2016; 16(7):1–6. https://doi.org/10.1007/s10459-015-9633.

85.

Perugini M, Gallucci M, Costantini G. Safeguard Power as a Protection Against Imprecise Power Estimates,. Perspect Psychol Sci. 2014; 9(3):319–32. https://doi.org/10.1177/1745691614528519.PubMedCrossRef

86.

Kordsmeyer T, Penke L. The association of three indicators of developmental instability with mating success in humans. Evol Hum Behav. 2017; 38:704–13.CrossRef

87.

Maxwell SE, Lau MY, Howard GS. Is psychology suffering from a replication crisis?: What does ’failure to replicate’ really mean?,. Am Psychol. 2015; 70(6):487–98. https://doi.org/10.1037/a0039400.PubMedCrossRef

88.

Rogers JL, Howard KI, Vessey JT. Using significance tests to evaluate equivalence between two experimental groups. Psychol Bull. 1993; 113(3):553–65. https://doi.org/10.1037/0033-2909.113.3.553.PubMedCrossRef

89.

McElreath R, Smaldino PE. Replication, communication, and the population dynamics of scientific discovery. PLoS ONE. 2015; 10(8):1–16. https://doi.org/10.1371/journal.pone.0136088.CrossRef

90.

Morey RD, Rouder JN. BayesFactor: Computation of Bayes Factors for Common Designs. R package version 0.9.12-4.2. 2018.

91.

R Core Team. R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing; 2020. https://www.r-project.org/.

92.

Lindley DV. A Statistical Paradox. Biometrika. 1957; 44(1):187–92.CrossRef

93.

Schuirmann DJ. On hypothesis testing to determine if the mean of a normal distribution is contained in a known interval. Biometrics. 1981; 37(617).

94.

Anderson S, Hauck WW. A New Procedure for Testing Equivalence in Comparative Bioavailability and Other Clinical Trials. Commun Stat Theory Methods. 1983; 12(23):2663–92. https://doi.org/10.1080/03610928308828634.CrossRef

95.

Hauck WW, Anderson S. A new statistical procedure for testing equivalence in two-group comparative bioavailability trials. J Pharmacokinet Biopharm. 1984; 12(1):83–91. https://doi.org/10.1007/BF01063612.PubMedCrossRef

96.

Rocke DM. On testing for bioequivalence. Biometrics. 1984; 40:225–30.PubMedCrossRef

97.

Berger RL, Hsu JC, Berger RL, Hsu JC. Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets. Stat Sci. 1996; 11(4):283–302.CrossRef

98.

Meyners M. Equivalence tests - A review. Food Qual Prefer. 2012; 26:231–45. https://doi.org/10.1016/j.foodqual.2012.05.003.CrossRef

99.

Chow S-C, Liu J-P. Design and Analysis of Bioavailability and Bioequivalence Studies, 3rd ed. Boca Raton: Chapman & Hall/CRC Press; 2008.CrossRef

100.

Wellek S. Testing Statistical Hypotheses of Equivalence and Noninferiority: CRC Press; 2010, p. 415. https://doi.org/10.1201/ebk1439808184.

101.

Blackwelder WC. “Proving the null hypothesis” in clinical trials. Control Clin Trials. 1982; 3(4):345–53. https://doi.org/10.1016/0197-2456(82)90024-1.PubMedCrossRef

Title: Bayesian Hodges-Lehmann tests for statistical equivalence in the two-sample setting: Power analysis, type I error rates and equivalence boundary selection in biomedical research
Author: Riko Kelter
Publication date: 01-12-2021
Publisher: BioMed Central
Published in: BMC Medical Research Methodology / Issue 1/2021
Electronic ISSN: 1471-2288
DOI: https://doi.org/10.1186/s12874-021-01341-7

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Bayesian Hodges-Lehmann tests for statistical equivalence in the two-sample setting: Power analysis, type I error rates and equivalence boundary selection in biomedical research

Abstract

Background

Methods

Results

Conclusion

Keynote webinar | Spotlight on medication adherence

Springer Medicine

Abstract

Background

Methods

Results

Conclusion

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