Optimal Sample Size Determinations for the Heteroscedastic Two One-Sided Tests of Mean Equivalence

Abstract

Equivalence assessment is becoming an increasingly important topic in many application areas including behavioral and social sciences research. Although there exist more powerful tests, the two one-sided tests (TOST) procedure is a technically transparent and widely accepted method for establishing statistical equivalence. Alternatively, a direct extension of Welch’s solution for the Behrens–Fisher problem is preferred in equivalence testing of means when the homogeneity of variance assumption is violated. For advance planning of equivalence studies, this article describes both exact and nearly exact power functions of the heteroscedastic TOST procedure and develops useful approaches to optimal sample size determinations under various allocation and cost considerations. Detailed numerical illustrations and simulation studies are presented to demonstrate the distinct features of the suggested techniques and the potential deficiency of existing method. Moreover, computer programs are provided to facilitate the implementation of the described sample size procedures. The proposed formulas and algorithms are recommended over the current results for their technical transparency, overall performance, and diverse utility.

Keywords

equivalence heteroscedasticity power sample size Welch’s statistic

Get full access to this article

View all access options for this article.

References

Allan

T. A.

Cribbie

R. A.

(2013). Evaluating the equivalence of, or difference between, psychological treatments: An exploration of recent intervention studies. Canadian Journal of Behavioral Science, 45, 320–328.

Berger

R. L.

Hsu

J. C.

(1996). Bioequivalence trials, intersection-union tests and equivalence confidence sets (with discussion). Statistical Science, 11, 283–319.

Blackwelder

W. C.

(1982). “Proving the null hypothesis” in clinical trails. Controlled Clinical Trials, 3, 345–353.

Bristol

D. R.

(1993). Probabilities and sample sizes for the two one-sided tests procedure. Communications in Statistics-Theory & Methods, 22, 1953–1961.

Cannon

D. S.

Bell

W. E.

Fowler

D. R.

Penk

W. E.

Finkelstein

A. S.

(1990). MMPI differences between alcoholics and drug abusers: Effect of age and race. Psychological Assessment: A Journal of Consulting and Clinical Psychology, 2, 51–55.

Chow

S. C.

Shao

Wang

(2002). A note on sample size calculation for mean comparisons based on noncentral t-statistics. Journal of Biopharmaceutical Statistics, 12, 441–456.

Chow

S. C.

Wang

(2001). On sample size calculation in bioequivalence trials. Journal of Pharmacokinetics and Pharmacodynamics, 28, 155–169.

Cribbie

R. A.

Arpin-Cribbie

C. A.

(2009). Evaluating clinical significance through equivalence testing: Extending the normative comparisons approach. Psychotherapy Research, 19, 677–686.

Cribbie

R. A.

Gruman

Arpin-Cribbie

(2004). Recommendations for applying tests of equivalence. Journal of Clinical Psychology, 60, 1–10.

10.

Dannenberg

Dette

Munk

(1994). An extension of Welch’s approximate t-solution to comparative bioequivalence trials. Biometrika, 81, 91–101.

11.

Dette

Munk

(1997). Optimum allocation of treatments for Welch’s test in equivalence assessment. Biometrics, 53, 1143–1150.

12.

Diletti

Hauschke

Steinijans

V. W.

(1991). Sample size determination for bioequivalence assessment by means of confidence intervals. International Journal of Clinical Pharmacology, Therapy and Toxicology, 29, 1–8.

13.

Golinski

Cribbie

R. A.

(2009). The expanding role of quantitative methodologists in advancing psychology. Canadian Psychology, 50, 83–90.

14.

Grissom

R. J.

(2000). Heterogeneity of variance in clinical data. Journal of Consulting and Clinical Psychology, 68, 155–165.

15.

Gruman

Cribbie

R. A.

Arpin-Cribbie

C. A.

(2007). The effects of heteroscedasticity on tests of equivalence. Journal of Modern Applied Statistical Methods, 6, 133–140.

16.

Guo

J. H.

Chen

H. J.

Luh

W. M.

(2011). Sample size planning with the cost constraint for testing superiority and equivalence of two independent groups. British Journal of Mathematical and Statistical Psychology, 64, 439–461.

17.

Heilbrun

L. K.

McGee

D. L.

(1985). Sample size determination for the comparison of normal means when one sample size is fixed. Computational Statistics & Data Analysis, 3, 99–102.

18.

Jan

S. L.

Shieh

(2011). Optimal sample sizes for Welch’s test under various allocation and cost considerations. Behavior Research Methods, 43, 1014–1022.

19.

Liu

J. P.

Chow

S. C.

(1992). Sample size determination for the two one-sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 20, 101–104.

20.

Meyners

(2012). Equivalence tests—A review. Food Quality and Preference, 26, 231–245.

21.

Parkhurst

D. F.

(2001). Statistical significance tests: Equivalence and reverse tests should reduce misinterpretation. BioScience, 51, 1051–1057.

22.

Peasarin

Salmaso

Carrozzo

Arboretti

(2016). Union-intersection permutation solution for two-sample equivalence testing. Statistics and Computing, 26, 693–701.

23.

Perlman

M. D.

(1999). The emperor’s new tests. Statistical Science, 14, 355–381.

24.

Phillips

K. F.

(1990). Power of the two one-sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 18, 137–144.

25.

Rogers

J. L.

Howard

K. I.

Vessey

J. T.

(1993). Using significance tests to evaluate equivalence between two experimental groups. Psychological Bulletin, 113, 553–565.

26.

Rosopa

P. J.

Schaffer

M. M.

Schroeder

A. N.

(2013). Managing heteroscedasticity in general linear models. Psychological Methods, 18, 335–351.

27.

Ruscio

Roche

(2012). Variance heterogeneity in published psychological research: A review and a new index. Methodology, 8, 1–11.

28.

SAS Institute. (2013). SAS/IML user’s guide, version 9.3. Cary, NC: Author.

29.

Schuirmann

D. J.

(1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15, 657–680.

30.

Schuirmann

D. L.

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

31.

Seaman

M. A.

Serlin

R. C.

(1998). Equivalence confidence intervals for two-group comparisons of means. Psychological Methods, 3, 403–411.

32.

Stegner

B. L.

Bostrom

A. G.

Greenfield

T. K.

(1996). Equivalence testing for use in psychological and service research: An introduction with examples. Evaluation and Program Planning, 19, 193–198.

33.

Steiner

P. M.

Wong

V. C.

(2016). Assessing correspondence between experimental and non-experimental results in within-study-comparisons. Unpublished manuscript.

34.

Wang

Chow

S. C.

(2002). On statistical power for average bioequivalence testing under replicated crossover designs. Journal of Biopharmaceutical Statistics, 12, 295–309.

35.

Welch

B. L.

(1938). The significance of the difference between two means when the population variances are unequal. Biometrika, 29, 350–362.

36.

Westlake

W. J.

(1981). Response to T.B.L. Kirkwood: Bioequivalence testing–a need to rethink. Biometrics, 3, 589–594.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.11 MB