非劣性試驗的目的是透過比較新藥和標準用藥的療效，證明新藥的療效達到標準用藥療效的某個水準以上，藉此說明新藥是具有療效的。試驗中，須定義一個新藥和標準用藥療效間可容忍的差距，稱為非劣性試驗的臨界值，該值通常是參考標準用藥和安慰劑試驗的歷史資料所得。允許新藥療效比標準用藥療效來的差一點，主要是因為新藥具有標準用藥所沒有的優點，例如：成本低、負作用少和容易使用等。將非劣性試驗的臨界值定義為新藥和標準用藥療效之差是常見且受歡迎的方式。基於這個定義，已有統計方法用以比較多組新藥是否非劣於標準用藥療效，同時驗證試驗是否具有檢測靈敏度。然而這個方法要求各組資料須滿足變異數一致性的假設。本研究首先討論當各組資料不滿足變異數一致性假設時，原本方法是否依然能夠有效地控制族系誤差率。針對各組資料不滿足變異數一致性假設下，我們提出兩個統計檢定方法，並且利用模擬的方式說明所提出的方法可以有效地控制族系誤差率。很多非劣性臨床試驗建構在比較多組新藥和多組標準用藥，同時驗證試驗是否具有檢測靈敏度。本研究也對此種設計提出一個新的檢定程序，並且提供各組試驗所需的樣本數計算，以使檢定達到所期望的檢定力。最後，本研究會以臨床試驗的例子，對所提出的檢定程序做說明。

The objective of a non-inferiority (NI) trial is to affirm the efficacy of a new treatment compared with a standard treatment by verifying that the new treatment maintains a considerable portion of the treatment effect of the standard. The marginal loss of efficacy (NI margin) of adopting the new treatment has to be justified by its other benefits, such as the alleviation of side effects, the lowering of costs, and the simplification of intricate treatment regimens. A popular approach is to express the NI margin in terms of the efficacy difference between the new treatment and the standard treatment. Based on this approach, statistical procedures that simultaneously conduct NI tests of several new treatments have been proposed. However, these procedures were formulated on the assumption of equal variances. In this thesis, the first objective is to discuss the undesirable effect on the familywise error rate using these procedures under the situation of heterogeneous variances. Two new procedures are proposed to alleviate the problem. A simulation study of the familywise error rate and power is conducted to compare different procedures. It is found that the proposed procedure that used a plug-in method is superior. The second objective of this thesis is to extend previous statistical methods for NI trials that only focus on testing of treatments with a single standard treatment. In practice, NI trials sometimes compare new treatments with multiple standard treatments and the appropriate testing procedure is derived. Further, the power function, together with the strategy to determine the optimal sample size, are discussed. Clinical trial examples are presented to illustrate our proposed procedure.

Barr RG, Bourbeau J, Camargo CA, Ram FS. (2006). Tiotropium for stable chronic obstructive pulmonary disease: a meta-analysis. Thorax, 61:854-862.

Bauwens O, Ninane V, Van de Maele B, Firth R, Dong F, Owen R, Higgins M. (2009). 24-hour bronchodilator efficacy of single doses of indacaterol in subjects with COPD: comparison with placebo and formoterol. Current Medical Research and Opinion, 25:463-470.

Beier J, Chanez P, Martinot JB, Schreurs AJM, Tkacova R, Bao W, Jack D, Higgins M. (2007). Safety, tolerability and effecacy of indacaterol, a novel once-daily beta-agonist, in patients with COPD: a 28-day randomised, placebo controlled clinical trial. Pulmonary Pharmacology and Therapeutics, 20:740-749.

Boyd MA, Cooper DA. (2012). Optimisation of HIV care and service delivery: doing more with less. Lancet, 380:1860-1866.

Burger HU, Beyer U, Abt M. (2011). Issues in the assessment of non-inferiority: perspectives drawn from case studies. Pharmaceutical Statistics, 10:433-439.

Casazza G, Solbiati M. (2013). Can we trust equivalence and non-inferiority trials? Internal and Emergency Medicine, 8:439-442.

Celli B, ZuWallack R, Wang S, Kesten S. (2003). Improvement in resting inspiratory capacity and hyperinflation with tiotropium in COPD patients with increased static lung volumes. Chest, 124:1743-1748.

Decramer M, Celli B, Kesten S, Lystig T, Mehra S, Tashkin DP. (2009). Effect of tiotropium on outcomes in patients with moderate chronic obstructive pulmonary disease (UPLIFT): a prespecified subgroup analysis of a randomised controlled trial. Lancet, 374:1171-1178.

Donohue JF, Fogarty C, Lotvall J, Mahler DA, Worth H, Yorgancioglu A, Iqbal A, Swales J, Owen R, Higgins M, Kramer B. (2010). Once-daily bronchodilators for chronic obstructive pulmonary disease: indacaterol versus tiotropium. American Journal of Respiratory and Critical Care Medicine, 182:155-162.

Dudoit S and van der Lann MJ. (2008). Multiple testing procedures with applications to genomics. New York: Springer, 2008.

Dunnett CW. (1955). A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association, 50:1096-1121.

European Medicines Agency. (2006). Guideline on the Choice of the Non-inferiority Margin. London: EMA.
http://www.ema.europa.eu/docs/en_GB/document_library/Scientific_guideline/2009/09/WC500003636.pdf

Fleming TR, Odem-Davis K, Rothmann MD, Shen YL. (2011). Some essential considerations in the design and conduct of non-inferiority trials. Clinical Trial, 8:432-439.

Gallwitz B, Rosenstock J, Rauch T, Bhattacharya S, Patel S, von Eynatten M, Dugi KA, and Woerle HJ. (2012). 2-year efficacy and safety of linagliptin compared with glimepiride in patients with type 2 diabetes inadequately controlled on metformin: a randomised, double-blind, non-inferiority trial. Lancet, 380:475-483.

Hasler M. (2012). Multiple comparisons to both a negative and a positive control. Pharmaceutical Statistics, 11:74-81.

Hasler M, Hothorn LA. (2008). Multiple contrast tests in the presence of heteroscedasticity. Biometrical Journal, 50:793-800.

Hasler M, Vonk R, Hothorn LA. (2008). Assessing non-inferiority of a new treatment in a three-arm trial in the presence of heteroscedasticity. Statistics in Medicine, 27:490-503.

Hida E, Tango T. (2011). On the three-arm non-inferiority trial including a placebo with a prespecified margin. Statistics in Medicine, 30:224-231.

Hochberg Y, Tamhane AC. (1987). Multiple Comparison Procedures. Wiley: New York.

Homma Y, Yamaguchi O. (2009). A randomized, double-blind, placebo- and propiverine-controlled trial of the novel antimuscarinic agent imidafenacin in Japanese patients with overactive bladder. International Journal of Urology, 16:499-506.

Houghton DM, Langley SJ, Singh SD, Holden J, Monici Preti AP, Acerbi D, Poli G, Woodcock A. (2004). Comparison of bronchoprotective and bronchodilator effects of a single dose of formoterol delivered by hypdrofluoroalkane and chlorofluorocarbon aerosols and dry powder in a double blind, placebo-controlled, crossover study. British Journal of Clinical Pharmacology, 58:359-366.

Kearney PM, Whelton M, Reynolds K, Muntner P, Whelton PK, He J. (2005) Global burden of hypertension: analysis of worldwide data. Lancet, 365:217-223.

Koch A, Rohmel J. (2004). Hypothesis testing in the "gold standard" design for proving the efficacy of an experimental treatment relative to placebo and a reference. Journal of Biopharmaceutical Statistics, 14:315-325.

Kwong KS, Cheung SH, Hayter AJ. (2013). Step-up procedures for non-inferiority tests with multiple experimental treatments. Statistical Methods in Medical Research, Online 0:1-13.

Kwong KS, Cheung SH, Hayter AJ, Wen MJ. (2012). Extension of three-arm non-inferiority studies to trials with multiple new treatments. Statistics in Medicine, 31:2833-2843.

Pigeot I, Schafer J, Rohmel J, Hauschke D. (2003). Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo. Statistics in Medicine, 22:883-899.

Rohmel J, Pigeot I. (2010). A comparison of multiple testing procedures for the gold standard non-inferiority trial. Journal of Biopharmaceutical Statistics, 20:911-926.

Rothmann MD, Wiens BL, Chan ISF. (2012). Design and Analysis of Non-Inferiority Trials. Chapman and Hall/CRC Biostatistics Series, Taylor and Francis Group.

Russell-Jones D, Vaag A, Schmitz O, Sethi BK, Lalic N, Antic S, Zdravkovic M, Ravn GM, Simo R. (2009). Liraglutide vs insulin glargine and placebo in combination with metformin and sulfonylurea therapy in type 2 diabetes mellitus (LEAD-5 met+SU): a randomised controlled trial. Diabetologia, 52:2046-2055.

Sica D, White WB, Weber MA, Bakris GL, Perez A, Cao C, Handley A, Kupfer S. (2011). Comparison of the novel angiotensin II receptor blocker azilsartan medoxomil vs valsartan by ambulatory blood pressure monitoring. The Journal of Clinical Hypertension, 13:467-472.

Stucke K, Kieser M. (2012). A general approach for sample size calculation for the three-arm `gold standard' non-inferiority design. Statistics in Medicine, 31:3579-3596.

Tamhane AC. (1979). A comparison of procedures for multiple comparisons of means with unequal variances. Journal of the American Statistical Association, 74:471-480.

Tashkin DP, Celli B, Senn S, Burkhart D, Kesten S, Menjoge S, Decramer M. (2008). A 4-year trial of tiotropium in chronic obstructive pulmonary disease. The New England Journal of Medicine, 359:1543-1554.

United States Food and Drug Administration. (2010). Guidance for Industry Non-inferiority Clinical Trials - Draft Guidance}. Rockville: FDA.
http://www.fda.gov/downloads/Drugs/GuidanceComplianceRegulatoryInformation/Guidances/UCM202140.pdf.

Ury HK, Wiggins AD. (1971). Large sample and other multiple comparisons among means. British Journal of Mathematical and Statistical Psychology, 24:174-194.

Vieta E, Cruz N. (2012). Head to head comparisons as an alternative to placebo-controlled trials. European Neuropsychopharmacology, 22:800-803.

Vilsboll T, Zdravkovic M, Le-Thi T, Krarup T, Schmitz O, Courreges JP, Verhoeven R, Buganova I, Madsbad S. (2007). Liraglutide, a long-acting human glucagon-like peptide-1 analog, given as monotherapy significantly improves glycemic control and lowers body weight without risk of hypoglycemia in patients with type 2 diabetes. Diabetes Care, 30:1608-1610.

Vogelmeier C, Ramos-Barbon D, Jack D, Piggott S, Owen R, Higgins M, Kramer B. INTIME study investigators. (2010). Indacaterol provides 24-hour bronchodilation in COPD: a placebo-controlled blinded comparison with tiotropium. Respiratory Research, 11:135-142.

White WB, Weber MA, Sica D, Bakris GL, Perez A, Cao C, Kupfer S. (2011). Effects of the angiotensin receptor blocker azilsartan medoxomil versus olmesartan and valsartan on ambulatory and clinic blood pressure in patients with stages 1 and stage 2 hypertension. Hypertension, 57:413-420.