近年來，由於多項創新藥專利到期，其他藥廠開始生產生物相似性產品。然而，評價創新藥與生物相似藥之生物相似性的統計方法尚不明確。此外，已知批次間變異性在評估生物相似性時扮演著重要的角色，但先前的研究很少提及批次間變異性的影響。因此，在這項研究中，我們假設療效反應可以用巢狀隨機效應模型來解釋，並開發了一個基於雙邊容忍區間的假設檢定。接下來，透過模擬研究調查本方法在不同參數狀況下的統計特性。最後，我們使用一個真實的例子來展示所提出的方法。

In recent years, due to the expiration of several innovative drug patents, other sponsors begin to manufacture the biosimilar products. However, the statistical rule for evaluating the biosimilarity between innovative biologics and biosimilar is not clear yet. Additionally, the between-batches variability is known playing a crucial role, and the previous studies seldom mentioned the between-batches variability. Therefore, in this study, we assume that the therapeutic response can be explained by a nested random effect model and develop a two-sided tolerance interval-based hypothesis test. The statistical properties are investigated by simulation studies with various parameter components. Finally, we use a real example to demonstrate the proposed approach.

Contents
1	Introduction                                1
2	Statistical inference                       4
2.1	Statistical model and hypothesis testing .. 4
2.2	Sample size determination ................. 8
3	Real example                               10
4	Simulation and numerical study             13
5	Conclusion and Discussion                  22
6	References                                 24
Appendix                                           36

Contents of Figures 
Figure 1. Histogram of the empirical coverage probabilities ....................... 14 
Figure 2. Histogram of the empirical powers ....................................... 15 
Figure 3. Histogram of the differences between the empirical power and 
asymptotical power ................................................................ 16 
Figure 4. Sample size per group with σB=σδ=σε=0.006, α=0.05, power=80%, 
γ=0.8, 0.85, 0.9, m_d is μD, and sd_a is σA ....................................... 17 
Figure 5. Sample size per group with μD=0, σB=σε=0.006, α=0.05, power=80%, 
γ=0.8, 0.85,0.9, sd_a is σA, and sd_d is σδ ....................................... 20 
Figure 6. Histogram of the empirical coverage probabilities between 0.97 
and 0.985 ......................................................................... 21

Contents of Tables
Table 1. Sample size per group, empirical coverage probability, asymptotical power, 
and empirical power with σδ=σB=σε=0.006, α=0.05, and power=80% for the minimum total 
sample size strategy .............................................................. 26 
Table 2. Sample size per group, empirical coverage probability, asymptotical power, 
and empirical power with σδ=σB=σε=0.006, α=0.05, and desired power=80% ............ 29 
Table 3. Sample size per group, empirical coverage probability, asymptotical power,
and empirical power with σδ=σB=σε=0.006, α=0.05, and power=80% for the minimum total
cost strategy ..................................................................... 30 
Table 4. Ratio of sample size and cost for per group, with σδ=σB=σε=0.006, α=0.05, 
and power=80% ..................................................................... 33

1.	U.S. Food and Drug Administration. (2022). Overview for Health Care Professionals. Retrieved June 18, 2023, from https://www.fda.gov/drugs/biosimilars/overview-health-care-professionals.
2.	U.S. Food and Drug Administration. (2023). Biosimilar Product Information. Retrieved June 18, 2023, from https://www.fda.gov/drugs/biosimilars/biosimilar-product-information.
3.	Zhang, A., Tzeng, J. Y., & Chow, S. C. (2013). Statistical considerations in biosimilar assessment using biosimilarity index. Journal of bioequivalence & bioavailability, 5(5), 209. https://www.walshmedicalmedia.com/open-access/statistical-considerations-in-biosimilar-assessment-using-biosimilarity-index-jbb.10000160.pdf.
4.	European Medicines Agency. (2005). GUIDELINE ON SIMILAR BIOLOGICAL MEDICINAL PRODUCTS. Retrieved June 18, 2023, from https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-similar-biological-medicinal-products-first-version_en.pdf.
5.	行政院衛生署藥政處 (2007). 行政院公報資訊網-每日即時刊登行政院及所屬各機關公布之法令規章等資訊 Retrieved June 18, 2023, from https://gazette.nat.gov.tw/EG_FileManager/eguploadpub/eg013053/ch08/type3/gov70/num21/Eg.htm. 
6.	World Health Organization. (2009). GUIDELINES ON EVALUATION OF SIMILAR BIOTHERAPEUTIC PRODUCTS (SBPs). Retrieved June 18, 2023, from https://www.nmra.gov.lk/images/PDF/guideline/who_guideline_01.pdf.
7.	U.S. Food and Drug Administration. (2016). FDA Withdraws Draft Guidance for Industry: Statistical Approaches to Evaluate Analytical Similarity Retrieved June 18, 2023, from https://www.fda.gov/drugs/guidance-compliance-regulatory-information/implementation-biologics-price-competition-and-innovation-act-2009.
8.	U.S. Food and Drug Administration. (2018) FDA Withdraws Draft Guidance for Industry: Statistical Approaches to Evaluate Analytical Similarity. Retrieved June 18, 2023, from  https://www.fda.gov/drugs/drug-safety-and-availability/fda-withdraws-draft-guidance-industry-statistical-approaches-evaluate-analytical-similarity.
9.	Oliva, A., & Llabrés, M. (2021). New Quality-Range-Setting Method Based on Between-and Within-Batch Variability for Biosimilarity Assessment. Pharmaceuticals, 14(6), 527. https://doi.org/10.3390/ph14060527. 
10.	Chow, S. C., Hsieh, T. C., Chi, E., & Yang, J. (2009). A comparison of moment-based and probability-based criteria for assessment of follow-on biologics. Journal of Biopharmaceutical Statistics, 20(1), 31-45. https://doi.org/10.1080/10543400903280308.
11.	Hsieh, T. C., Chow, S. C., Liu, J. P., Hsiao, C. F., & Chi, E. (2009). Statistical test for evaluation of biosimilarity in variability of follow-on biologics. Journal of Biopharmaceutical Statistics, 20(1), 75-89. https://doi.org/10.1080/10543400903367097. 
12.	Liao, J. J., & Darken, P. F. (2013). Comparability of critical quality attributes for establishing biosimilarity. Statistics in medicine, 32(3), 462-469. https://doi.org/10.1002/sim.5564.
13.	Tsou, H. H., Chang, W. J., Hwang, W. S., & Lai, Y. H. (2013). A consistency approach for evaluation of biosimilar products. Journal of Biopharmaceutical Statistics, 23(5), 1054-1066. https://doi.org/10.1080/10543406.2013.813518.
14.	Zhang, N., Yang, J., Chow, S. C., & Chi, E. (2014). Nonparametric tests for evaluation of biosimilarity in variability of follow-on biologics. Journal of biopharmaceutical statistics, 24(6), 1239-1253. https://doi.org/10.1080/10543406.2014.941991. 
15.	Chen, C. T., Tsou, H. H., Hsiao, C. F., Lai, Y. H., Chang, W. J., & Liu, J. T. (2017). A tolerance interval approach to assessing the biosimilarity of follow-on biologics. Statistics in Biopharmaceutical Research, 9(3), 286-292. https://doi.org/10.1080/19466315.2017.1323669.
16.	Chiang, C., Chen, C. T., & Hsiao, C. F. (2021). Use of a two‐sided tolerance interval in the design and evaluation of biosimilarity in clinical studies. Pharmaceutical Statistics, 20(1), 175-184. https://doi.org/10.1002/pst.2065.
17.	Hoffman, D., & Kringle, R. (2005). Two-sided tolerance intervals for balanced and unbalanced random effects models. Journal of biopharmaceutical statistics, 15(2), 283-293. https://doi.org/10.1081/BIP-200048826.