1. Biostatistics Case Studies 2015 Youngju Pak, PhD. Biostatistician ypak@labiomed.org Session 2: Sample Size & Power for Inequality and Equivalence Studies II

2. What we have leaned in session 1? Info Needed for Study Size: Comparing Means (Inequality test) 1.Effect (clinically meaningful difference) 2.Subject variability 3.Type I error (1.96 for α=0.05; 2.58 for α=0.01) 4.Power (0.842 for 80% power; 1.645 for 95% power)  Free sample size calculations: www.stat.uiowa.edu/~rlenth/Power (1.96 + 0.842) 2 2SD 2 Δ 2 N =

3. Case Study Ophthalmology 2006; 113:70-76.

4. Abstract

5. Primary Outcome and Study Size Study Size - Page 72 bottom of column 1: Primary Outcome - Page 72 middle of column 1: Needs Consensus PI’s Gamble

6. Testing inequality vs. equivalence. Hypotheses for testing inequality: H a : | mean(treatment ) - mean (control ) | ≠ 0 H 0 : | mean(treatment ) - mean (control ) | = 0 Hypotheses for testing inequality: H a : δ 1 < mean(trt1) – mean (trt2) < δ 2 H 0 : mean(trt1) – mean (trt2) ≤ δ 1 Non-Inferiority or mean(trt1) – mean (trt2) ≥ δ 2

7. Graphical presentation of equivalence test With our regular t-tests, to conclude there is a substantial difference you must observe a difference large enough to conclude it is not due to sampling error To conclude there is not a substantial difference you must observe a difference small enough to reject that closeness is not due to sampling error from distributions centered on large effects

8. Non-Inferiority Study Usually a new treatment or regimen is compared with an accepted treatment or regimen or standard of care. The new treatment is assumed inferior to the standard and the study is designed to show overwhelming evidence that it is at least nearly as good, i.e., non- inferior. It usually has other advantages, e.g., oral vs. inj. A negative inferiority study fails to detect inferiority, but does not necessarily give evidence for non-inferiority. The accepted treatment is usually known to be efficacious already, but an added placebo group may also be used.

9. How to determine Sample Size? For IOP study, we have –H a : mean IOP change uf – mean IOP change f < 1.5 –H 0 : mean IOP change uf – mean IOP change f ≥ 1.5 thus, we are only interested in the upper limit of the difference  Non-inferiority  one-sided T-test Thus we reject the H 0 if Signal/ Noise < some clinical value. But N for a non-inferiority test require more complicated parameters such as the non- centrality parameter of the t-distribution (a Two One Sided T-test is usually used for the equivalence test ).

10. Let’s run a software from www.stat.uiowa.edu/~rlenth/Power Information you will need –Equivalence Margin Non-Inferiority Margin(NIM) =1.5 for the IOP study –Assumed mean difference in change of IOP between two groups -> usually zero difference assumed but it is assumed 0.5 for the IOP study –SD of changes of IOP = 3.5 –α (usually set to 2.5%) since the confidence level of the confidence interval is (100-2 x α) %

11. Sample size for IOP study

12. Three dimensional power curve for a non-inferiority test

13. How do we determine if the fixed method is non-inferior to the unfixed method? Regardless of study aim – to prove treatments equivalent or to prove them different - inference can be based on: Primary Outcome: IOP reduction D= D uf – D f, where D f = mean IOP reduction with fixed therapy Typical superiority/inferiority study: Compare to 0. Non-inferiority study: Compare to δ 2, a pre-specified margin of equivalence (1.5 here). = 95% CI for D(= D uf – D f ) = “true (population) values for D”

14. Typical Analysis: Inferiority or Superiority H 0 : D uf – D f = 0 H 1 : D uf – D f ≠ 0 Aim: H 1 → therapies differ α = 0.05 & N=2194 Power = 80% when Δ=1, SD=3.5 Fixed is inferior = 95% CI for D = “true (population) values for D” Fixed is superior 0 0 D u – D f [Not used in this paper] 0 No difference detected D u – D f

15. Typical Analysis: Inferiority Only H 0 : D u – D f ≤ 0 H 1 : D u – D f > 0 Aim: H 1 → fixed is inferior α = 0.025 & N=2194 Power = 80% for when Δ=1, SD=3.5 Fixed is inferior = 95% CI for D u – D f = “true (population) values for D” 0 0 D uf – Df [Not used in this paper] 0 Inferiority not detected D uf – D f ( α = 0.05 → N=2153 )

16. Non-Inferiority H 0 : D u – D f ≥ 1.5 H 1 : D u – D f < 1.5 Aim: H 1 → fixed is non-inferior α = 0.025 & N=2194 Power = 80% for When Δ= 0.5, NIM=1.5 Fixed is non-inferior = 95% CI for D u – D f = “true (population) values for D” 0 0 D uf – D f [As in this paper] 0 Non-Inferiority not detected D uf – D f 1.5 Fixed is inferior 1.5

17. Inferiority and Non-Inferiority Fixed is non-inferior = 95% CI for D u – D f = “true (population) values for D” 0 0 0 Neither is detected D uf – D f 1.5 Fixed is inferior 01.5 Fixed is “non-clinically” inferior D^ uf = 9.0 D^ f = 8.7 D^ = 0.3 95% CI = -0.1 to 0.7 Observed Results: Fixed is non-inferior 01.5

18. Conclusions: General “Negligibly inferior” would be a better term than non- inferior. All inference can be based on confidence intervals. Pre-specify the comparisons to be made. Cannot test for both non-inferiority and superiority. Power for only one or for multiple comparisons, e.g., non-inferiority and inferiority. Power can be different for different comparisons. Very careful consideration must be given to choice of margin of equivalence (1.5 here). You can be risky and gamble on what expected differences will be (0.5 here), but the study is worthless if others in the field would find your margin too large.

19. FDA Guidelines : http://www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformati on/guidances/ucm202140.pdf http://www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformati on/guidances/ucm202140.pdf Where, M 1 = Full effect of the active control compare with the test drug M 2 = NI Margin

20. Self-Quiz 1.Give an example in your specialty area for a superiority /inferiority study. Now modify it to an equivalence study. Now modify it to a non-inferiority study. 2.T or F: The main point about non-inferiority studies is that we are asking whether a treatment is as good or better vs. worse than another treatment, so it uses a one-sided test. 3.Power for a typical superiority test is the likelihood that you will declare treatment differences (p<0.05) if treatments really differ by some magnitude Δ. Explain what power means for a non-inferiority study. 4.T or F: Last-value-carried-forward is a good way to handle drop-outs in a non-inferiority study. Explain. continued

21. Self-Quiz 5.T or F: In a non-inferiority study, you should first test for non- inferiority with a confidence interval, and then use a t-test to test for superiority, but only if non-inferiority was established at the first step. 6.What is the meaning of the equivalence margin, and how do you determine it? continued

22. Self-Quiz 7.Suppose the primary outcome for a study is a serum inflammatory marker. If it’s assay is poor (low reproducibility), then it is more difficult to find treatment differences in a typical superiority/inferiority study than for a better assay, due to this noise. Would it be easier or more difficult to find non-inferiority with this assay, compared to a better assay? 8.Does the assumed treatment difference (0.5 here) for power calculations have the same meaning as the difference used for power calculations in a typical superiority/inferiority study?

23. Self-Quiz 1.Give an example in your specialty area for a superiority /inferiority study. Now modify it to an equivalence study. Now modify it to a non-inferiority study.

24. Self-Quiz 1.Answer Vaccine Testing: Superiority: New candidate vaccine vs. placebo Equivalence: Antigen potency between two manufacturing plants or lots. Non-Inferiority: New candidate vaccine vs. old one.

25. Self-Quiz 2.T or F: The main point about non-inferiority studies is that we are asking whether a treatment is as good or better vs. worse than another treatment, so it uses a one-sided test.

26. Self-Quiz 2.Answer False. That is a feature of these studies, but not their distinguishing feature. They and equivalence studies are used to try to prove sameness, as opposed to typical studies that try to prove differences.

27. Self-Quiz 3.Power for a typical superiority test is the likelihood that you will declare treatment differences (p<0.05) if treatments really differ by some magnitude Δ. Explain what power means for a non-inferiority study.

28. Self-Quiz 3.Answer Power for a non-inferiority study is the likelihood that you will declare one treatment (A) to be no worse than a pre- specified magnitude δ from the other treatment(B): accept Ha if treatments really differ by some Δ. Of course, Δ is less than δ, and is often 0.

29. Self-Quiz 4.T or F: Last-value-carried-forward is a good way to handle drop-outs in a non-inferiority study. Explain.

30. Self-Quiz 4.Answer False. LVCF biases results toward less of a difference if the projected difference is to be increased over time. This makes typical superiority studies conservative, but increases the chance of a falsely “proving” the aim. True if the projected difference is to be decreased over time.

31. Self-Quiz 5.T or F: In a non-inferiority study, you should first test for non-inferiority with a confidence interval, and then use a t-test to test for superiority, but only if non-inferiority was established at the first step.

32. Self-Quiz 5.Answer False. You must specify a-priori superiority, in order to have a legitimate claim of proving it (beyond a reasonable (5%) doubt). The stated sequential strategy will only allow you to claim an observed result, without a statement about it’s certainty.

33. Self-Quiz 6.What is the meaning of the equivalence margin, and how do you determine it?

34. Self-Quiz 6.Answer The equivalence margin is the maximum difference between treatments that is considered to be negligible or unimportant. It must be pre-specified in order to prove equivalence or non-inferiority to that degree, rather than just noting it as an observation. Thus, it is ideally determined by peer- agreement or FDA concurrence prior to starting the study.

35. Self-Quiz 7.Suppose the primary outcome for a study is a serum inflammatory marker. If it’s assay is poor (low reproducibility), then it is more difficult to find treatment differences in a typical superiority/inferiority study than for a better assay, due to this noise. Would it be easier or more difficult to find non-inferiority with this assay, compared to a better assay?

36. Self-Quiz 7.Answer It would still be more difficult to show the aim, non- inferiority here, since CIs will be wider, but there will be no bias due to it toward either treatment. Generally, a poorer study conduct is penalized in superiority studies and rewarded in non-inferiority studies, but that is not true for this type of poorer measurement error.

37. Self-Quiz 10.Does the assumed treatment difference (0.5 for the IOP study) for power calculations have the same meaning as the difference to be detected for a power calculation in a typical superiority/inferiority study?

38. Self-Quiz 10.Answer No. Here, it is our best estimate of true treatment differences. For superiority studies, the difference is ideally the minimal difference that is “clinically relevant”, not the expected difference, closer in meaning to the equivalence margin here. In practice, it is the smallest difference that logistics, money, time, and effort will allow us to detect with specified certainty.