1. Lecture 20: Study Design and Sample Size Estimation in TTE Studies

2. Study Design and Sample Size Ideally we are involved in a study from the beginning As statisticians (an epidemiologist) part of our role is to ensure the study is designed to address the primary hypothesis under consideration – Ensure proper study design – Ensure appropriate sample size Power and significance level

3. Study Design We’ve already talked (informally) about study design… Our job is to ask appropriate questions – What is the study population? – Primary hypothesis? – Is the event recurrent? – Competing risks? – …

4. Components of Sample Size General Considerations in any hypothesis test 1.Hypothesis to be tested 2.Test statistic 3.Size of the test (i.e.  ) 4.Desired power 5.Assumed effect size

5. Specific for Time To Event Data Additional considerations 1.Probability of an event during the study 2.Expected rate of loss (i.e. censoring) 3.Enrollment rate 4.Competing risks

6. Basic Considerations Ensure precise specification of the hypothesis Select a significance level and power appropriate for the study What is the test statistic that will be used to test the hypothesis? – Many statistics have well known properties upon which we base our calculations – Deviations from assumptions complicate these calculations…

7. Null Hypothesis to be Tested Recall for log-rank test Where HR is assumed to be proportional for all t

8. Test-Statistic Can use either – Log-rank test (or some variation there of) – HR estimated from Cox PHM To test the null hypothesis of no difference consider the log of the hazard ratio – log(HR) ~ N when comparing two groups

9. Significance Level The probability that a statistical test will reject H 0 when H 0 is actually true – Significance =  Interpretation: For a given value under the null hypothesis, we’re going to reject the null in favor of the alternative in error (  )100% of the time.

10. Power The probability that a statistical test will reject H 0 when H 0 is false – Power=1-  Interpretation: For a given value under the alternative hypothesis, we’re going to correctly reject the null in favor of the alternative (1-  )100% of the time.

11. Choice of  and 1-  Generally choose  = 0.05 Other values can be used but should be justified – Say choose  = 0.01, we want very string evidence of a treatment effect – Alternatively  = 0.10 or 0.20 might be chosen for something like a pilot study Power generally set to 80-99%

12. Effect Size Generally assume proportional hazards Hazard ratio: – Null: HR = 1 – HR < 1 implies longer survival in treatment B – HR > 1 implies longer survival in treatment A Base sample size calculation on having sufficient power to detect minimum clinically important effect – For example, maybe a 30% reduction in incidence for Trt A vs. Trt B (i.e. HR = 0.7) is clinically meaningful

13. Minimum Scientifically Important Difference Definition: the smallest difference which would mandate, in the absence of serious side effects and/or excessive cost, a change in scientific practice/ understanding. This is a scientific question, not a statistical question.

14. Significance, Power, and Sample Size Sample size impacted not only by significance level, power, and research question but also practical considerations – Number of available patients – Study duration – Cost

15. Basic Sample Size Formula Generic formula for total # subjects/group Under restrictions

16. Design Consideration in TTE Studies How will sample size be collected? – Enroll fixed number of patients and follow for some specified period of time – Continue study until a sufficient number of events have been observed Other considerations – Expected event rate in each group – How much loss to follow-up expected

17. Study Type Type I study – All subjects experience an event by the end of the study Type II study – A study terminates at fixed time T resulting in administratively censored subjects. Administrative censoring – Right censoring that occurs of subject fails to experience the event prior to the end of the study Loss to follow-up – Occurs when a subject fails to complete the study for reasons unrelated to the event of interest

18. Sample Size in Time to Event Data For many power calculations we specify significance level, power, variability, and our minimum clinically relevant difference to get our total N In TTE studies, it is easier to specify the number of events we need to observe rather than the total number of people

19. Simplest Case for TTE Trial comparing Treatment A to Treatment B Simplest case assumes all subjects are followed to the end of the study Also assumes all subjects in within treatment group have the same probability of experiencing the event Finally assumes hazard rates are proportional

20. Required Number of Events Z  /2 and Z  are standard normal percentiles p A and p B are the proportion of subjects allocated to each treatment group – i.e. equal allocation both are ½  is the minimum clinically relevant difference we want to detect – In this case it is the log of the hazard ratio under the alternative hypothesis

21. Example: Required # Events Suppose  = 0.05 and  = 0.10 (90% power) – z  /2 = 1.96 and z  = 1.282 Equal allocation – p A p B = ¼ Always round up. Hazard Ratio# Events Required 0.8845 0.7331 0.588

22. How Many People? We’ve not yet discussed event rate(s) We’ve determined for 90% power at a significance level  = 0.05 – To detect a 50% reduction in hazard, we need to observe 88 events But… How many people do we actually need to enroll to get that many events?

23. A Simplification Consider only the case in which each patient is followed for some specified time period, T More general case – Patients recruited during accrual period, a – After recruitment, there is an additional follow-up period, f – First patient followed a + f – Last patient followed for f – Requires slightly more complex power calculations

24. Calculating Number of People Need to consider probability of an event during the study period Once we have this, estimating the total number of people needed is easy NOTE: This still ignores loss to follow-up

25. Overall Probability of Event Recall p A and p B are the proportion of subjects allocated to each treatment group S A (T) and S B (T) are the survival distributions for the two treatment groups How do we get values for S A (T) and S B (T) ?

26. Estimating Survival Distributions Crude rate – How many events are expected in each group over the course of the study? Alternatively, we could assume S(T) ~Exp( ) Recall S(t) = exp{- t} Use the assumed distribution to calculate failure probability – e.g. if = 0.1/ unit time, then S(1) = 0.905 – Thus, assume by t = 1, 9.5% of subjects will have experienced the event

27. Example 1 One-year study ( T = 1 ) with p A = p B,  = 0.1/yr for control group, and   = 0.5

28. Example 2 5 year study ( T = 5 ) with p A = p B,  = 0.1/yr for control group, and  A = 0.5

29. More General Case Consider if subjects enroll over time – First patient followed a + f – Last patient followed for f Proportion of patients that will survive is the average survival curve from time f to a + f P(event) can be estimated by

30. Example 2 Revisited 5 year study with p A = p B,  = 0.1/yr for control group, and  = 0.5 But accrue patients for 2 years and follow for the remaining 3 years

31. Beyond Basic Considerations Many factors may cause statistic to deviate from expected behavior – Loss to follow-up (non-administrative censoring) – Failure to comply with treatment – Non-Uniform patient entry – Non-constant hazard ratio Failure time differ greatly from exponential – Competing risks… Violations generally require an increase in sample size to achieve desired power

32. Non-Uniform Entry Previously assumed uniform entry of subjects into the trial, which could be an error – For example patients enter trial in staggered fashion – Sample size related to total number of person years observed Alternative “general” sample size equation proposed by Lachin and Foulkes

33. Lachin and Foulkes Formula cont and trt are hazard rates for each group is overall hazard rate  ( ) is a component of the variance of  2 ( ) where

34. Non-Uniform Entry using L-F Formulation Assume patient entry times follow g(t) – If it is Uniform (for recruitment period a ) – If it is not Uniform If recruitment is faster than expected, power will be greater If recruitment is slower than expected, power will be reduced

35. Example Alternate Distribution Truncated exponential entry distribution

36. Example of Impact on Power  Probability of Event in Trt E(d | l trt ) Probability of Event in Trt E(  |  l trt ) Power (assuming Uniform entry) N to maintain 90% power 00.4960.6380.900378 0.4260.5630.859430 -20.3870.5200.832468 -30.3710.5000.820490 -40.3590.4880.813502 -50.3520.4800.805510 -60.3480.4770.801516 Trial Conditions: trt = 0.2 & cont = 0.3 a = 3 & f = 2  = 0.05 (one-sided) & 1-  = 0.9

37. Loss To Follow-Up Again, our early expressions assumed only administrative censoring BUT clearly this is not always the case. The Lakin-Foulkes expression can also be adapted to address random right censoring

38. Right Censoring Formulation Easiest “case” – censoring times ~Exp(  ) – Uniform entry into trial

39. Impact of LTFU on Sample Size Trial Conditions: trt = 0.2, cont = 0.3, &  trt =  cont =  a = 3 & f = 2  = 0.05 (one-sided) & 1-  = 0.9 entry times ~U(a)  E(  | trt )E(  | cont )E(  | trt,  )E(  | cont,  ) PowerSample Size 00.4960.638000.901378 0.050.4590.5940.1150.0990.881406 0.100.4250.5540.2130.1850.860436 0.150.3960.5180.2970.2590.839468 0.200.3690.4860.3690.3240.817500

40. Competing Risk Setting Same idea- we want to compare 2 treatments but now we have some competing risk(s) Latouche et. al (2004) developed an approach for estimating sample size in data with competing risks – Extension of Schoenfeld formula – Also based on Fine and Gray model for competing risks

41. Recall the Fine and Gray Model Model and Partial likelihood

42. Competing Risk Sample Size Number of events Number of subjects

43. Additional Considerations There are extensions for >2 groups If interim analyses are planned, these also need to be accounted for in the sample size analysis Multi-center trial? Non-proportional hazards – Methods by Halpern and Lakatos allows for non- proportional hazards – Specify event rates for groups within specific time intervals

44. Additional Considerations Non-compliance – Differs from LTFU in that these people don’t comply to treatment but are still followed – There are adjustments that can be made to the Lakin- Foulkes formula if non-compliance expected Covariate adjustments? – If balanced, adjustment for covariate shouldn’t impact power – However, in cases of extreme in-balance, formulas are not valid

45. If Complications Exist Consider a 3-stage approach – Use basic formula as first estimate – Refine sample size calculation based on likely deviations from assumptions – If necessary, develop simulation to address more complicated deviations

46. Implementation The Lakin and Foulkes implemented in the gsDesign package in R – Includes ability to power for stratified analysis – Can power for interim analyses Can use proc power in SAS – TwoSampleSurvival statement Based on log-rank but allows for different weights (Gehan, log-rank, or Tarone-Ware)

47. Implementation Alternative software? – PASS Can power for binary or non-binary covariate Can power for scenario where an additional correlated covariate is considered – Nquery Basic calculation based on Schoenfeld equation

48. All R Packages Related to Survival Analysis https://cran.r-project.org/web/views/Survival.html

49. Statistician Vs. Lab Researcher https://www.youtube.com/watch?v=PbODigCZqL8

50. Next Time 1.Nate O’Connell Sequential Designs for Phase I Clinical Trials with Late-Onset Toxicities 2. Lutffiya Muhammad Covariate-adjusted non-parametric survival curve estimation 3. James Small Simulating biologically plausible complex survival data 4. Cameron Miller Tutorial in biostatistics: Competing risks and multi-state models 5. Jamie Speiser Bagging survival trees