1. Stratified Randomization in Clinical Trials Katherine L. Monti, Ph.D. Senior Statistical Scientist and Director of the Massachusetts Office, Rho, Inc. Adjunct Associate Professor, Biostatistics University of North Carolina

2. 2  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Outline Introduction Motivation for this work: “Stratification can’t hurt” Literature search –Why stratify? Advantages –Why not stratify? Disadvantages –Considerations if stratification is going to take place –Alternatives –Limitations of the literature

3. 3  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Outline Explore the notion that “stratification can’t hurt” –Description of the simulation –Results Conclusions

4. 4  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Introduction Stratification is generally undertaken so that treatment comparisons can be made within relatively homogenous groups of experimental units. Stratification in clinical trials is different from classical stratification in survey sampling, or from blocking in experimental design.

5. 5  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Introduction In survey sampling, the population is divided into subgroups (strata). There is a defined sampling frame. Each stratum is randomly sampled with a known sample size. In experimental design, treatments are assigned within blocks, which are defined by factors that are generally determinable and often controllable (e.g., temp, water level in a greenhouse setting). Again, the sample size in each block is part of the design.

6. 6  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Introduction In clinical trials… The sample size for each factor level is often unknown until the end of the study. Exception: when sampling is halted differentially by strata to force balanced strata. The “blocking” factors are generally not controllable (e.g., stage of disease, concomitant medication usage).

7. 7  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Introduction Sometimes stratification is beneficial in clinical trials. Some trialists maintain that it is never harmful. –Is that the case?

8. 8  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Motivation A drug company’s design: –120 subjects – 4 treatments (placebo, three drug doses) – 30 sites – 1 prognostic factor with 2 levels (hi and low levels, continuous covariate) Randomization: –At each site, NOT centralized –In blocks of 4 within factor level within site

9. 9  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Motivation 120 subjects / (30 sites) = 4 subjects per site With 4 treatments, perfect balance overall would occur without stratification. 120 subjects / (30 sites x 2 levels) = 2 subjects per site for each level With 4 treatments, balance is not assured if randomization occurs within level.

10. 10  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Motivation Do we really expect 4 subjects to enroll per site? No There will be some imbalance among the treatments even if randomization is performed just within site, without regard to level.

11. 11  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Motivation Those designing the study thought that randomizing within factor level –would increase balance in the design –“couldn’t hurt” Others argued that randomizing within factor level would increase the overall imbalance in the design.

12. 12  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Literature What does the literature have to say about stratification in clinical trials ? –When is stratification beneficial? –When is stratification harmful? –Does the literature suggest that stratification “couldn’t hurt”?

13. 13  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Why stratify? Advantages Keep variability of subjects within strata as small as possible and between-strata variability as large as possible in order to have the most precision of the treatment effect (Chow and Liu, 1998) Avoid imbalance in the distribution of treatment groups within strata –Efficiency, credibility

14. 14  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Why stratify? Advantages Protect against Type I and Type II errors Avoid confounding Satisfy prevailing investigator preconceptions about study design Provide credibility to choice of analysis covariates –Stratification variables are definitely specified a priori.

15. 15  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Why not stratify? Disadvantages Gains (power/efficiency) that can occur with stratification is often small, particularly once (# subjects) / (# treatments) > 50 More costly More complicated trial –Greater opportunity to introduce randomization error

16. 16  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. One Alternative: Adaptive allocation Dynamic allocation / adaptive allocation -Minimization by Taves -Pocock and Simon’s method -Zelen’s method -Begg and Iglewicz -Others

17. 17  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Minimization Keep track of the current imbalance and assign the treatment to a new subject to reduce the existing imbalance between strata Advantages: -Produces less imbalance than simple permuted blocks -Can accommodate more factors

18. 18  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Minimization Disadvantages: -Need to keep track of current imbalance (central randomization) -None of the assignments are completely random -Since it only aims to balance marginal totals of multiple factors, precision is only increased if the interaction between prognostic factors is not pronounced. (Tu et. al., 2000)

19. 19  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Another Alternative: Post-stratification If stratification is not done at randomization, covariate analysis can be performed. -Easier and less costly to implement -Often nearly as efficient -May be less convincing, particularly if covariate was not mentioned in the protocol -Cannot correct for cases of extreme imbalance or confounding of covariates

20. 20  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. If you want to stratify …

21. 21  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Consider How well is the stratification variable measured? –If the covariates used to stratify are imprecisely assessed, then may introduce error. Is the stratification variable related to outcome? –If not, the gain in efficiency may be small or negative. How many strata will there be?

22. 22  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Number of Strata The number of strata to allow depends on: -Total number of subjects in the trial -Expected number to be in each stratum -Predictive capability of prognostic factors -Type of allocation scheme (permuted blocks vs. dynamic allocation)

23. 23  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Number of Strata The number of strata should be less than (total sample size) / (block size). (Hallstrom and Davis, 1988) –In our case, N=120, B=4, Recommendation: < 30 strata Design: 60 strata Stratification begins to fail (in terms of balance) if the total number of strata is greater than approximately N/2 (for 2 treatments). (Therneau, 1993) –or N/k, k= number of treatments

24. 24  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Number of Strata “One can inadvertently counteract the balancing effects of blocking by having too many strata.” “…, most blocks should be filled because unfilled blocks permit imbalances.” (Piantadosi,1997)

25. 25  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Number of Strata “If ‘institution effect’ were to be introduced as a further prognostic factor, …, the total number of strata may then be in the hundreds and one would have achieved little more than purely random treatment assignment.” (Pocock and Simon, 1975)

26. 26  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Number of Strata And thus we see that there are some warnings in the literature about employing too many strata. However….

27. 27  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Conclusions of the Literature Search Authors are still concluding that “Stratification is … harmless always, useful frequently, and important rarely”. (Kernan et al., 1999) (Caveat: Elsewhere in the article, Kernan et al recommend against overstratification, but this is the topic sentence of their discussion section.)

28. 28  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Limitations of the Literature Literature refers mostly to trials of two treatments. In the statistical literature, little attention is paid to operational disadvantages of more complex designs.

29. 29  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Conclusions of the Literature Search Consider stratifying only if: Prognostic factors are known to be related to the outcome and are easy to collect prior to randomization. Operational costs justify any gain. Sample size is small ( N < 100), but the stratified design does not induce imbalance. - The number of strata should be less than (total sample size) / (block size). (Hallstrom and Davis, 1988)

30. 30  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. “Stratification can’t hurt.” The notion that stratification “couldn’t hurt” –remains in current literature –is being advanced by some trialists This conclusion should be reconsidered.

31. 31  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. “Stratification can’t hurt.” The remainder of the talk will –Review the motivating example –Describe a simulation to explore the notion that “stratification can’t hurt” –Summarize the results –Provide conclusions

32. 32  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Motivation A drug company’s design: –120 subjects – 4 treatments (placebo, three drug doses) – 30 sites – 1 prognostic factor with 2 levels (hi and low levels, continuous covariate) Randomization: –At each site, NOT centralized –In blocks of 4 within factor level within site

33. 33  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Operational Difficulties Randomization was actually done within strata within site Drug supply requirements increased. (~ 33%) Packaging/shipping costs increased. Additional training visits to sites were needed in order to explain the more complex randomization scheme. The project management burden increased considerably. The misassignment of subjects to treatment was more likely.

34. 34  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Imbalance 120 subjects/ (30 sites) = 4 subjects per site Perfect balance with 4 treatments However, 4 subjects enrolling at each site is not really expected! Perfect balance overall is not expected.

35. 35  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Imbalance Additional restriction on randomization to within level of the prognostic factor within site could only increase the imbalance. How much worse does it get?

36. 36  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation I set out to compare the magnitude of the treatment imbalance if randomization were performed in permuted blocks of 4: –within site (WS) (30 strata) –within level of the factor within site (WLWS) (60 strata) I used simulation.

37. 37  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation The general approach was this: –I don’t now how many subjects would enroll in each site, so I used a variety of guesses regarding the enrollment pattern. –I don’t know how many subjects in each strata are going to enroll, but I assumed an underlying 50:50 ratio and forced 50:50 enrollment in some cases. –I don’t know in what order subjects will enter the trial, but I assumed that subjects enter randomly with respect to their strata status in each site.

38. 38  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation –Once I “identified” the number of subjects in each strata at each site and the order in which they enrolled, I randomized them to treatment twice: once randomly WS and once randomly WLWS. –Did that 10,000 times for each enrollment pattern. –Compared the balance of WS to WLWS randomization.

39. 39  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Enrollment Patterns To assign subjects to treatments, we need to know the the number of subjects ( N ij ) –in site i (i=1-30) who have –factor level j (j=1,2) The N ij are unknown….

40. 40  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Enrollment Patterns …So we make assumptions However, instead of prescribing the exact N ij for each i and j in the simulation, I defined 9 different enrollment patterns for the 60 site/level combinations.

41. 41  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Enrollment Patterns Each enrollment pattern assumed a distribution of the number of subjects in the 60 site/levels, so that there were: –30 sites, 2 levels per site –0-8 subjects in each of the 60 site/level strata –120 subjects Some enrollment patterns forced balance between the factor levels: N.1 = N.2 = 120/2 = 60 subjects per stratum

42. * Plans forced a balance between strata across sites (60 subjects per strata overall). No. Subjects Enrollment Plan 12,3*4,5*6,7*89 0 610530 1 20141211 2602016 15 3 2014128 4 610630 5 2 6 1 7 1 8 1 Entries are the number of strata having the indicated number of subjects.

43. 43  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. An Enrollment Pattern EX: Pattern 2: –20 site/levels strata w/ 3 subjects –20 site/levels strata w/ 2 subjects –20 site/levels strata w/ 1 subject Total of 60 site/level strata w/ 60+40+20 = 120 subjects Here, N ij = 1, 2 or 3 Given that pattern –The 60 strata were randomly paired to construct 30 sites. –N i1 / N i2 for site i could be any of the following: 3/3, 3/2, 3/1, 2/3, 2/2, 2/1, 1/3, 1/2, or 1/1

44. 44  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. An Enrollment Pattern In Pattern 2, it is unlikely that the factor levels will be exactly balanced overall. EX: Pattern 3 forces a 60/60 split –20 site/levels strata w/ 3 subjects (10/10 split) –20 site/levels strata w/ 2 subjects (10/10 split) –20 site/levels strata w/ 1 subject (10/10 split)

45. 45  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Enrollment Patterns To compare the balance in treatment assignment when randomizing WS and WLWS, I assigned subjects to treatments –at each site and then reassigned –in each level at each site

46. 46  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation Used SAS PROC PLAN to generate treatment assignments in permuted blocks of 4 –for 30 sites and –for 30 sites with 2 factor levels per site

47. 47  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Evaluation Criteria For each enrollment pattern, we want to be able to compare the treatment balance when randomization is performed WS and WLWS using permuted blocks of 4 treatments.

48. 48  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Evaluation Criteria There are two types of treatment balance: –Overall balance The extent to which the treatment assignments are balanced overall. –Within-level balance The extent to which the treatment assignments are balanced within each of the 2 factor levels.

49. 49  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Evaluation Criteria 4 criteria used to assess the randomization results of each simulated run are reported here: –N[1] = smallest N of the 4 treatments –N[1] + N[2] = smallest total sample size for any comparison of treatments –% loss of power compared to a completely balanced design for the comparison based on N[1] + N[2] for a study designed for 90% power –N[4] – N[1] = maximum difference in sample sizes

50. 50  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Evaluation Criteria Overall, treatment balance would be achieved if –N1 = N2 = N3 = N4 = 120/4 = 30 –N[1] = 30 –N[1] + N[2] = 60 –No loss of power relative to complete balance –N[4] – N[1] = 0

51. 51  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation Scheme Do for enrollment pattern (EP) = 1 to 9 –Do 10,000 replications: Generate 60 strata of subjects (per EP). –Assign random numbers to each subject (to determine order of enrollment at the site and the order of enrollment in the level at the site). –Assign a random number to each stratum (to identify the levels, 1 vs 2). Randomly pair the 60 strata into 30 sites. (The stratum with the lower random number is level 1.)

52. 52  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Simulation Scheme Randomly assign treatments WS based on the order of enrollment in the site, then -determine the sample size of each treatment -compute the evaluation criteria Randomly assign treatments WLWS based on the order of enrollment in the site/level, then -determine the sample size of each treatment -compute the evaluation criteria –Retain all the evaluation criteria, go to next iteration Go to next enrollment pattern

53. 53  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Results Reported For each EP there are 10,000 values of the evaluation criterion –when randomizing WS –when randomizing WLWS

54. * Plans forced a balance between strata across sites (60 subjects per strata overall). No. Subjects Enrollment Plan 12,3*4,5*6,7*89 0 610530 1 20141211 2602016 15 3 2014128 4 610630 5 2 6 1 7 1 8 1 Entries are the number of strata having the indicated number of subjects.

55. 55  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent.

56. 56  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent.

57. 57  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent.

58. 58  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent.

59. 59  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. * For EP 9, there are 4 subjects for each stratum, but the strata are not balanced, so it is not necessarily the case that each treatment has 15 observations. Therefore, the sum N{1}+N[2} can be less than 30. *

60. 60  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent.

61. 61  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Conclusions With a relatively large number of treatments: –Randomizing within numerous small sites can lead to some treatment imbalance and loss of power. –Randomization within levels of a prognostic factor in those small sites will generally Increase the treatment imbalance overall Increase the loss of power in overall pairwise comparisons Do little to reduce the treatment imbalance within the levels of the prognostic factor Increase cost of conducting the trial Increase the complexity of the trial and the chance of errors in randomization

62. 62  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Conclusions Is it the case that stratification by prognostic factor “can’t hurt”? –NO: In some cases, stratification can hurt Statistically (power) Operationally (money)

63. 63  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Contact Information KMonti@RhoWorld.com Slides: www.rhoworld.com

64. 64  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. Acknowledgements Gretchen Marcucci, M.S. Stephen Gilbert, Ph.D.

65. 65  2004 Rho, Inc. All rights reserved. No part of this document may be copied without express written consent. References Begg CB, Iglewicz B. A treatment allocation procedure for sequential clinical trials. Biometrics 36 : 81-90, 1980 Chow SC, Liu JP. Design and Analysis of Clinical Trials. John Wiley and Sons; 1998. Hallstrom A, Davis K. Imbalance in treatment assignments in stratified block randomization. Control Clin Trials 1988; 9:375-382. Kernan WN, Viscoli CM, Makuch RW, et al. Stratified randomization for clinical trials. J Clin Epidemiolol 1999; 52: 19-26. Piantadosi, S. Clinical Trials. A methodologic perspective. John Wiley and Sons; 1997. Pocock SJ, Simon R. Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics 1975; 31:103-115. Taves DR. Minimization: A new method in assigning patient to treatment and control group. Clinical Pharmacology and Therapeutics 15: 443-453, 1974. Therneau TM. How many stratification factors is "too many" to use in a randomization plan? Control Clinical Trial 14: 98-108, 1993. Tu D, Shalay K, Pater J Adjustment of treatment effect for covariates in clinical trials: Statistical and Regulatory Issues Drug Info Journal 34:511-523, 2000. Zelen M. The randomization and stratification of patients to clinical trials. Journal of Chronic Dis, 27:365-375, 1974.