When Is Stratification Detrimental to a Clinical Trial Design? Part I Gretchen Marcucci, M.S. Biostatistician, Rho, Inc. and Katherine L. Monti, Ph.D. Rho, Inc. and University of North Carolina

2  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Outline Introduction Motivation for the literature search Why stratify? Advantages Why not stratify? Disadvantages If you want to stratify …

3  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Outline How many strata? Alternatives Limitations of the literature Conclusions

4  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Introduction This paper presents the results of a literature search on the use of stratified randomization, with particular interest in clinical trials.

5  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Introduction Stratification in clinical trials is different from classical stratification in survey sampling, or from blocking in experimental design. In stratified sampling, “the population is divided into subgroups, or strata, each of which is sampled randomly with a known sample size.” In experimental design, treatments are assigned within blocks, which are defined by factors that are largely determinable and controllable (e.g., temp, water level in a greenhouse setting). Again, the sample size in each block is part of the design.

6  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Introduction In clinical trials, stratification refers to the assignment of treatment to homogeneous groups defined by patient-related characteristics that may affect outcome. –These factors are generally not controllable (e.g., stage of disease, age within the allowable range). –Until the end of the study, the sample size of each factor level is generally unknown.

7  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Introduction Sometimes stratification is beneficial. Some trialists maintain that it is never harmful. –Is that the case?

8  2003 Gretchen Marcucci. 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  2003 Gretchen Marcucci. 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 imbalance in the design.

10  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Motivation 120 subjects / (30 sites) = 4 subjects per site Perfect balance with 4 treatments. 120 subjects / (30 sites x 2 levels) = 2 subjects per site for each level Balance not assured with 4 treatments.

11  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Motivation Although 2 subjects/level/site is not a realistic enrollment pattern, it is unlikely that stratification would help balance the design.

12  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Motivation What does the literature have to say about stratification in clinical trials ? –When is stratification beneficial? –When is stratification harmful? –Is it true that it “couldn’t hurt”?

13  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Why stratify? Advantages To 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. Increase efficiency. Protect against Type I and Type II errors.

14  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Why stratify? Advantages Avoid confounding. Satisfy prevailing investigator assumptions. Provide credibility to choice of analysis covariates.

15  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Why not stratify? Disadvantages More costly and complicated trial. More opportunity to introduce error. Power loss from unstratified randomization may be very small in many cases. Gain in precision of estimates is small once (number of subjects) / (treatment) > 50.

16  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. If you want to stratify …

17  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Consider If the covariates are imprecisely assessed, then may introduce error. If there are too many covariates, –then there is a higher chance of imbalance, or –the effect is the same as simple randomization. If covariates are not related to outcome, then the gain in efficiency will be small or negative. If there are too many strata with small number of subjects / stratum, the analysis model may be overparameterized.

18  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Consider How many strata depends on: -Total number of subjects in the trial. -Expected number to be in each stratum. -Importance of prognostic factors. -Type of allocation scheme (permuted blocks vs. dynamic allocation).

19  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Consider 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

20  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Number of strata: notes “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)

21  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Number of strata: notes “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)

22  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Alternatives Dynamic allocation / adaptive stratification -Minimization by Taves. -Pocock and Simon’s method. -Zelen’s method. -Begg and Iglewics. -Others. Post-stratification (ANCOVA).

23  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Minimization Keeps track of the current imbalance and assigns the treatment that reduces the imbalance. Advantages: -Produces less imbalance than conventional stratification. -Can accommodate more factors.

24  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Minimization Disadvantages: -Need to keep track of current imbalance. -None of the assignments are completely random. -Since it only aims to balance marginal totals, precision is only increased if the interaction between prognostic factors is not pronounced. (Tu et. al., 2000)

25  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. 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, in particular if covariate was not mentioned in the protocol. -Cannot correct for cases of extreme imbalance.

26  2003 Gretchen Marcucci. 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. Little attention is paid to operational disadvantages of more complex designs.

27  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Conclusions 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)

28  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Conclusions Authors are still (1999) concluding that “Stratification is … harmless always, useful frequently, and important rarely”. (Kernan et. al., 1999) The preconception that stratification would improve the balance and could not hurt should be reconsidered.

29  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. Contact Information Slides:

30  2003 Gretchen Marcucci. 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; Hallstrom A, Davis K. Imbalance in treatment assignments in stratified block randomization. Control Clin Trials 1988; 9: Kernan WN, Viscoli CM, Makuch RW, et al. Stratified randomization for clinical trials. J Clin Epidemiolol 1999; 52: Piantadosi, S. Clinical Trials. A methodologic perspective. John Wiley and Sons; 1997.

31  2003 Gretchen Marcucci. All rights reserved. No part of this document may be copied without express written consent. References Pocock SJ, Simon R. Sequential tretment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics 1975; 31: Taves DR. Minimization: A new method in assigning patient to treatment and control group. Clinical Pharmacology and Therapeutics 15: , Therneau TM. How many stratification factors is "too many" to use in a randomization plan? Control Clinical Trial 14: , Tu D, Shalay K, Pater J Adjustment of treatment effect for covariates in clinical trials: Statistical and Regulatory Issues Drug Info Journal 34: , Zelen M. The randomization and stratification of patients to clinical trials. Journal of Chronic Dis, 27: , 1974.