Chapter 5 Randomization Methods

RANDOMIZATION Why randomize What a random series is How to randomize

Randomization (1) Rationale Reference: Byar et al (1976) NEJM 274:74-80. Best way to find out which therapy is best Reduce risk of current and future patients of being on harmful treatment

Randomization (2) Basic Benefits of Randomization Basic Methods Eliminates assignment basis Tends to produce comparable groups Produces valid statistical tests Basic Methods Ref: Zelen JCD 27:365-375, 1974. Pocock Biometrics 35:183-197, 1979

Beta-Blocker Heart Attack Trial Goal: Achieve Comparable Groups to Allow Unbiased Estimate of Treatment Beta-Blocker Heart Attack Trial Baseline Comparisons Propranolol Placebo (N-1,916) (N-1,921) Average Age (yrs) 55.2 55.5 Male (%) 83.8 85.2 White (%) 89.3 88.4 Systolic BP 112.3 111.7 Diastolic BP 72.6 72.3 Heart rate 76.2 75.7 Cholesterol 212.7 213.6 Current smoker (%) 57.3 56.8

Nature of Random Numbers and Randomness A completely random sequence of digits is a mathematical idealization Each digit occurs equally frequently in the whole sequence Adjacent (set of) digits are completely independent of one another Moderately long sections of the whole show substantial regularity A table of random digits Produced by a process which will give results closely approximating to the mathematical idealization Tested to check that it behaves as a finite section from a completely random series should Randomness is a property of the table as a whole Different numbers in the table are independent

Allocation Procedures to Achieve Balance Simple randomization Biased coin randomization Permuted block randomization Balanced permuted block randomization Stratified randomization Minimization method

Randomization & Balance (1) p = ½ s = #heads V(s) = npq = 100 · ½ · ½ = 25 E(s) = n · p = 50

Randomization & Balance (2) p = ½ E(s) = 10 V(s) = np = 20/4 = 5

Simple Random Allocation A specified probability, usually equal, of patients assigned to each treatment arm, remains constant or may change but not a function of covariates or response a. Fixed Random Allocation n known in advance, exactly n/2 selected at random & assigned to Trt A, rest to Trt B b. Complete Randomization (most common) n not exactly known marginal and conditional probability of assignment = 1/2 analogous to a coin flip (random digits)

Simple Randomization Advantage: simple and easy to implement Disadvantage: At any point in time, there may be an imbalance in the number of subjects on each treatment With n = 20 on two treatments A and B, the chance of a 12:8 split or worse is approximately 0.5 With n = 100, the chance of a 60:40 split or worse is approximately 0.025 Balance improves as the sample size n increases Thus desirable to restrict randomization to ensure balance throughout the trial

Simple Randomization For two treatments assign A for digits 0-4 B for digits 5-9 For three treatments assign A for digits 1-3 B for digits 4-6 C for digits 7-9 and ignore 0

Simple Randomization For four treatments assign A for digits 1-2 B for digits 3-4 C for digits 5-6 D for digits 7-8 and ignore 0 and 9

Restricted Randomization Simple randomization does not guarantee balance over time in each realization Patient characteristics can change during recruitment (e.g. early pts sicker than later) Restricted randomizations guarantee balance 1. Permuted-block 2. Biased coin (Efron) 3. Urn design (LJ Wei)

Permuted-Block Randomization (1) Simple randomization does not guarantee balance in numbers during trial If patient characteristics change with time, early imbalances can't be corrected Need to avoid runs in Trt assignment Permuted Block insures balance over time Basic Idea Divide potential patients into B groups or blocks of size 2m Randomize each block such that m patients are allocated to A and m to B Total sample size of 2m B For each block, there are 2mCm possible realizations (assuming 2 treatments, A & B) Maximum imbalance at any time = 2m/2 = m

Permuted-Block Randomization (2) Method 1: Example Block size 2m = 4 2 Trts A,B }  4C2 = 6 possible Write down all possible assignments For each block, randomly choose one of the six possible arrangements {AABB, ABAB, BAAB, BABA, BBAA, ABBA} ABAB BABA ...... Pts 1 2 3 4 5 6 7 8 9 10 11 12

Permuted-Block Randomization (3) Method 2: In each block, generate a uniform random number for each treatment (Trt), then rank the treatments in order Trt in Random Trt in any order Number Rank rank order A 0.07 1 A A 0.73 3 B B 0.87 4 A B 0.31 2 B

Permuted-Block Randomization (4) Concerns - If blocking is not masked, the sequence become somewhat predictable (e.g. 2m = 4) A B A B B A B ? Must be A. A A Must be B B. - This could lead to selection bias Simple Solution to Selection Bias Do not reveal blocking mechanism Use random block sizes If treatment is double blind, no selection bias

Biased Coin Design (BCD) Efron (1971) Biometrika Allocation probability to Treatment A changes to keep balance in each group nearly equal BCD (p) Assume two treatments A & B D = nA -nB "running difference" n = nA + nB Define p = prob of assigning Trt > 1/2 e.g. PA = prob of assigning Trt A If D = 0, PA = 1/2 D > 0, PA = 1 - p Excess A's D < 0, PA = p Excess B's Efron suggests p=2/3 D > 0 PA = 1/3 D < 0 PA = 2/3

Urn Randomization Wei & Lachin: Controlled Clinical Trials, 1988 A generalization of Biased Coin Designs BCD correction probability (e.g. 2/3) remains constant regardless of the degree of imbalance Urn design modifies p as a function of the degree of imbalance U(, ) & two Trts (A,B) 0. Urn with  white,  red balls to start 1. Ball is drawn at random & replaced 2. If red, assign B If white, assign A 3. Add  balls of opposite color (e.g. If red, add  white) 4. Go to 1. Permutational tests are available, but software not as easy.

Analysis & Inference Most analyses do not incorporate blocking Need to consider effects of ignoring blocks Actually, most important question is whether we should use complete randomization and take a chance of imbalance or use permuted-block and ignore blocks Homogeneous or Heterogeneous Time Pop. Model Homogeneous in Time Blocking probably not needed, but if blocking ignored, no problem Heterogeneoous in Time Blocking useful, intrablock correlations induced Ignoring blocking most likely conservative Model based inferences not affected by treatment allocation scheme. Ref: Begg & Kalish (Biometrics, 1984)

Kalish & Begg Controlled Clinical Trials, 1985 Time Trend Impact of typical time trends (based on ECOG pts) on nominal p-values likely to be negligible A very strong time trend can have non-negligible effect on p-value If time trends cause a wide range of response rates, adjust for time strata as a co-variate. This variation likely to be noticed during interim analysis.

Balancing on Baseline Covariates Stratified Randomization Covariate Adaptive Minimization Pocock & Simon

Stratified Randomization (1) May desire to have treatment groups balanced with respect to prognostic or risk factors (co-variates) For large studies, randomization “tends” to give balance For smaller studies a better guarantee may be needed Divide each risk factor into discrete categories Number of strata f = # risk factors; li = number of categories in factor i Randomize within each stratum For stratified randomization, randomization must be restricted! Otherwise, (if CRD was used), no balance is guaranteed despite the effort.

Example Sex (M,F) 1 2 Factors and X 2 2 Levels in each Risk (H,L)  4 Strata 3 4 H L M F H L For stratified randomization, randomization must be restricted! Otherwise, (if CRD was used), no balance is guaranteed despite the effort!

Stratified Randomization (2) Define strata Randomization is performed within each stratum and is usually blocked Example: Age, < 40, 41-60, >60; Sex, M, F Total number of strata = 3 x 2 = 6 Age Male Female 40 ABBA, BAAB, … BABA, BAAB, ... 41-60 BBAA, ABAB, ... ABAB, BBAA, ... >60 AABB, ABBA, ... BAAB, ABAB, ..

Stratified Randomization (3) The block size should be relative small to maintain balance in small strata, and to insure that the overall imbalance is not too great With several strata, predictability should not be a problem Increased number of stratification variables or increased number of levels within strata lead to fewer patients per stratum In small sample size studies, sparse data in many cells defeats the purpose of stratification Stratification factors should be used in the analysis Otherwise, the overall test will be conservative

Comment For multicenter trials, clinic should be a factor Gives replication of “same” experiment. Strictly speaking, analysis should take the particular randomization process into account; usually ignored (especially blocking) & thereby losing some sensitivity. Stratification can be used only to a limited extent, especially for small trials where it's the most useful; i.e. many empty or partly filled strata. If stratification is used, restricted randomization within strata must be used.

Minimization Method (1) An attempt to resolve the problem of empty strata when trying to balance on many factors with a small number of subjects Balances Trt assignment simultaneously over many strata Used when the number of strata is large relative to sample size as stratified randomization would yield sparse strata A multiple risk factors need to be incorporated into a score for degree of imbalance Need to keep a running total of allocation by strata Also known as the dynamic allocation Logistically more complicated Does not balance within cross-classified stratum cells; balances over the marginal totals of each stratum, separately

Example: Minimization Method (a) Three stratification factors: Sex (2 levels), age (3 levels), and disease stage (3 levels) Suppose there are 50 patients enrolled and the 51st patient is male, age 63, and stage III Trt A Trt B Sex Male 16 14 Female 10 10 Age < 40 13 12 41-60 9 6 > 60 4 6 Disease Stage I 6 4 Stage II 13 16 Stage III 7 4 Total 26 24

Example: Minimization Method (b) Method: Keep a current list of the total patients on each treatment for each stratification factor level Consider the lines from the table above for that patient's stratification levels only Sign of Trt A Trt B Difference Male 16 14 + Age > 60 4 6 - Stage III 7 4 + Total 27 24 2 +s and 1 -

Example: Minimization Method (c) Two possible criteria: Count only the direction (sign) of the difference in each category. Trt A is “ahead” in two categories out of three, so assign the patient to Trt B Add the total overall categories (27 As vs 24 Bs). Since Trt A is “ahead,” assign the patient to Trt B

Minimization Method (2) These two criteria will usually agree, but not always Choose one of the two criteria to be used for the entire study Both criteria will lead to reasonable balance When there is a tie, use simple randomization Generalization is possible Balance by margins does not guarantee overall treatment balance, or balance within stratum cells

Covariate Adaptive Allocation (Sequential Balanced Stratification) Pocock & Simon, Biometrics, 1975; Efron, Biometrika, 1971 Goal is to balance on a number of factors but with "small" numbers of subjects In a simple case, if at some point Trt A has more older patients that Trt B, next few older patients should more likely be given Trt B until "balance" is achieved Several risk factors can be incorporated into a score for degree of imbalance B(t) for placing next patient on treatment t (A or B) Place patient on treatment with probability p > 1/2 which causes the smallest B(t), or the least imbalance More complicated to implement - usually requires a small "desk top" computer

Example: Baseline Adaptive Randomization Assume 2 treatments (1 & 2) 2 prognostic factors (1 & 2) (Gender & Risk Group) Factor 1 - 2 levels (M & F) Factor 2 - 3 levels (High, Medium & Low Risk) Let B(t) = Wi Range (xit1, xit2) wi = weight for each factor e.g. w1 = 3 w1/w2 = 1.5 w2 = 2 xij = number of patients in ith factor and jth treatment xitj = change in xij if next patient assigned treatment t Let P = 2/3 for smallest B(t) Pi = (2/3, 1/3) Assume we have already randomized 50 patients Now 51st pt. Male (1st level, factor 1) Low Risk (3rd level, factor 2)

Now determine B(1) and B(2) for patient #51.… If assigned Treatment 1 (t = 1) (a) Calculate B(t) (Assign Pt #51 to trt 1) t = 1 (1) Factor 1, Level 1 (Male) Now Proposed K X1K  X11K Trt Group 1 16 17 2 14 14 Range =|17-14| = 3

(a) Calculate B(t) (Assign Pt #51 to trt 1) t=1 (2) Factor 2, Level 3 (Low Risk) K X2K X12K Trt Group 1 4  5 2 6 6 Range = |5-6|, = 1 * B(1) = 3(3) + 2(1) = 11

(b) Calculate B(2) (Assign Pt #51 to trt 2) t=2 (1) Factor 1, Level 1 (Male) K X1K X21K Group 1 16 16 2 14 15 Range = |16-15| = 1 (2) Factor 2, Level 3 (Low Risk) K X2k X22k Group 1 4 4 2 6 7 Range = |4-7| = 3 * B(2) = 3(1) + 2(3) = 9

(c) Rank B(1) and B(2), measures of imbalance Assign t t B(t) with probability 2 9 2/3 1 11 1/3 * Note: “minimization” would assign treatment 2 for sure

Response Adaptive Allocation Procedures Use outcome data obtained during trial to influence allocation of patient to treatment Data-driven i.e. dependent on outcome of previous patients Assumes patient response known before next patient The goal is to allocate as few patients as possible to a seemingly inferior treatment Issues of proper analyses quite complicated Not widely used though much written about Very controversial

Play-the-Winner Rule Zelen (1969) Treatment assignment depends on the outcome of previous patients Response adaptive assignment When response is determined quickly 1st subject: toss a coin, H = Trt A, T = Trt B On subsequent subjects, assign previous treatment if it was successful Otherwise, switch treatment assignment for next patient Advantage: Potentially more patients receive the better treatment Disadvantage: Investigator knows the next assignment

Response Adaptive Randomization Example "Play-the-winner” Zelen (1969) JASA TRT A S S F S S S F TRT B S F Patient 1 2 3 4 5 6 7 8 9 ......

Two-armed Bandit or Randomized Play-the-Winner Rule Treatment assignment probabilities depend on observed success probabilities at each time point Advantage: Attempts to maximize the number of subjects on the “superior” treatment Disadvantage: When unequal treatment numbers result, there is loss of statistical power in the treatment comparison

ECMO Example References Michigan 1a. Bartlett R., Roloff D., et al.; Pediatrics (1985) 1b. Begg C.; Biometrika (1990) Harvard 2a. O’Rourke P., Crone R., et al.; Pediatrics (1989) 2b. Ware J.; Statistical Science (1989) 2c. Royall R.; Statistical Science (1991) Extracoporeal Membrane Oxygenator(ECMO) treat newborn infants with respiratory failure or hypertension ECMO vs. conventional care

Michigan ECMO Trial Bartlett Pediatrics (1985) Modified “play-the-winner” Urn model A ball  ECMO B ball  Standard control If success on A, add another A ball .… Wei & Durham JASA (1978) Randomized Consent Design Results *sickest patient P-Values, depending on method, values ranged .001 6 .05 6 .28

Harvard ECMO Trial (1) O’Rourke, et al.; Pediatrics (1989) ECMO for pulmonary hypertension Background Controversy of Michigan Trial Harvard experience of standard 11/13 died Randomized Consent Design Two stage 1st Randomization (permuted block) switch to superior treatment after 4 deaths in worst arm 2nd Stay with best treatment

Harvard ECMO Trial (2) Results Survival P = .054 (Fisher) * less severe patients P = .054 (Fisher)

Multi-institutional Trials Often in multi-institutional trials, there is a marked institution effect on outcome measures Using permuted blocks within strata, adding institution as yet another stratification factor will probably lead to sparse cells (and potentially more cells than patients!) Use permuted block randomization balanced within institutions Or use the minimization method, using institution as a stratification factor

Mechanics of Randomization (1) Basic Principle - “Analyze What is Randomized” * Timing Actual randomization should be delayed until just prior to initiation of therapy Example Alprenolol Trial, Ahlmark et al (1976) 393 patients randomized two weeks before therapy Only 162 patients treated, 69 alprenolol & 93 placebo

Mechanics of Randomization (2) * Operational 1. Sequenced sealed envelopes (prone to tampering!) 2. Sequenced bottles/packets 3. Phone call to central location - Live response - Voice Response System 4. One site PC system 5. Web based Best plans can easily be messed up in the implementation

Example of Previous Methods (1) 20 subjects, treatment A or B, risk H or L Subject Risk 1 H Randomize Using 2 L 3 L 1. Simple 4 H 5 L 2. Blocked (Size=4) 6 L 7 L 3. Stratify by risk + use simple 8 L 9 H 4. Stratify by risk + block 10 L 11 H 12 H For each compute 13 H 14 H 1. Percent pts on A 15 L 16 L 2. For each risk group, percent of pts on A 17 H 18 H 19 L 20 H 10 subjects with H 10 subjects with L

Example of Previous Methods (2) 1. Simple 1st Try 2nd Try (a) 9/20 A's 7/20 A's OVERALL BY (b) H: 5/10 A's 3/10 A's SUBGROUP L: 4/10 A's 4/10 A's 2. Blocked (No stratification) (a) 10 A's & 10 B's (b) H: 4 A's & 6 B's L: 6 A's & 4 B's 3. Stratified with simple randomization (a) 5 A's & 15 B's (b) H: 1 A & 9 B's L: 4 A's & 6 B's 4. Stratified with blocking (a) 10 A's & 10 B's MUST BLOCK TO MAKE STRATIFICATION PAY (b) H: 5 A's & 5 B's OFF L: 5 A's & 5 B's