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Chapter 5Chapter 5

Randomization MethodsRandomization Methods

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RANDOMIZATIONRANDOMIZATION

• Why randomize

• What a random series is

• How to randomize

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Randomization (1)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

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Randomization (2)Randomization (2)

Basic Benefits of Randomization• Eliminates assignment basis• Tends to produce comparable groups• Produces valid statistical tests

Basic MethodsRef: Zelen JCD 27:365-375, 1974.

Pocock Biometrics 35:183-197, 1979

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Goal: Achieve Comparable Groups to Allow Unbiased Estimate of

Treatment

Beta-Blocker Heart Attack TrialBaseline Comparisons

Propranolol Placebo (N-1,916) (N-1,921)

Average Age (yrs) 55.2 55.5Male (%) 83.8 85.2White (%) 89.3 88.4Systolic BP 112.3 111.7Diastolic BP 72.6 72.3Heart rate 76.2 75.7Cholesterol 212.7 213.6Current smoker (%) 57.3 56.8

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Nature of Random Numbers Nature of Random Numbers and Randomnessand Randomness

• A completely random sequence of digits is a mathematicalidealization

• 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

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Allocation Procedures Allocation Procedures to Achieve Balanceto Achieve Balance

• Simple randomization

• Biased coin randomization

• Permuted block randomization

• Balanced permuted block randomization

• Stratified randomization

• Minimization method

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Randomization & Balance (1)Randomization & Balance (1)

n = 100

p = ½

s = #heads V(s) = npq = 100 · ½ · ½ = 25

E(s) = n · p = 50

60SP 50-60 np-SP

25

5060

25

npSP

5

10 ΖP

.0252 ΖP

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Randomization & Balance (2)Randomization & Balance (2)

n = 20

p = ½

E(s) = 10

V(s) = np = 20/4 = 5

12SP

5

1012

V(s)

npSP

5

2 ΖP

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Simple Random AllocationSimple Random AllocationA 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)

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Simple RandomizationSimple 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

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Simple Randomization Simple Randomization

For two treatmentsassign A for digits 0-4

B for digits 5-9For three treatments

assign A for digits 1-3 B for digits 4-6 C for digits 7-9 and ignore 0

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Simple Randomization Simple Randomization

For four treatmentsassign 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

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Restricted RandomizationRestricted 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 balance1. Permuted-block2. Biased coin (Efron)3. Urn design (LJ Wei)

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Permuted-Block Randomization (1)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

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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 (2)Permuted-Block Randomization (2)

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• 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 AA 0.73 3 B B 0.87 4 AB 0.31 2 B

Permuted-Block Randomization (3)Permuted-Block Randomization (3)

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• 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

Permuted-Block Randomization (4)Permuted-Block Randomization (4)

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Biased Coin Design (BCD)Biased Coin Design (BCD)Efron (1971) Efron (1971) BiometrikaBiometrika

• 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

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Urn RandomizationUrn RandomizationWei & 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.

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Analysis & InferenceAnalysis & 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)

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Kalish & Begg Kalish & Begg Controlled Clinical Trials, 1985Controlled 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.

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Balancing on Balancing on Baseline CovariatesBaseline Covariates

• Stratified Randomization

• Covariate Adaptive– Minimization– Pocock & Simon

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• 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.

i

f

i

1

Stratified Randomization (1)Stratified Randomization (1)

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Example

Sex (M,F)

and

Risk (H,L)

1 2 FactorsX

2 2 Levels in each

4 Strata3

4

H

L

H

L

M

F

For stratified randomization, randomization must be restricted!Otherwise, (if CRD was used), no balance is guaranteed despite the effort!

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Stratified Randomization (2)Stratified Randomization (2)• Define strata

• Randomization is performed within each stratum and is usually blocked

• Example: Age, < 40, 41-60, >60; Sex, M, FTotal 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, ..

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Stratified Randomization (3)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

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CommentComment

• 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.

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Minimization Method (1)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

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Example: Minimization Method (a)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 BSex Male 16 14

Female 10 10Age < 40 13 12

41-60 9 6> 60 4 6

Disease Stage I 6 4Stage II 13 16

Stage III 7 4Total 26 24

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Example: Minimization Method (b)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 DifferenceMale 16 14 +Age > 60 4 6 -Stage III 7 4 +Total 27 24 2 +s and 1 -

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• 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

Example: Minimization Method (c)Example: Minimization Method (c)

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Minimization Method (2)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

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Covariate Adaptive AllocationCovariate Adaptive Allocation(Sequential Balanced Stratification)(Sequential Balanced Stratification)

Pocock & Simon, Pocock & Simon, Biometrics,Biometrics, 1975; 1975; Efron, Efron, BiometrikaBiometrika, 1971, 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

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Example: Baseline Adaptive RandomizationExample: 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, xi

t2) 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)

2

1i

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Factor 1(Sex) 2(Risk)Level 1(M) 2(F) 1(H) 2(M) 3(L) Total

1 16* 10 13 9 4* 26PatientGroup 2 14 10 12 6 6 24Total 30 20 25 15 10 50

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

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(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

Factor 1(Sex) 2(Risk)Level 1(M) 2(F) 1(H) 2(M) 3(L) Total

1 16* 10 13 9 4* 26PatientGroup 2 14 10 12 6 6 24Total 30 20 25 15 10 50

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(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 15Range = |16-15| = 1

(2) Factor 2, Level 3(Low Risk)

K X2k X22k

Group 1 4 42 6 7Range = |4-7| = 3

* B(2) = 3(1) + 2(3) = 9

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(c) Rank B(1) and B(2), measures of imbalance

Assign tt B(t) with probability

2 9 2/3

1 11 1/3

* Note: “minimization” would assign treatment 2 for sure

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Response Adaptive Response Adaptive Allocation ProceduresAllocation Procedures

• Use outcome data obtained during trial to influence allocation of patient to treatment

• Data-driveni.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

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Play-the-Winner RulePlay-the-Winner Rule Zelen (1969)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

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Response Adaptive Response Adaptive RandomizationRandomization

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 ......

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Two-armed Bandit or Two-armed Bandit or Randomized Play-the-Winner RuleRandomized 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

#44

ECMO ExampleECMO 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

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Michigan ECMO TrialMichigan ECMO Trial• Bartlett Pediatrics (1985)

• Modified “play-the-winner”– Urn model

A ball ECMOB ball Standard controlIf 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 .05 .28

1 2* 3 4 5 6 7 8 9 10ECMO S S S S S S S S SCONTROL F

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Harvard ECMO Trial (1)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 stage1st Randomization (permuted block) switch to

superior treatment after 4 deaths in worst arm

2nd Stay with best treatment

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Harvard ECMO Trial (2)Harvard ECMO Trial (2)

• Results

Survival

* less severe patients

P = .054 (Fisher)

1st 2nd*ECMO 9/9 19/20CONTROL 6/10

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Multi-institutional TrialsMulti-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 astratification factor

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Mechanics of Randomization (1)Mechanics of Randomization (1)

Basic Principle - “Analyze What is Randomized”

* Timing

• Actual randomization should be delayed until just prior to initiation of therapy

• ExampleAlprenolol Trial, Ahlmark et al (1976)– 393 patients randomized two weeks before therapy

– Only 162 patients treated, 69 alprenolol & 93 placebo

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Mechanics of Randomization (2)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

#51

Example of Previous Methods Example of Previous Methods (1)(1)

20 subjects, treatment A or B, risk H or L

Subject Risk1 H Randomize Using2 L3 L 1. Simple4 H5 L 2. Blocked (Size=4)6 L7 L 3. Stratify by risk + use simple8 L9 H 4. Stratify by risk + block10 L11 H12 H For each compute13 H14 H 1. Percent pts on A15 L16 L 2. For each risk group, percent of pts on A17 H18 H19 L20 H

10 subjects with H10 subjects with L

#52

Example of Previous Methods Example of Previous Methods (2)(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