1. Chapter 5 Randomization Methods

2. RANDOMIZATION
Why randomize
What a random series is
How to randomize

3. 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

4. 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: , 1974.
Pocock Biometrics 35: , 1979

5. 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)
Male (%)
White (%)
Systolic BP
Diastolic BP
Heart rate
Cholesterol
Current smoker (%)

6. 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

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

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

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

10. 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)

11. 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

12. 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

13. 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

14. 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)

15. 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

16. 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

17. 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 A A B B A B B

18. 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

19. 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

20. 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.

21. 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)

22. 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.

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

24. 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.

25. 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!

26. 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 = Age Male Female ABBA, BAAB, … BABA, BAAB, BBAA, ABAB, ... ABAB, BBAA, ... >60 AABB, ABBA, ... BAAB, ABAB, ..

27. 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

28. 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.

29. 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

30. 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
Female
Age <
>
Disease Stage I Stage II
Stage III
Total

31. 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
Age >
Stage III
Total s and 1 -

32. 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

33. 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

34. 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

35. Example: Baseline Adaptive Randomization
Assume 2 treatments (1 & 2)
2 prognostic factors (1 & 2) (Gender & Risk Group)
Factor levels (M & F)
Factor 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)

36. 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
Range =|17-14| = 3

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

38. (b) Calculate B(2) (Assign Pt #51 to trt 2) t=2
(1) Factor 1, Level 1
(Male)
K X1K X21K
Group
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

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

40. 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

41. 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

42. Response Adaptive Randomization
Example
"Play-the-winner” Zelen (1969) JASA
TRT A S S F S S S F
TRT B S F
Patient

43. 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

44. 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

45. 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

46. 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

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

48. 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

49. 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

50. 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 (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

52. 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