Transcript Randomization

Chapter 5
Randomization Methods
#1
RANDOMIZATION
• Why randomize
• What a random series is
• How to randomize
#2
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
#3
Randomization (2)
Basic Benefits of Randomization
• 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
#4
Goal: Achieve Comparable Groups to Allow
Unbiased Estimate of Treatment
Beta-Blocker Heart Attack Trial
Baseline Comparisons
Propranolol
(N-1,916)
Average Age (yrs)
Male (%)
White (%)
Systolic BP
Diastolic BP
Heart rate
Cholesterol
Current smoker (%)
55.2
83.8
89.3
112.3
72.6
76.2
212.7
57.3
Placebo
(N-1,921)
55.5
85.2
88.4
111.7
72.3
75.7
213.6
56.8
#5
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
#6
Allocation Procedures
to Achieve Balance
• Simple randomization
• Biased coin randomization
• Permuted block randomization
• Balanced permuted block randomization
• Stratified randomization
• Minimization method
#7
Randomization & Balance (1)
n = 100
p=½
s = #heads
E(s) = n · p = 50
V(s) = np(1-p) = 100 · ½ · ½ = 25
PS  60  PS - np  60 - 50
 S  np 60  50 
 P


25 
 25
10 

 P Ζ 

5 

 PΖ  2  .025
#8
Randomization & Balance (2)
n = 20
p=½
E(s) = 10
V(s) = np(1-p) = 20/4 = 5
 S  np 12  10 


PS  12  P
5 
 V(s)
2 

 P Ζ 

5

#9
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)
#10
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
#11
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
#12
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
#13
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)
#14
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
2 m ! possible realizations
For each block, there are
m  2!
– (assuming 2 treatments, A & B)
– Maximum imbalance at any time = 2m/2 = m
#15
Permuted-Block Randomization (2)
Method 1: Example
• Block size 2m = 4
2 Trts A,B
}
4!
= 6 possible
2  2!
• Write down all possible assignments
• For each block, randomly choose one of the six
possible arrangements
• {AABB, ABAB, BAAB, BABA, BBAA, ABBA}
ABAB
Pts
1234
BABA
5678
......
9 10 11 12
#16
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
any order
A
A
B
B
Random
Number
Rank
0.07
0.73
0.87
0.31
1
3
4
2
Trt in
rank order
A
B
A
B
#17
Permuted-Block Randomization (4)
• Concerns
- If blocking is not masked, the sequence become
somewhat predictable (e.g. 2m = 4)
ABAB BAB?
Must be A.
AA
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
#18
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
#19
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.
#20
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)
#21
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.
#22
Balancing on
Baseline Covariates
• Stratified Randomization
• Covariate Adaptive
– Minimization
– Pocock & Simon
#23
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
f
Number of strata

i 1
i
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.
#24
Example
1
H
Sex (M,F)
M
L
F
H
2
and
Risk (H,L)
L
2 Factors
X
2 Levels in
each
 4 Strata
3
4
For stratified randomization, randomization must be restricted!
Otherwise, (if CRD was used), no balance is guaranteed despite
the effort!
#25
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
41-60
>60
ABBA, BAAB, …
BBAA, ABAB, ...
AABB, ABBA, ...
BABA, BAAB, ...
ABAB, BBAA, ...
BAAB, ABAB, ..
#26
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
#27
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.
#28
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
#29
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
Sex
Age
Disease
Total
Male
Female
< 40
41-60
> 60
Stage I
Stage II
Stage III
Trt A
16
10
13
9
4
6
13
7
26
Trt B
14
10
12
6
6
4
16
4
24
#30
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 #31
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
#32
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
#33
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
than 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
#34
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)
2
Let B(t) =  Wi Range (xit1, xit2) wi = weight for each factor
i 1
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)
#35
Factor
Level
Patient 1
Group 2
Total
1(Sex)
1(M) 2(F)
16* 10
14
10
30
20
2(Risk)
1(H) 2(M) 3(L)
13
9
4*
12
6
6
25
15
10
Total
26
24
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)
Trt Group
K
1
2
Now Proposed
X1K  X11K
16
17
14
14
Range =|17-14| = 3
#36
Factor
Level
Patient 1
Group
2
Total
1(Sex)
1(M) 2(F)
16* 10
14
10
30
20
2(Risk)
1(H) 2(M) 3(L)
13
9
4*
12
6
6
25
15
10
(a) Calculate B(t) (Assign Pt #51 to trt 1)
Total
26
24
50
t=1
(2) Factor 2, Level 3
(Low Risk)
Trt Group
K
1
2
X2K
X12K
4  5
6
6
Range = |5-6|, = 1
* B(1) = 3(3) + 2(1) = 11
#37
(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
#38
(c) Rank B(1) and B(2), measures of imbalance
Assign t
with probability
t
B(t)
2
9
2/3
1
11
1/3
* Note: “minimization” would assign treatment
2 for sure
#39
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
#40
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
#41
Response Adaptive
Randomization
Example
"Play-the-winner” Zelen (1969) JASA
TRT A
TRT B
SSF
SSSF
SF
Patient 1 2 3 4 5 6 7 8 9 ......
#42
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
#43
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
#44
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
ECMO
CONTROL
1
S
2*
3
S
4
S
5
S
6
S
7
S
8
S
9
S
10
S
F
*sickest patient
• P-Values, depending on method, values ranged
.001 6 .05 6 .28
#45
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
#46
Harvard ECMO Trial (2)
• Results
Survival
ECMO
CONTROL
1st 2nd*
9/9 19/20
6/10
* less severe patients
P = .054 (Fisher)
#47
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
#48
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
#49
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
#50
Example of Previous Methods (1)
20 subjects, treatment A or B, risk H or L
Subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Risk
H
L
L
H
L
L
L
L
H
L
H
H
H
H
L
L
H
H
L
H
Randomize Using
1. Simple
2. Blocked (Size=4)
3. Stratify by risk + use simple
4. Stratify by risk + block
For each compute
1. Percent pts on A
2. For each risk group, percent of pts on A
10 subjects with H
10 subjects with L
#51
Example of Previous Methods (2)
1. Simple
(a)
(b)
1st Try
9/20 A's
H: 5/10 A's
L: 4/10 A's
2nd Try
7/20 A's
3/10 A's
4/10 A's
OVERALL BY
SUBGROUP
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
(b)
H: 5 A's & 5 B's
L: 5 A's & 5 B's
MUST BLOCK TO MAKE
STRATIFICATION PAY
OFF
#52