Outline of
Randomization Lectures
1. Background and definitions
2. Generation of schedules
3. Implementation (to ensure allocation
concealment, sometimes called blinded
randomization)
4. Theory behind randomization
Readings
• Chapter 6 of Friedman, Furberg and DeMets
• Supplemental notes for week 3 on the class
web site
• Other papers are cited in the notes
Key Points
• A random process should be used to
generate treatment allocations or
assignments
• Treatment allocations should be concealed
until the time of randomization – “allocation
concealment” is critical to prevent selection
bias. Some refer to this as “blinded
randomization”. (It should not be confused
with blinding of treatments).
Randomization
Assignment of experimental units to
treatment by a random process such that
neither investigator nor patient knows the
treatment to be assigned at the time the
patient is registered.
Timing of Randomized Trials Considerations
• 1st patient (Chalmers)
• Strong degree of equipoise (collective
and individual uncertainty) exists
• Feasibility and timing
See Freedman B N Engl J Med 1987.
Advantages of Randomization
Bradford Hill:
1. Eliminates bias from treatment assignment
2. Balances known and unknown differences
between groups on average
3. More credible study
RA Fisher:
1. Assures validity of statistical tests (type 1
error)
Are the Groups Comparable (Table 1 in trial reports)
ESPRIT Study: Baseline Characteristics
(N Engl J Med 2009; 361: 1548-59)
IL-2
Control
40
40
Female (%)
19%
19%
19%
Non-white race (%)
25%
24%
25%
Median CD4+ cells/mm3 (IQR)
464
450
457 (372, 584)
Nadir CD4+ cells/mm3 (IQR)
200
194
197 (91, 306)
HIV-RNA < 500 copies (%)
79%
80%
80%
Prior clinical AIDS (%)
25%
27%
26%
Years prior ART (IQR)
4.1
4.3
2071
2040
Age (median years)
No. Randomized
Total
40
4.2 (2.2, 6.4)
Steps in Patient (Study Participant)
Registration (Randomization)
1. Patient requires treatment (participants screened for
risk factor eligibility)
2. Patient (participant) eligible for inclusion in trial
3. Clinician willing to randomize patient (participant)
4. Patient (participant) is willing to be randomized
(consent is obtained)
5. Patient (participant) formally entered in the trial
– Treatment assignment obtained from randomization list
(schedule)
– Case-report and other records completed to document
randomization
6. Treatment commences as soon as possible
Usual Sequence of Events in a
Randomized Clinical Trial
Determine eligibility+
Not eligible
Eligible
Obtain informed consent
yes
randomize
A
follow-up
no
B
+ In many trials consent must
also be obtained for screening.
Key Elements
of Informed Consent
• A fair explanation of the procedures to be followed,
and their purposes, including identification of any
procedures which are experimental
• A description of any participant discomforts and risks
reasonably to be expected
• A description of any benefits to the subject or to
others which may be expected
• A disclosure of any appropriate alternative procedures
that might be advantageous for the subject
Key Elements
of Informed Consent (cont.)
• An offer to answer any inquiries concerning the procedures
• Instructions to a subject concerning the freedom to withdraw
his/her consent and to discontinue participation in the project or
activity at any time without prejudice or explanation (this should
be balanced by a statement that emphasizes participation until
the end of the study to preserve integrity of research question)
• Reasons study may be stopped
• An explanation as to whether compensation and medical
treatment are available if physical injury occurs and, if so, what
it consists of or where further information may be obtained
• A statement describing the extent, if any, to which
confidentiality of records identifying the subject will be
maintained
• A commitment to share new findings that emerge.
See also Chapter 2 of Friedman, Furberg and DeMets
Informed Consent (cont.)
• Length of sample informed consents:
–
–
–
–
ESPRIT (8 pages) (experimental treatment: interleukin-2)
SMART (12 pages) (treatment strategy trial using approved drugs)
START (14 pages) (treatment strategy trial using approved drugs)
MRFIT (1 page)
• Comprehension (when assessed) by participants is low on
key items suggesting simpler, not longer, forms may be
better.
• Separate consent documents for stored specimens and
substudies
• Multiple reviews of consent: Institutional Review Board
(IRB), Ethics Committee and sponsor
Bad Allocation Schemes
1. Ward 1 receives Drug A; ward 2, Drug B
2. M, T, W - Drug A
TH, F - Drug B
3. Every other patient receives A
4. Drug A on odd days
5. Drug A to patients born Jan. - Jun.; Drug B, Jul. Dec.
AVOID SYSTEMATIC ALLOCATION
ALLOCATION CONCEALMENT (BLINDED
RANDOMIZATION IS CRITICAL
The Unit of Randomization is Not Always
the Individual Study Participant
• Right and left eye
• Kidneys from deceased donors
• Clusters of participants
– Clinical sites (e.g., interventions aimed at
adherence, informed consent, counseling to
avoid high risk behaviors)
– Households
– Schools
– Communities
Cluster Randomization
J Acquir Immune Defic Syndr 2006; 43 (Suppl):S41-S47
38 clusters of clinical sites
Randomized
Medication
Manager
+
Electronic
reminder
(N = 9)
Medication
Manager
Alone
(N = 10)
Electronic
reminder
Alone
(N = 10)
• 200-250 patients/cluster
• Embedded in treatment trial
Control
(N = 9)
Cluster Randomized Study of Short
Versus Standard (Long) Consent
Form
150+ clusters of clinical sites
Randomized
Short consent form
(N = 2,000)
Long consent form
( N=2,000)
Embedded in START study
A similar design is being used to study of on-site monitoring: research on research!
Cluster Randomization Considerations
• It is important that the unit of randomization be taken into
account both in the design and analysis (e.g., matched
pairs).
• Like randomization of individual participants, allocation
concealment (blinded randomization) is important.
• In some studies, individual consent is not required.
• Observations/measurements on different participants
within clusters cannot be considered independent.
• It is important to account for all members of the cluster in
the analysis (clusters are randomized but measurements
are usually on individuals).
• The between-group (cluster) variability has to be
accounted for in the analysis.
Example: Possible Bias
in Treatment Assignment
Distribution of Prognostic Variables
According to Treatment Assignment
At Least
One Variable
No.
Studies Maldistributed* (%)
Blinded randomization
57
14.0
Unblinded
randomization
45
26.7
Non-random assignment
or historical controls
43
58.1
* p<.05
Source: Chalmers et al., NEJM, 1983.
Treatment Results
by Type of Assignment
Percent
p<0.05
Average Treatment
Difference
(Case-Fatality Rate)
8.8
0.003 ± 0.008
Unblinded
randomization
24.4
0.052 ± 0.016
Nonrandom assignment
or historical controls
58.1
0.105 ± 0.017
Blinded randomization
Source: Chalmers et al., NEJM, 1983.
Outline of
Randomization Lectures
1. Background and definitions
2. Generation of schedules
3. Implementation (to ensure allocation
concealment, sometimes called blinded
randomization)
4. Theory behind randomization
Randomization Schedule
A list showing the order in which
subjects are to be assigned to the
various treatment groups
Categorization of Randomization
Schemes
1. Fixed Allocation
a. Simple randomization
b. Permuted block (restricted)
c. Permuted blocks of different sizes randomly mixed
(restricted)
2. Adaptive Allocation Methods
Treatments are assigned with probabilities which
change during the course of the trial
a. Baseline adaptive procedure
b. Response adaptive
Simple Randomization
The number and order of patients receiving
treatments A and B is determined by chance.
Example: Equal allocation
– Toss a coin: A = head, B = tails
– Random number table:
A = odd, B = even
(see next slide taken from Pocock, page 74)
– Uniform random number generator (equally probable
numbers between 0.0 and 1.0):
A if < 0.5
B if > 0.5
Produced by a process that gives results that can be viewed
as a finite piece of a completely random series of numbers – roughly
equal numbers of the digits 0-9..
Example: Non-uniform treatment
allocation
British Aspirin Study
2 treatments with allocation ratio 2:1 (Aspirin:no Aspirin)
Source: Br Med J, 296:313-16, 1988.
Treatment
Aspirin
No Aspirin
Don’t Use
Random Numbers
1,2,3,4,5,6
7,8,9
0
Disadvantage of Simple
Randomization
• Chance imbalance in numbers assigned to
each treatment
– At end of study
– At periodic looks (e.g., interim analyses)
Could result in loss of power and logistical
problems
Probability of Specified Treatment
Allocations Using Simple Randomization
(10 Patients)
Treatment A
0 (10)
1 (9)
2 (8)
3 (7)
4 (6)
5
Treatment B
10
9
8
7
6
5
(0)
(1)
(2)
(3)
(4)
Binomial
Probability
0.002
0.02
0.09
0.23
0.41
0.25
Binomial Probability
Total Number
of Patients
10
20
50
100
200
500
1000
Difference in Numbers
or More Extreme
Prob. ≥ 0.05
Prob. ≥ 0.01
2: 8
6: 14
18: 32
40: 60
86: 114
228: 272
469: 531
1: 9
4: 16
16: 34
37: 63
82: 118
221: 279
459: 541
To find N to obtain allocation ratio which
results in a Prob ≥ 0.05
For large sample sizes can use
normal approximation
Loss of Power Due to Chance Imbalance
Comparison of 2 Means
H0 : µA = µ B
HA : µA ≠ µB ;
µA - µB = 
N = NA + NB and r = NA / N
Z 1- =
Z 1- =


1+ 1
NA NB

1

Nr (1-r)
1/2
1/2
-Z
1-/2
-Z
1-/2
r
Power
0.5
0.6
0.7
0.8
0.9
0.90
0.88
0.84
0.74
0.49
1. Fixed Allocation Methods
Treatments are assigned with a pre-specified
probability
A. Simple randomization
B. Permuted blocks
Restricted
{
C. Permuted blocks
randomly mixed
Permuted Block Randomization
1) Divide patients into blocks of equal size
according to time they enter the study
2) Choose a block size
3) Write down all possible permutations
4) Randomly choose one
Advantages of Permuted Block
Randomization
1. Forces balance at end of study and during
patient accession
2. Reduces the likelihood of bias due to
changing patient characteristics during
course of study
3. Facilitates planning with regard to treatment
administration (resource planning)
Disadvantages
1. If investigators become aware of block size
(may be hard to mask in a non-blind study),
some assignments are known within
certainty,
e.g., block size 4:
A
A
The next assignments have to be B
2. From a theoretical point of view, analysis
more cumbersome (more on this later)
Example: Permuted Block
Randomization: Block Size = 4
Write down the 6 possible different sequences
of 2As and 2Bs and randomly choose one for
the 1st 4 patients, next 4, etc.
1
2
3
4
5
6
A
A
B
B
B
B
A
A
A
B
A
B
B
A
B
A
A
B
B
A
B
A
A
B
Unequal Allocation
To determine block size, consider the sum of the integers
which define the allocation ratio:
Example: Mt. Sinai Hypertension Trial (MSHT); 3 treatment
groups (K+, placebo, control) randomized 2:2:1
Use block size of 5 or multiples of 5
1. Generate all possible arrangements of numbers
1-5
2. Choose one at random
3. 1,2 = A; 3,4 = B; 5 = C
4. Repeat steps 2 and 3 as often as necessary
Randomly Mixed Permuted
Blocks
A solution to the problem of easily guessing
future assignments (particularly important in
non-blind trials.
Example: The Multiple Risk Factor
Intervention Trial (MRFIT) used randomly
mixed block sizes of 2, 4 and 6.
Mixing Blocks of Sizes 2 and 4
Block Size
Permutations
1
2
3
4
5
6
2
AB
BA
–
–
–
–
Two Step Procedure
4
AABB
BBAA
ABAB
BABA
ABBA
BAAB
1. Randomly choose block size
2. Randomly choose permutation within block size
Computer Program for Generating
Random Permuted Blocks of Different
Size
1. Set number of treatments.
2. Set number of stratum.
3. Set block sizes to be used considering allocation
ratio.
4. Randomly choose a block size (K).
5. Generate K uniform random numbers.
Computer Program for Generating
Random Permuted Blocks of Different
Size (cont.)
6. Order the random numbers carrying along the original
index (1-K).
7. Associate treatment codes with ordered index array.
8. Print the K random assignments.
9. Go to Step 4 and continue until desired number of
allocations have been generated.
Other Variations
• Mix block sizes with different probabilities. For
example,
Block size
2
4
6
Prob
1/4
1/4
1/2
• Flip a coin for 1st assignment and mix block sizes
afterwards
• Use a large block size initially (e.g., >8) and then
smaller block sizes (e.g., mix 2, 4, and 6)
Baseline Adaptive Randomization
Procedure
Def.: The probability of the next treatment
assignment is altered on the basis of the
previous assignments in order to achieve
better balance (biased coin).
Considerations:
1. Implementation (central)
2. Multiple treatments
3. Definition of lack of balance
Let D = No. of patients assigned to A - No. assigned to B
D=0
Simple randomization (P = 1/2)
D>0
Assign to B with prob. (P) > 1/2
D<0
Assign to B with prob. (1 - P) < 1/2
What should P be set equal to?
Note P = 1 corresponds to permuted blocks of size 2.
Example Baseline Adaptive
Randomization
• 20 patients are to be randomized, 1:1
allocation. After 10 patients, we have:
– 5A and 5B (D=0): use schedule with 1:1
allocation (e.g., created with simple
randomization)
– A is assigned
– 6A and 5B (D=1): use schedule with Prob (B) =
2/3 (could also be created with simple
randomization)
– B is assigned
– 6A and 6B (D=0): continue as above
This could be implemented by preparing 3 schedules in advance: 1) 1:1 allocation; 2) 2:1 allocation
favoring B; and 3) 1:2 allocation Favoring A. All could be prepared with simple randomization.
Suppose there are k treatments:
1) Rank treatments according to number of patients
(fewest to largest);
2) Assign next patient with probability
Response Adaptive
Randomization
The probability of the next treatment assignment is altered on
the basis of the responses of previous patients enrolled.
Motivation: More patients receive the “best” treatment
Rosenberger, Cont Clinical Trials, 1999;20:328-342.
Play the Winner Rule
• Assign 1st patient to either treatment
with probability ½.
• An observed success generates a
future trial using the successful
treatment on the next patient
• A failure generates a future trial on the
alternative treatment
Zelen, 1969 JASA
Play the Winner Rule
Patient
Accession No.
Treatment
Success/
Failure
1
A
S
2
A
S
3
A
F
4
B
S
5
B
F
6
A
S
7
A
S
8
A
S
Some Problems Are Apparent
• Not a randomized design
• Selection bias - investigator knows next
assignment
• Outcome may not be known when the next
patient is enrolled
Randomized
Play the Winner Rule
• Start MA and MB balls in urn (e.g., MA = MB for equal
probability of A or B.
• Draw a ball for each assignment and replace it
• Success on A: Add  balls of Type A and  balls of Type B
• Success on B: Add  balls of Type B and  balls of Type A
• Failure on A: Add  balls of Type A and  balls of Type B
• Failure on B: Add  balls of Type B and  balls of Type A
• >
less randomization (e.g.,  = 1 and  = 0)
• =
simple randomization
Wei and Durham, 1978 JASA
Conditions Under Which the RPW
Rule is Reasonable
• The therapies have been evaluated previously for toxicity
• Response is binary
• Delay in response is moderate, allowing adapting to take place
• Sample sizes are moderate (at least 50 subjects)
• Duration of the trial is limited and recruitment can take place during
the entire trial
• The trial is carefully planned with extensive computations done
under different models and initial urn compositions
• The experimental therapy is expected to have significant benefits
to public health if it proves effective
See Rosenberger, Cont Clinical Trials, 1999;20:328-342.
Extracorporeal Membrane Oxygenation
(ECMO Trials)
• Used RPW with MA = MB = 1;  = 1; and  = 0.
• Total 12 patients – 11 given ECMO (A balls) and all
survived; 1 given control (B balls) and died (1st patient
received ECMO and survived; the 2nd control and died; all
subsequent patients received ECMO) (Pediatrics 1985;
76:479-87)
• Two other trials followed:
– A trial done in Boston that used 1:1 randomization until 4 deaths on
an arm (phase 1); in phase 2, all patients assigned successful
treatment until 4 deaths or significant result. Combined phase 1
and 2 results were 1/29 deaths on ECMO and 4/10 deaths on
control (Pediatrics 1989; 84:957-63).
– A conventional trial in the UK done (30/93 deaths on ECMO and
54/92 on control after one year) (Pediatrics 1998; 101:1-10).
Other Considerations in Using a
Response Adaptive Randomization
Scheme
1. Multiple endpoints
2. Changing patient characteristics
3. Implementation, timing of response
4. Ethical concerns of investigators, e.g., suppose
allocation probability for treatment decreases from 1/2
to 1/10
5. More difficult to describe; less acceptable (credible) to
others
6. Most useful for large differences
Adaptive Designs in Clinical Trials
• Currently, research on adaptive designs, defined
as modifications to trials procedures and/or
statistical methods during the conduct of a study,
is a hot area.
• Play-the-winner is probably the earliest type of
adaptive design
• The term “adaptive design” is used very broadly
and includes:
–
–
–
–
–
–
–
Covariate adaptive randomization (e.g., minimization)
Sample size re-estimation
Group sequential designs
Drop-the-loser designs
Adaptive dose-finding
“Switchover designs” or dynamic treatment regimes
Phase II/III seamless designs
Requirements for Sound Treatment
Allocation Scheme According to Meinert
1. Assignment remains masked to the patient, physician and all other
clinic personnel until it is needed for initiation of treatment.
2. Future assignments cannot be predicted from past assignments.
3. Order of allocation is reproducible.
4. Methods for generation and administration of schedules are
documented.
5. Process used for generation has known mathematical properties.
6. Process provides a clear audit trail.
7. Departures from the established sequence of assignments can be
detected.
Reference: Meinert, Chapter 8
Key Requirements of a Randomization
Schedule
1. Unpredictability (allocation concealment)
2. Approximate balance (desired allocation
ratio) within strata
3. Extendability …
4. Well documented; reproducible
Why Paranoid – Mistakes Are
Embarrassing!
• Herpes simplex vaccine trial (N Engl J Med 2012):
“…owing to a programming error, the initial subjects were
randomized at a 3:1 ratio in favor of the HSV-vaccine
group.” (Instead of 1:1; N’s 4,577 vs 3,746)
• HIV trial (Arch Int Med 1995): “…an error occurred in the
randomization program. This error resulted in a
nonrandom excess of patients assigned to the zidovudine
arm…” (Analysis restricted to 617 instead of 830 patients)
• Anemia trial (AJKD 2002): no mention of error in trial
report. In FDA Medical Officer review, it is noted that
sponsor reported to FDA in 1999 that allocation ratio of 2:1
darbepoetin alpha vs epoetin had been reversed.
Preparation of Randomization
Schedules: Typical Situation
• Schedules are prepared in advance and kept in a safe
place (e.g., secured computer files)
• Block randomization used to ensure desired allocation
ratio
• Procedure should not permit investigators to determine
treatment assignments in advance (randomly mixed
permuted blocks) (less of an issue in double-blind
studies)
• Separate schedules prepared for different treatment
sites (clinics) and strata (more on this later)
• Entire procedure should be documented in writing and
audited