A brief introduction to
randomisation methods
Peter T. Donnan
Professor of Epidemiology and Biostatistics
Treatment Allocation Methods
Overview
Fixed Methods:
Simple randomisation
Stratification
Minimisation
Adaptive Methods:
Urn randomisation
Biased Coin
Play-the-winner
Parallel-group
Randomised Controlled Trial
Eligible subjects
RANDOMISED
Intervention
Control
RANDOMISED CONTROLLED
TRIAL (RCT)
Random allocation to
intervention or control so
likely balance of all factors
affecting outcome
Hence any difference in outcome
‘caused’ by the intervention
Treatment Allocation Methods
Randomisation is main allocation
method in scientific experiments
First proposed by Fisher (1935)
‘The Design of Experiments’
Two Properties :
1. Unbiased allocation
2. Balances covariates, known
and unknown
Treatment Allocation Methods
Properties required for unbiased and efficient
treatment comparison:
1. Equal distribution of known covariates
2. Equal distribution of unknown, or
unmeasured, covariates
3. Balanced group size
Random allocation is the best means of
ensuring equal distribution of unknown
covariates
Random Allocation Methods
Simple Randomisation – 0,1 computer
generated list:
• Coin toss pr (A) = 0.5
• Least predictable method – can have
long runs of same treatment
• Risk of covariate imbalance especially
with short sequences i.e. small trials
PROCESS OF
RANDOMISATION
•Example - 5-digit Random numbers below
75792, 80169, 94071, 67970, 91577, 84334
03778, 58563, 29068, 90047, 54870, 23327
With two treatments can be converted to:
A
B
B
A
B
A
A
B
A
B
A
B
Where last digit even = A and odd = B
RANDOMISATION
Power of RCT is RANDOMISATION
•Facilitates blind objective unbiased assessment
of outcome – removes selection bias
•But note not necessarily what the patient wants
•Nor what the physician prefers
•In a sense trial patients acting
altruistically
PROCESS OF
RANDOMISATION
•Usually generate random numbers (statistical
software or spreadsheet)
•Note that to be GCP–compliance requires:
1. Record of seed used to generate the random
numbers so that list is replicable
2. Record of patient allocation
OUTCOME OF
RANDOMISATION
•Often just a randomised list ABBAAABB….
•To ensure treatment balance use randomised
blocks e.g. size 4 ABBA ABAB BAAB BABA
•Electronic 24 hr telephone randomisation may
be necessary or web-based
•Usually provided by a trials unit
Example of Parallel-group RCT
Sebag-Montefiore et al Lancet 2009; 373: 811-820
•Trial of short course preoperative radiotherapy vs. initial
surgery with selective postoperative chemotherapy for
operable rectal cancer (n=1350)
•Reduction of 61% of risk of local recurrence with
preoperative radiotherapy
•HR = 0.39 (95% CI 0.27, 0.58)
•Consistent evidence that short course preoperative
radiotherapy is effective treatment for operable rectal
cancer
Random Allocation Methods
Restricted methods improve balance but
more predictable:
• Permuted blocks – e.g BAAB ABAB ….
• Stratification
• Minimisation
• Biased coin
• Urn randomisation
• Optimal biased coin
Permuted Blocks
AABABBBA
BAABABBA
BBAAABAB
Guarantees balance in group size, at end
of each block
n
• Predictable, especially if block size
2
10
known
• Predictability depends on block size 20
• Randomly vary block size
• Start at random point in first block
• Details in protocol? No!
p
0.75
0.65
0.62
Stratified Randomisation
Randomise separately within each strata
• Each randomisation list should use restricted
methods e.g. permuted blocks
• Ensures balance of known prognostic factors
• Limited to two or three factors, strata multiply
• ICH-E9 recommends stratification by centre
• Group small centres
Stratified Parallel-group
Randomised Controlled Trial
Eligible subjects
Stratum
Mild
Moderate
Severe
RANDOMISED
Active
Control
Active
Control
Active
Control
Minimisation
Balances a number of known prognostic factors
Deterministic case of Pocock-Simon (1975) method
Start with simple randomisation and after, say n=56 we
end up with ……
Smoker
ARM A
ARM B
9
7
Non-smoker 17
23
Male
13
16
Female
13
14
Total
26
30
Minimisation
Minimisation
Next patient is female smoker then
GA = 1 x | 10-7 | + 1 x | 13-14 | = 4
GB = 1 x | 9-8 | + 1 x |12-15 | = 4
Smoker
ARM A
ARM B
9
7
Non-smoker 17
23
Male
14
16
Female
12
14
Total
26
30
Hence imbalance is equal so patient is allocated to
treatment by chance pr=0.5
Minimisation
What happens if give more weight to smoking (2)?
GA = 2 x | 10-7 | + 1 x | 13-14 | = 7
GB = 2 x | 9-8 | + 1 x |12-15 | = 5
ARM A
ARM B
Smoker
9
7
Non-smoker
17
23
Male
14
16
Female
12
14
Total
26
30
Hence patient is deterministically allocated to B
Minimisation
• Dynamic – uses allocations and patient
characteristics
• Deterministic – predictability high in theory,
low in practice?
• Uses categorical covariates
• Does require more complex programming
• But ensures balance on known factors
• TCTU system will incorporate minimisation
• ICH-E9 recommends a random element be
added pr (0.7-0.8)
Treatment- or ResponseAdaptive Randomisation
Biased Coin Randomisation
Urn randomisation
Play-The-Winner
Biased Coin
Adaptive technique, which is a modified version
of flipping a coin
• Start as simple randomisation, with an
unbiased (fair) randomisation process (pr =
0.5)
• If group size becomes unequal then the
probability of treatment allocation changes to
FAVOUR THE SMALLER GROUP
• Alter prob. so if arm A has nA < nB
• Then pr(A) > 0.5
• If arms are balanced then use equal
• Prob i.e. 0.5
Biased Coin
• Absolute difference generally used
e.g. if group size differs by >3, use ratio of 2:1
to favour smaller group
• `Big stick’ randomisation, force next patient
into smaller group
• Improves balance but becomes more can
become predictable
Urn Randomisation
More flexible Adaptive method is Urn
Randomisation
Probability of treatment assignment depends
on the magnitude of imbalance
Start with two coloured balls in an urn
• Sample with replacement
• Add extra ball of opposite colour to the one
selected each time
Urn Randomisation
Draw
Allocation
Etc…..
Urn Randomisation
Improves balance at start of trial
• simple randomisation as trial progresses
• Effect depends on no. balls at start and no.
balls added
e.g. start with 10 red, 10 blue for less effect,
add more balls each time for greater effect
• Useful for smaller trials
In larger trials urn randomisation eventually
behaves like complete simple randomisaton
Other Adaptive Methods –
Play-The-Winner
If one treatment is clearly inferior over time,
many patients getting the weaker drug
Zelen (1969) suggested classify each patient’s
outcome as ‘success’ or ‘failure’
Starts as simple randomisation
If patient ‘success’ then allocate same
treatment to next patient.
If patient ‘failure’ then allocate different
treatment to next patient
Play-The-Winner
Prob of patient’s allocation is unknown at start
and depends on prob of ‘success’
Trial stops when fixed pre-specified number of
‘failures’ observed or predetermined sample
size is reached
Benefit to the patient is more patients get
‘successful’ treatment
Play-The-Winner
Drawbacks:
Balance not an aim of method and imbalance
can occur
High susceptibility to selection bias
Recent outcomes determine subsequent
allocation and researcher can guess what next
assignment will be
Randomisation prob. changes over time
So time trend in outcome confounds treatment
effect and biases its estimate
Can magnify early random differences
Optimal Biased Coin
Dynamic, uses allocations and patient
characteristics
• Coin is biased so that next allocation
minimises treatment effect variance , based on
OLS regression model
• Continuous covariates
• Continuous outcome
• Non-deterministic
• Complex, matrix computation
Atkinson A.C, Statistics in Medicine, 1982
Summary
•
•
•
•
•
•
For large trials differences in methods have little
effect so stick to simple randomisation
Methods described mainly for smaller trials
Permuted –block randomisation prone to selection
bias – keeping these unknown reduces this potential
How much detail in protocol?
Omit size of blocks in protocol
Urn randomisation – use instead of blocks?
Urn randomisation reduces selection bias,
and protects against high imbalance better
than simple randomisation with small samples
Summary
• Optimal biased coin – an alternative to minimisation
but difficult to program – rarely used
• Adaptive methods more complex to program and
difficult to implement
• Stratification factors – No evidence base for
stratifying by many factors
• Always stratify by centre? Probably
• So fixed randomisation methods
generally preferable
Treatment allocation methods in clinical trials: a review.
Leslie A. Kalish and Colin B. Begg. Stats in Med, Vol.4, 129-144 (1985)
Remember
1.Primary goal of randomisation is
to guarantee independence
between treatment assignment
and outcome
2.Any other goals are secondary
(covariate balance, equal group
sizes,…)
TCTU
• TCTU randomisation system provides
simple, stratified and minimisation methods
of randomisation