Determining the Parameter
Settings of Different
Randomization Methods for
Specific Study Designs
Petra Ofner-Kopeinig, Maximilian Errath
and Andrea Berghold
Institute for Medical Informatics, Statistics and Documentation
Medical University of Graz, Austria
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Motivating Example
• 200 patients to be included into the study
• Stratified by
– Gender (male, female)
– Treatment history (past, recent, none)
• Which randomization method should be used?
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Randomization Methods
•
•
•
•
•
•
•
Complete randomization
Biased Coin (Efron)
Big Stick (Soares & Wu)
Minimization (Taves; Pocock & Simon)
Urn Design (Wei)
Permuted Block Randomization (Matts & Lachin)
…
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Randomization Methods
• Allocation of treatment at random
• Achieve treatment group balance
• Potential for Selection Bias
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Choice of Randomization Method
• Smallest treatment imbalance at the end of the
study
• Maximum imbalance ever achieved over the
course of the study
• Compare different parameter settings of the
methods
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Definition of Imbalance
• Two treatment groups: relative frequency of the
absolute differences between groups
• Different treatment group sizes: differences
between expected and observed frequencies
• More than 2 treatment groups: maximum of
differences between expected and observed
frequencies
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Randomizer – Simulation Tool
• Developed at the Institute for Medical
Informatics, Statistics and Documentation,
Medical University Graz
• Web based software for randomization of multicenter clinical trials
• Trial Management
www.randomizer.at
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Simulation tool
The simulation tool can be used for:
• Generation of static randomization lists
• Validation
– FDA-Guidelines
– GCP-compliant  AGES Pharmed
• Simulation of different study designs
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Simulation tool
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Simulations
• Complete randomization
• Urn design with different parameters
– Ud011 (initial urn = 0, with replacement, balls to add = 1)
– Ud002 (initial urn = 0, without replacem balls to add = 2)
• Permuted block randomization with different
block lengths
– Pb6 (block length = 6)
– Pb20 (block length = 20)
• 1000 Trials
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Complete Randomization
• Balance behaviour can not be controlled in any
way
• Big differences between treatment groups are
possible
• Stratified Randomization: Randomization is
done within subgroups, that means for small
patient numbers
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Urn Design (1)
• Generalization of the Biased Coin Method.
• UD (, ), with or without replacement
–  = initial urn
–  = balls to add
• Inital urn contains for 2 treatments  white und  red
balls.
• Drawing a red ball means allocation of treatment X,
drawing a white ball allocation of treatment Y.
• After each drawing the ball is replaced to the urn or not
and  balls of the opposite colour are added to the urn.
• For each randomization step this procedure is repeaded.
•  > 0;  = 0  corresponds to complete randomization
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Urn Design (2)
•  = 0,  > 0: no difference in imbalance for any 
•  /   0: ud approaches cr
•  /   : urn randomization preserves balance
within small strata
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Permuted Block Randomization
• M blocks containing m = n/M patients
• M and n/M are positive integers
• Within block i, m/2 patients are assigned to
treatment A, m/2 patients are assigned to
treatment B
• Randomization is performed within blocks
• Maximum imbalance m/2
• Randomizer: length of blocks must be a multiple
of the number of treatments
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Example 1
- 200 patients
- Patients are stratified by gender (male, female)
and their treatment history (none, past, recent)
- Distribution of factors is not known, we expect a
uniform distribution
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Simulated Means and Variances of
the Treatment Group Imbalance at
study end
Method Mean Variance
,5005 ,0076
Pb6
,5002 ,0004
Pb20
,5000 ,0010
Ud011
,5006 ,0025
Ud002
,5003 ,0010
0,8
Probability
Cr
1
0,6
cr
pb6
pb20
ud011
ud002
0,4
0,2
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Absolute Differences
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Example 2
Treatment history
none
Gender Female
Male
past
recent
93
33
37
163
21
8
8
37
114
41
45
200
- 200 patients
- Patients are stratified by gender (male, female) and
their treatment history (none, past, recent)
- Distribution of strata is known
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Simulated Means and Variances of
the Treatment Group Imbalance
Method Mean Variance
,5025
,0138
Pb6
,5011
,0023
Pb20
,5007
,0066
Ud011
,4997
,0050
Ud002
,4998
,0022
Probability
Cr
1
0,8
0,6
0,4
cr
pb6
pb20
ud011
ud002
0,2
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Absolute Differences
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stratum
none, female
none, male
recent, female
recent, male
past, female
past, male
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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Maximal Imbalance ever achieved
stratum
none, female
none, male
recent, female
recent, male
past, female
past, male
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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Imbalance at Study End unstratified
stratum: none, female
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: none, female
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
pb 20
ud 011
ud 002
Randomization Method
stratum: none, female
stratum: none, female
Maximum Imbalance unstratified
35
Maximum Imbalance
pb 6
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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Imbalance at Study End unstratified
stratum: past, male
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: past, male
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
pb 20
ud 011
ud 002
Randomization Method
stratum: past, male
Maximum Imbalance unstratified
stratum: past, male
35
Maximum Imbalance
pb 6
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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Summary
• Effects of imbalances on power are small unless
imbalance is considered substantial (0.6 or 0.7 to one of
the two groups)
• For trials with n > 200 substantial treatment imbalances
are unlikely with complete randomization or urn design.
• Stratified block randomization: can result in treatment
imbalances in the trial due to incomplete blocks in some
strata.
• Urn design: balls to add / initial urn determines to what
degree balance is enforced
• Multicenter studies
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References
•
•
•
•
•
•
•
•
•
Efron, B., Forcing a sequential experiment to be balanced, Biometrika 57: 403-417,
1971
Lachin J.M., Statistical Properties of Randomization in Clinical Trials, Controlled
Clinical Trials 9: 289-311 (1988))
Lachin, J.M., Properties of Simple Randomization in Clinical Trials, Controlled Clinical
Trials 9: 312-326, 1988
Matts, J.P., Lachin, J.M., Properties of Permuted-Block Randomization in Clinical
Trials, Controlled Clinical Trials 9: 327-344, 1988
Wei, L.J., Lachin, J.M., Properties of the Urn Randomization in Clinical Trials,
Controlled Clinical Trials 9: 345-365, 1988
Taves, D.R., Minimization: a new method of assigning patients to treatment and
control groups, Clinical Pharmacol. Ther. 15: 443-453, 1974
Pocock, S.J., Simon, R., Sequential treatment assignment with balancing for
prognostic factors in the controlled clinical trial, Biometrics 31, 103-115, 1975
Soares, J.F., Wu, C.F.J., Some Restricted Randomization Rules in Sequential
Designs, Communications in Statistics: Theory and Methods 17, 2017-2034, 1983
…
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Imbalance at Study End unstratified
stratum: none, male
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: none, male
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
pb 20
ud 011
ud 002
stratum: none, male
Maximum Imbalance unstratified
stratum: none, male
35
Maximum Imbalance
pb 6
Randomization Method
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
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IMI
ud 002
Imbalance at Study End unstratified
stratum: recent, female
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: recent, female
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
stratum: recent, female
pb 20
ud 011
ud 002
Maximum Imbalance unstratified
stratum: recent, female
35
Maximum Imbalance
pb 6
Randomization Method
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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IMI
Imbalance at Study End unstratified
stratum: recent, male
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: recent, male
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
pb 20
ud 011
ud 002
Randomization Method
stratum: recent, male
Maximum Imbalance unstratified
stratum: recent, male
35
Maximum Imbalance
pb 6
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
Ofner,
IMI
Imbalance at Study End unstratified
stratum: past, female
Imbalance at Study End
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
stratum: past, female
35
30
25
20
15
10
5
0
ud 002
cr
Randomization Method
pb 20
ud 011
ud 002
Randomization Method
stratum: past, female
Maximum Imbalance unstratified
stratum: past, female
35
Maximum Imbalance
pb 6
30
25
20
15
10
5
0
35
30
25
20
15
10
5
0
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
cr
pb 6
pb 20
ud 011
Randomization Method
ud 002
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