Ofner, IMI
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
Ofner, IMI
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 multi-center 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
Cr ,5005 ,0076
Pb6 ,5002 ,0004
Pb20 ,5000 ,0010
Ud011 ,5006 ,0025
Ud002 ,5003 ,00100
0,2
0,4
0,6
0,8
1
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Absolute Differences
Pro
ba
bili
ty
cr
pb6
pb20
ud011
ud002
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Example 2
- 200 patients- Patients are stratified by gender (male, female) and
their treatment history (none, past, recent)- Distribution of strata is known
Treatment history
none past recent
Gender Female 93 33 37 163
Male 21 8 8 37
114 41 45 200
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Simulated Means and Variances of the Treatment Group Imbalance
Method Mean Variance
Cr ,5025 ,0138
Pb6 ,5011 ,0023
Pb20 ,5007 ,0066
Ud011 ,4997 ,0050
Ud002 ,4998 ,0022
0
0,2
0,4
0,6
0,8
1
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Absolute Differences
Pro
bab
ility
cr
pb6
pb20
ud011
ud002
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ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
past, malepast, femalerecent, malerecent, femalenone, malenone, female
stratum
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imal
Imb
alan
ce e
ver
ach
ieve
d
past, malepast, femalerecent, malerecent, femalenone, malenone, female
stratum
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: none, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: none, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: none, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: none, female
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: past, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: past, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: past, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: past, male
Ofner, IMI
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
• …
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: none, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: none, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: none, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: none, male
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: recent, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: recent, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: recent, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: recent, female
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: recent, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: recent, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: recent, male
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: recent, male
Ofner, IMI
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Imb
alan
ce a
t S
tud
y E
nd
stratum: past, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0Imb
alan
ce a
t S
tud
y E
nd
un
stra
tifi
ed
stratum: past, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
stratum: past, female
ud 002ud 011pb 20pb 6cr
Randomization Method
35
30
25
20
15
10
5
0
Max
imu
m Im
bal
ance
un
stra
tifi
ed
stratum: past, female