1. A New Algorithm for Unequal Allocation Randomization
Wenle Zhao, PhD
Medical University of South Carolina, Charleston, SC, 29425, USA
Society for Clinical Trials 36th Annual Meeting
Arlington, VA, USA - May 17-20, 2015

2. Contents What do We Need? For unequal allocation randomization:
Why unequal allocation
What features do we need?
What algorithms do we have?
What are the problems?
A solution

3. What Features do We Need?
Accurately target the desired allocation
Want 1:√2:√3, not 1:2:3, 2:3:4, or 3:4:5.
Consistent imbalance control
|Achieved – Target | < MTI
High allocation randomness
Low allocation predictability
Simple formula for easy implementation
P(Ti = j) = F(X, Y, Z)

4. What Algorithms do We Have
Accurate Target
Consistent
Balancing
Random
Allocation
Easy Implement
Permuted Block Randomization No
Yes No
Yes
Block Urn Design - Zhao & Weng No
Yes No
Yes
Maximal Procedure -- Berger No
Yes
Yes No
Modified Urn Design
-- Rosenberger & Lachin No No
Yes
Yes
Brick Tunnel Randomization
-- Kuznetsova & Tymofyeyev
Yes
Yes No No
Complete Randomization
Yes No
Yes
Yes

5. The Problem of Permuted Block Randomization
Allocation 7:9
IM ≤ 1.5
Δ = 1.5
1:1
Block size = 4
DA = 33.3%
IM = n
2:3
Block size = 5
DA = 30.0%
IM= n
3:4
Block size = 7
DA = 22.9%
IM= n
7:9
Block size = 16
DA = 11.4%
IM = 5.53
Small block size 
high precision
low accuracy
Large block size 
low precision
high accuracy

6. Current Unequal Allocation Randomization Methods

7. Trade-off Between Desired Features
Large block size
Small block size
Accuracy
Precision
Accuracy
Precision
Accuracy
Precision

8. Mass-Weighted Urn Design
A Solution
Mass-Weighted Urn Design

9. The Procedure Target Allocation 1 : √2 : √3 = 0.2412 : 0.3411 : 0.4177
The urn starts with one ball for each arm. Mass of each ball is proportional to the target allocation ratio, in continuous format. Total mass in the urn = α, a pre-specified value.
Randomly draw a ball with probability proportional to the mass of each ball.
Assign the subject accordingly.
The mass of the ball is reduced by 1 unit.
This 1 unit mass is re-distributed to each arm based on the target allocation ratio.
Repeat steps 3 to 6 until the end of the study.

10. Conditional Probability
The Conditional Allocation Probability
as target allocation .
as treatment allocation sequence.
as achieved allocation after Ti.
as mass for the balls in the urn after Ti.
as conditional allocation probability for Ti.
Conditional Probability
Target Probability
Observed
Number
Expected
Number

11. Conditional Allocation Probability
Subject Treatment Assignment
1.0
Conditional Allocation Probability
0.342
0.278
0.382
R = 0.794
R = 0.542
R = 0.252
Random Variable
U(0,1)
Treatment
Allocation

12. Imbalance & Randomness in Unequal Allocation
Target Allocation Ratio: B
Achieved Distribution:
Target Distribution:
Treatment Imbalance:
Conditional Allocation Probability
Allocation Randomness A

13. Conditional Allocation Probability
Possible Scenario of Negative Mass
1.0
P3 = 0.696
P1= 0.304
-0.128
P3 = 0.786
P1= 0.342
Conditional Allocation Probability
R = 0.468
R = 0.252
Random Variable
U(0,1)
Treatment
Allocation

14. Performance Comparison

15. Summary Accurately target the desired allocation
Directly target any unequal allocation
Consistent imbalance control
Contained by parameter alpha
High allocation randomness
Achieved by parameter alpha
Simple formula for easy implementation

16. Summary Mass Weighted Urn Design Yes Yes Yes Yes
Accurate Target
Consistent
Balancing
Random
Allocation
Easy Implement
Mass Weighted Urn Design
Yes
Yes
Yes
Yes
Permuted Block Randomization No
Yes No
Yes
Block Urn Design - Zhao & Weng No
Yes No
Yes
Maximal Procedure -- Berger No
Yes
Yes No
Modified Urn Design
-- Rosenberger & Lachin No No
Yes
Yes
Brick Tunnel Randomization
-- Kuznetsova & Tymofyeyev
Yes
Yes No No
Complete Randomization
Yes No
Yes
Yes

17. Discussion Unconditional allocation probability for each assignment
Formula for the maximal imbalance
Asymptotic allocation convenience
Knowledge dissemination

18. Acknowledgement
This research is partly supported by following NIH/NINDS grants:
U01NS (NETT Palesch, Y. & Durkalski, V.)
U01NS (StrokeNet, Palesch, Y. & Zhao, W.)

19. Thank You!
Contact me at:

20. Performance Comparison
Table 2. Performance comparison between MWUD and alternative designs
Optimal allocation = 1: √2 : √3 , sample size = 100, simulation = 50,000 per scenario
Randomization design
Target allocation
Parameter
Average Allocation Predictability *
Average Treatment Imbalance †
Complete randomization
4.8072
Modified Urn Design
α = 1, β = 1
0.0586
3.9141
Permuted Block Randomization
(block size = b)
2 : 3 : 4
b = 9
0.2841
1.9584
5 : 7 : 8
b = 20
0.2121
1.7374
10 : 14 : 17
b = 41
0.1378
1.8466
Mass Weighted Urn Design
α = 2
0.3480
0.7747
α = 4
0.2501
1.0268
α = 6
0.2032
1.2359
α = 8
0.1747
1.4134
†: Calculated based on the Euclidean distance between the desired treatment distribution and the achieved treatment distribution, averaged across the randomization sequence and the simulation runs.
*: Calculated based on the Euclidean distance between the desired allocation probability and the conditional allocation probability, averaged across the randomization sequence and the simulation runs.

22. Equipoise + Maximum Power  Equal Allocation

23. Maximal Treatment Imbalance

24. Equipoise + Maximum Power  Equal Allocation
0.564
0.522

25. Change Allocation Ratio
Bayesian Adaptive Design  Unequal Allocation
Change Allocation Ratio