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Randomization: Urn Designs

Office of AIDS Research, NIHICSSC, FHI Goa, India,

September 2009

Kenneth F. Schulz, PhD

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Biased-coin approaches Alter the allocation probability during the trial – rectify imbalances

Begin with simple randomization with equal probability of assignment– 0.50/0.50 in a 2-arm trial

Continue as long as the disparity between the numbers assigned to the treatment groups remains below a prespecified limit

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Biased-coin approaches (cont.)

If the disparity reaches the limit, increase the probability of assignment to the group with the least participants, e.g. to 0.60

Implemented properly, achieves balance while preserving most of the unpredictability of simple randomisation

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Adaptive bias-coin designs, the urn design

The most widely studiedAlter the probability of assignment based on the magnitude of the imbalanceUrn design designated as UD (α,β) – α the number of blue and green balls initially– β the number of balls added to the urn of the

opposite color to the ball chosen– α and β being any reasonable nonnegative

numbers

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Example of an urn design UD(2,1), an urn contains two blue balls and two green balls – i.e., 0.50/0.50 probabilities to begin (α=2)

Balls are drawn at random and replaced for treatment assignments– blue for treatment A– green for treatment B

One additional ball (β=1) of the opposite color of the ball chosen is added to the urn

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Start

URN

AABB

Assignments

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If a blue ball was chosen first in this urn design

Two blue balls and three green balls would be in the urn after the first assignment– 0.40/0.60 for the next assignment

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1st A Selected

URN

AABBB

Assignments

1. A

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If a blue ball was chosen second in this urn design

If another blue was chosen second, then two blue balls and four green balls would be in the urn after the second assignment– 0.33/0.67 for the next assignment

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2nd A Selected

URN

AABBBB

Assignments

1. A2. A

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If a green ball was chosen third

If a green ball was selected third, then three blue and four green balls would be in the urn after the third assignment– 0.43/0.57 for the next assignment

That drawing procedure repeats with each assignmentAllocation probabilities fluctuate with the previous assignments

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3rd B Selected

URN

AAABBBB

Assignments

1. A2. A3. B

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Urn Designs

“Urn designs" function particularly well to promote balance without forcing it

Urn designs usually have adequate balancing properties while still being less susceptible to selection bias than permuted-block designs– Never deterministic

Approaches simple randomisation with increasing trial size

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End