ofner, imi determining the parameter settings of different randomization methods for specific study...

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

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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

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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

• …

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