Optimal Allocation Ratio in Multi-arm Clinical Trials

MotivationMethods

ResultsDiscussion

Can we improve clinical trials by our choice of

allocation ratio?

Martin Law

Martin Law Can we improve clinical trials by our choice of allocation ratio?

MotivationMethods

ResultsDiscussion

Number of new chemical or molecular entities:

1991-95:

2112006-11: 151


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1991-95: 211

2006-11: 151


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1991-95: 2112006-11:

151


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1991-95: 2112006-11: 151


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Estimated cost of bringing a new chemical or biological entity to

market:

1975:

$138 million

2005: $1,318 million


MotivationMethods

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market:

1975: $138 million

2005: $1,318 million


MotivationMethods

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market:

1975: $138 million

2005:

$1,318 million


MotivationMethods

ResultsDiscussion


market:

1975: $138 million

2005: $1,318 million


MotivationMethods

ResultsDiscussion


market:

1975: $138 million

2005: $1,318 million


MotivationMethods

ResultsDiscussion

Multi-arm trials


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Multi-arm trials


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Two-arm trials


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Multi-arm trials


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Multi-arm trial: Details

Total sample size N = Rn + Kn

Mean results of treatment i : X̄i ∼ N(µi ,σ2

n ), i = 1, . . . ,K

Test Statistic for treatment i :

Zi = X̄i−X̄0

σ√

(1/n)+(1/Rn)=

Sin− S0

Rn

σ√

(1/n)+(1/Rn)

Proceed to next phase if max{Zi} > C , for some critical value

C.


MotivationMethods

ResultsDiscussion

Type I error, Power

Type I error α = P(Proceed with any treatmentTi |H0 : µ0 =

µ1 = · · · = µK = 0)

Power (1− β) = P(Proceed with treatmentTK |HA : µ0 =

0, µ1 = · · · = µK−1 = δ0, µK = δ),

where δ > δ0 ≥ 0. δ is a clinically relevant improvement over the

control, while δ0 represents the largest improvement over control

which is not clinically relevant. This alternative hypothesis is

known as the least favourable configuration [Thall et al ].


MotivationMethods

ResultsDiscussion

Type I error, Power

α = P(Zi > C |H0) for any i ≥ 1

(1− β) = P(Zk > C , max(Zi ) = ZK |HA)

Recall that Zi = X̄i−X̄0

σ√

(1/n)+(1/Rn); Is there an optimal value of R,

which minimizes N for a given Type I error and power?


MotivationMethods

ResultsDiscussion

Type I error, Power

α = P(Zi > C |H0) for any i ≥ 1

(1− β) = P(Zk > C , max(Zi ) = ZK |HA)

Recall that Zi = X̄i−X̄0

σ√

(1/n)+(1/Rn); Is there an optimal value of R,

which minimizes N for a given Type I error and power?


MotivationMethods

ResultsDiscussion

A brief aside...

Dunnett [1964] states without proof that the optimal allocation

ratio is R =√

K .

Why?


MotivationMethods

ResultsDiscussion

A brief aside...

Dunnett [1964] states without proof that the optimal allocation

ratio is R =√

K .

Why?


MotivationMethods

ResultsDiscussion

Dunnett’s R

Consider the following simplification: The power is

P(ZK > C |HA) = P( X̄K−X̄0

σ√

(1/n)+(1/Rn)> C |HA).

To maximise power, must minimise 1n + 1

Rn , or equivalently,

minimise R+KN + R+K

RN , (as N = Rn + Kn).

ddN (R+K

N + R+KRN ) = 0⇒ R =

√K .


MotivationMethods

ResultsDiscussion

Recall

α = P(Zi > C |H0)

(1− β) = P(Zk > C ,max(Zi ) = ZK |HA)

Replacing Zi withSin− S0

Rn

σ√

(1/n)+(1/Rn), we may derive the following

equations:


MotivationMethods

ResultsDiscussion

1− α =

∫ ∞−∞

[Φ

(C

√R + 1

R+

x√R

)]Kφ(x)dx

1− β =

∫ ∞−∞

[Φ

(w +

√n

σ(δ − δ0)

)]K−1

Φ

(w√R +

√Rnδ

σ− C√R + 1

)φ(w) dw

Type I error depends only on K ,R and C . Thus by fixing the

number of active treatments and allocation ratio, we can find the

smallest critical value which satisfies a required type I error

probability. These values can then be used along with σ, δ, δ0, to

find the smallest total sample size which also satisfies a required

power.


MotivationMethods

ResultsDiscussion

Example


MotivationMethods

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Example


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Example


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Example


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Example


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Optimal allocation ratio was found for a fixed set of arguments,

then each argument was varied in turn, and the gains made were

noted.


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ResultsDiscussion

1 2 3 4 5

150

200

250

300

350

400

Active treatments = 2

R

tota

l sam

ple

size

Alpha0.0250.050.10.2


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1 2 3 4 5

250

300

350

400

450

500

550


R

tota

l sam

ple

size


MotivationMethods

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1 2 3 4 5

350

400

450

500

550

600

650


R

tota

l sam

ple

size


MotivationMethods

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1 2 3 4 5

400

450

500

550

600

650

700

750


R

tota

l sam

ple

size


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ResultsDiscussion

α

0.2 0.1 0.05 0.025

2 0.99 (2) 0.99 (2) 0.99 (2) 0.97 (8)

K 3 0.99 (3) 0.97 (9) 0.96 (14) 0.95(23)

4 0.98 (6) 0.95 (19) 0.94 (30) 0.93 (41)

5 0.96 (16) 0.95 (28) 0.92 (48) 0.90 (70)

Table: Sample size reduction achieved by choosing optimal allocation

ratio, as a proportion of sample size for 1 : 1 ratio (actual reduction in

brackets)


MotivationMethods

ResultsDiscussion

In many cases, optimal allocation ratio is <√

K ;

All cases, decrease in sample size achieved is ≤ 10%. Though

this was often equivalent to greater than 30 patients, similar

decreases may be made by simply choosing R = 2;

Reductions increase - in raw numbers and proportion - as

number of active treatments increase and as Type I error is

decreased;

Altering power, variance, δ and δ0 shows no proportional

increase in gains.


MotivationMethods

ResultsDiscussion

Can we increase R to greatly decrease total sample size?

Sometimes! Conditions which would be conducive:

The total sample size would otherwise be large;

There are many (≥ 5) active treatments;

The required Type I error is small.


MotivationMethods

ResultsDiscussion

Can we increase R to greatly decrease total sample size?

Sometimes! Conditions which would be conducive:

The total sample size would otherwise be large;

There are many (≥ 5) active treatments;

The required Type I error is small.


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ResultsDiscussion

What about the rest of the time?


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While may not always markedly decrease sample size, increasing R

can still result in an ethical or financial improvement in clinical

trials.


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ResultsDiscussion

References

Thall, P. F., Simon, R., and Ellenberg, S. S. (1988). Two-stage selection and

testing designs for comparative clinical trials. Biometrika 75, 303-310.

Dunnett, C.W. (1964). New Tables for Multiple Comparisons with a Control.

Biometrics 20, No. 3, (pp. 482-491)


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Optimal Allocation Ratio in Multi-arm Clinical Trials