optimal allocation ratio in multi-arm clinical trials
TRANSCRIPT
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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?
![Page 2: Optimal Allocation Ratio in Multi-arm Clinical Trials](https://reader036.vdocuments.mx/reader036/viewer/2022062406/55b117a0bb61eb2d1e8b482d/html5/thumbnails/2.jpg)
MotivationMethods
ResultsDiscussion
Number of new chemical or molecular entities:
1991-95:
2112006-11: 151
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Number of new chemical or molecular entities:
1991-95: 211
2006-11: 151
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Number of new chemical or molecular entities:
1991-95: 2112006-11:
151
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Number of new chemical or molecular entities:
1991-95: 2112006-11: 151
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Estimated cost of bringing a new chemical or biological entity to
market:
1975:
$138 million
2005: $1,318 million
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Estimated cost of bringing a new chemical or biological entity to
market:
1975: $138 million
2005: $1,318 million
Martin Law Can we improve clinical trials by our choice of allocation ratio?
![Page 8: Optimal Allocation Ratio in Multi-arm Clinical Trials](https://reader036.vdocuments.mx/reader036/viewer/2022062406/55b117a0bb61eb2d1e8b482d/html5/thumbnails/8.jpg)
MotivationMethods
ResultsDiscussion
Estimated cost of bringing a new chemical or biological entity to
market:
1975: $138 million
2005:
$1,318 million
Martin Law Can we improve clinical trials by our choice of allocation ratio?
![Page 9: Optimal Allocation Ratio in Multi-arm Clinical Trials](https://reader036.vdocuments.mx/reader036/viewer/2022062406/55b117a0bb61eb2d1e8b482d/html5/thumbnails/9.jpg)
MotivationMethods
ResultsDiscussion
Estimated cost of bringing a new chemical or biological entity to
market:
1975: $138 million
2005: $1,318 million
Martin Law Can we improve clinical trials by our choice of allocation ratio?
![Page 10: Optimal Allocation Ratio in Multi-arm Clinical Trials](https://reader036.vdocuments.mx/reader036/viewer/2022062406/55b117a0bb61eb2d1e8b482d/html5/thumbnails/10.jpg)
MotivationMethods
ResultsDiscussion
Estimated cost of bringing a new chemical or biological entity to
market:
1975: $138 million
2005: $1,318 million
Martin Law Can we improve clinical trials by our choice of allocation ratio?
![Page 11: Optimal Allocation Ratio in Multi-arm Clinical Trials](https://reader036.vdocuments.mx/reader036/viewer/2022062406/55b117a0bb61eb2d1e8b482d/html5/thumbnails/11.jpg)
MotivationMethods
ResultsDiscussion
Multi-arm trials
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Multi-arm trials
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Two-arm trials
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Multi-arm trials
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
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.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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 ].
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
A brief aside...
Dunnett [1964] states without proof that the optimal allocation
ratio is R =√
K .
Why?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
A brief aside...
Dunnett [1964] states without proof that the optimal allocation
ratio is R =√
K .
Why?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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 .
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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:
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Example
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Example
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Example
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Example
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Example
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
Optimal allocation ratio was found for a fixed set of arguments,
then each argument was varied in turn, and the gains made were
noted.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
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
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
1 2 3 4 5
250
300
350
400
450
500
550
Active treatments = 3
R
tota
l sam
ple
size
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
1 2 3 4 5
350
400
450
500
550
600
650
Active treatments = 4
R
tota
l sam
ple
size
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
1 2 3 4 5
400
450
500
550
600
650
700
750
Active treatments = 5
R
tota
l sam
ple
size
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
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)
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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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.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
What about the rest of the time?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
What about the rest of the time?
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
ResultsDiscussion
What about the rest of the time?
While may not always markedly decrease sample size, increasing R
can still result in an ethical or financial improvement in clinical
trials.
Martin Law Can we improve clinical trials by our choice of allocation ratio?
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MotivationMethods
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)
Martin Law Can we improve clinical trials by our choice of allocation ratio?