Interim Treatment Selection In Clinical Trials Zhenming Shun, - - PowerPoint PPT Presentation

Interim Treatment Selection In Clinical Trials

Zhenming Shun, Gordon Lan, Yuhwen Soo Sanofi-Aventis, J & J, Regeneron

April, 2008 Sanofi-Aventis 2

References

 “Interim Treatment Selection Using Normal

Approximation Approach in Clinical Trials”, 2008, Zhenming Shun, Gordon Lan, and Yuhwen Soo, Statistics in Medicine, 27:597–618

 “Three-Pick-One Two-Stage Winner Design”,

2006, Yuhwen Soo, Lin Wang, and Zhenming Shun, Technical Report #4, Sanofi-Aventis

 “Normal Approximation for Two-Stage

Winner Design”, 2006, Gordon Lan, Yuhwen Soo, and Zhenming Shun. Random Walks, Sequential Analysis and Related Topics. World Scientific: Singapore: 28–43.

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Introduction

 Background  Statistical concepts of “Two Stage Winner

Design”

 Distribution and mathematical details  Practical solutions  Example  Some key points  Summary  Computation demo

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Why We Need Interim Dose Selection?

 Do we have a good surrogate for heart failure

clinical event?

– Not really

 Can we afford a large scale, long-term phase II

study for dose selection?

– Not in general

 What dose we propose for a phase III study?  Is one dose enough for marketing need?

– No need to have multiple doses in some cases

 This approach is not for phase II dose ranging

studies

April, 2008 Sanofi-Aventis 5

A Strategy for Phase III

 “Two-Stage Winner Design”

– Start with 2 doses and a Control – One interim analysis is planned – Drop the worse dose at interim – Continue the better (winner) dose to the second period – Final analysis based on all data in Winner and Control groups

 Pros and Cons

– Pros

• Combine dose selection and adaptive design at interim
• Efficient: Save time and resources

– Cons --- our research is intended to provide solutions

• Type-I error is inflated
• What is the distribution of the test statistic?

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Combining Three Concepts

 Three statistical concepts

– Adaptive design: Modify study design based on interim data – Multiple comparison adjustment (Dunnett’s test): Many-to-one comparison – Winner selection (Simon, 1985)

• Selection is based on numerical comparison
• Not intended to reject null hypothesis (control type-I error)

 Nature of “Two-Stage Winner Design”

– Earlier the interim analysis, more “adaptability” – Later the interim analysis, more “multiplicity” adjustment

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Data Assumptions and Notation (1)

 Assumptions

– Two treatment groups (j = 1, 2) and one control (j = 0) – Primary data has a distribution of N(Y

j,Y),

– Interim data has a distribution of N(X

j,X),

– The correlation between X and Y is ρ (ρ = 1 when X is part of Y) – May apply to any normally distributed test statistics

 Hypothesis

– Two pairs of treatment differences:

1 = Y

1 - Y 0 and 2 = Y 2 - Y

– H0: 1 = 2 = 0 versus Ha: 1 > 0 or 2 > 0 – We assume j = δj under Ha for j = 1 and 2

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Data Assumptions and Notation (2)

 Test Statistics

– Interim analysis is performed at time  – Final sample means: Yn

(0), Yn (1), and Yn (2)

– Interim sample means: Xn1

(1) and Xn1 (2) (don’t care the

control) – Final test statistics for pair j: Z(j) = √(n/2Y

2) (Yn (j) - Yn (0))

– “Winner” test statistic W is defined as W = Z(1), if Xn1

(1) > Xn1 (2),

W = Z(2), if Xn1

(1) < Xn1 (2)

 W is not longer normally distributed

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Type-I Error Inflation

 If use the same rejection region as specified in the situation

without interim dose selection, the type-I error will be inflated

 Must consider 1-sided test: Two-sided test will not change

the type-I error rate

 Inflation will depend on  and ρ

Type-I error rate (nominal α = 0.025)

Correlation ρ Time τ 0.00 0.25 0.50 0.75 1.00 25% 0.025 0.028 0.031 0.034 0.036 50% 0.025 0.029 0.033 0.037 0.040 75% 0.025 0.030 0.035 0.039 0.043 100% 0.025 0.031 0.036 0.041 0.045

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Type-I Error Inflation

Information Time Type I Error

0.0 0.2 0.4 0.6 0.8 1.0 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

rho = 0

Figure 2A: Overall Alpha With Interim Dose Selection Different Measurements: One-Sided Alpha = 0.025

rho = 0.1 rho = 0.2 rho = 0.3 rho = 0.4 rho = 0.5 rho = 0.6 rho = 0.7 rho = 0.8 rho = 0.9 rho = 1

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Type-I Error Inflation

Information Time Adjusted Type I Error

0.0 0.2 0.4 0.6 0.8 1.0 0.010 0.015 0.020 0.025

rho = 0

Figure 2B: Adjusted Alpha With Interim Dose Selection Different Measurements: One-Sided Alpha = 0.025

rho = 0.1 rho = 0.2 rho = 0.3 rho = 0.4 rho = 0.5 rho = 0.6 rho = 0.7 rho = 0.8 rho = 0.9 rho = 1

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Distribution of W

Density function of W is fW(w) = p f1(w-w1) + q f2(w-w2) Where p = Pr (Xn1

(1) > Xn1 (2)), q = 1 - p

wj = √(n/2σY²) δj for j = 1, 2 f1(w) = (1/p) Φ(k0 + kw) υ(w) f2(w) = (1/q) Φ(- k0 + kw) υ(w) with k0 = λ / √(1-η²) and k = η / √(1-η²) and η = 0.5ρ√τ λ = √(n1/(2σX²)) ( X

1 - X 2)

Note: fi(w) here are “skewed normal”

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Normal Approximation

 The mean and S.D. for fj are

μ1 = (Λ / p), σ1² = 1 – λ η μ1 - μ1² μ2 = (Λ / q), σ2² = 1 + λ η μ2 – μ2² Λ = η / √(2π) exp{-(1/2) λ²}

 fj can be approximated by N(μj , σj 2) for j = 1, 2  Therefore, fW can be “replaced” by a linear

combination of two normal density functions

 When δ1 = δ2 = 0 (assume X

1 = X 2 ) or under H0

μ1 = μ2 = μ0 = η√(2/π) σ1 = σ2 = σ0 = 1 - (2/π)η²

April, 2008 Sanofi-Aventis 14

Normal Approximation: Density Function

• 2

2 4 6 8 w 0.0 0.1 0.2 0.3 0.4 Density

• 2

2 4 6 8 w

• 0.0010
• 0.0005

0.0000 0.0005 0.0010 Difference in Density

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Normal Approximation:

Type-I Error (Tail probability under H0)

• 4
• 2

2 4 w 0.0 0.2 0.4 0.6 0.8 1.0 Tail Probability Under H0

• 4
• 2

2 4 w

• 0.0006
• 0.0004
• 0.0002

0.0000 0.0002 Pr(W*>w) - Pr(Z>w)

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Normal Approximation:

Power (Tail probability under Ha)

1 2 3 4 5 6 w 0.0 0.2 0.4 0.6 0.8 1.0 Tail Probability 1 2 3 4 5 6 w

• 0.0008
• 0.0006
• 0.0004
• 0.0002

0.0000 0.0002 Difference in Tail Prob. of W and Its Normal Approximation

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Design a Study Using Normal Approximation

 Interim sample size n1: it can be determined by assumed

treatment difference and p from p=Φ(λ). n1 = 2σX²zp²/(X

1 - X 2)²  Sample size under overall power

– For δ1 = δ2:

• Given p

n = (zβ + zα)² (Y / δ)²{1 + √[1 - (n1ρ²) / (π (zβ + zα)² (Y / δ)²)]}

• Given 

n = 2(zβ + zα)² (σY / δ)²{1 - ρ²  / 2π } – For δ1  δ2 n =2 (μ0 +σ0 zα – μ1 + σ1 zβ1 )² (Y / δ1)² with two constraints

• 1. δ1 / δ2 = (μ0 +σ0 zα – μ1 + σ1 zβ1) / (μ0 +σ0 zα – μ2 + σ2 zβ2),
• 2. β = p β1 + q β2

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Confidence Interval and p-value

 Confidence interval

– For δ1 = δ2

Δn - √((2σY²) / n) (μ0 + σ0 zp) < δ < Δn - √((2σY²) / n) (μ0 + σ0 zp)

– For δ1  δ2, only “conditional” confidence interval can be provided

Δj

n - √((2σY²) / n) (μj + σj zp) < δ < Δj n - √((2σY²) / n) (μj + σj zp)

where j = 1 or 2 depending on the interim selection. – Using interim estimates for λ in the calculation

 P-value calculation: Assuming w is the observed

value of W (either z1 or z2), then

Pr (W > w) = Pr [(W - μ0) / σ0 > (w - μ0) / σ0] ≈ 1 – Φ[(w - μ0) / σ0]

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Practical Solutions

 Choose p or τ first

– Choose p first to maintain targeted probability for selecting a good dose – Choose τ first to perform the interim analysis at right time

 Sample size calculation

– Naïve method: Using δ = (δ1 + δ2) / 2 – Iterative method: no closed-form solution from the formula of sample size

April, 2008 Sanofi-Aventis 20

Algorithm for Iterative Method

 Step 1:

– Choose n(0) and n1 based on the targeted p using the “naive method” with δ(0) = (δ1 + δ2)/2 . – Then calculate β(0)

1 and β(0) 2 using δ1, δ2, n(0), and τ(0) = n1 / n(0).

– If β(0) = p β(0)

1 + q β(0) 2 ≈ β, stop and choose n = n(0). Otherwise,

move to next step.

 Step 2:

– Calculate n(1) using either δ(1) = (δ(0) + δ2)/2 or δ(1) = (δ1 + δ(0))/2 depending on 1- β(0) < 1 - β or 1- β(0) > 1 - β. – Then calculate β(1)

1 and β(1) 2 using n(1) and τ(1) = n1 / n(1).

– If β(1) = p β(1)

1 + q β(1) 2 ≈ β, stop and choose n = n(1). Otherwise,

repeat this step to adjust n(1) to n(2).

 Continue these steps until the power reaches the targeted

level.

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Example

 Assumptions:

– δ1 = 0.7 and δ2 = 0.5, σY = 1 – Power = 90%, α = 0.025 (one sided) – p = 0.75

 Sample size

– Naïve

• n1 = (2σY²zp²) / (δ₁- δ₂)² = 22.8 ≈ 23
• n = 54.5 ≈ 55

– Iterative

• n = 50
• τ = 0.46 (n1 = 23)

 We can also fixed  = 0.5

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Example (Continued)

 Confidence interval and p-value

– Sample sizes: n = 50, n₁= 23 (p = 0.75 and τ = 0.46) – Interim: X23

(1) - X23 (2) = Y23 (1) - Y23 (2) = 1.136 - 0.80 =

0.336 > 0

• Using 0.336 in the calculation of λ

– Final: Δδ1 = Y50

(1) – Y50 (0) = 1.214 - 0.601 = 0.613

– P-value: Pr(W > w) ≈ 1 - Φ[(w - μ0) / σ0] = 0.00185 < 0.025 – 90% Confidence Interval: LL1 = 0.212 and UL1 = 1.014

Some Key Points (1)



Overall power, not the power for individual arm are considered

Power, Alpha, and Sample Size of Dunnett's Test (Overall power = 90%) #TrTs δ/σ Individual α Individual Power m n 1 0.5 0.050 90.0% 59 59 2 0.5 0.027 78.0% 68 50 3 0.5 0.019 70.1% 74 46 4 0.5 0.015 64.6% 78 45

_________________________________________________________________________________________________________________________________________________

Note: m = Sample size for individual power of 90% n = Sample size for overall power of 90%

Some Key Points (2)

 Conflict between p and τ

Interim Sample Size vs Treatment Effects and Probability of Winner Selection

(δ1 ,δ2) (0.55, 0.50) (0.70, 0.50) (1.00, 0.50) p 0.55 / 0.65 / 0.75 0.55 / 0.65 / 0.75 0.55 / 0.65 / 0.75 n₁ 13 / 119 / 364 1 / 8 / 23 1 / 2 / 4 τ 0.17 / >1 / NA 0.02 / 0.14 / 0.46 0.02 / 0.04 / 0.10 n 75 / 46* / NA 64 / 58 / 50 60 / 51 / 39

_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

*The sample size for the whole study is not correct due to large n1 for the interim analysis

Some Key Points (3)

 Difference from conventional “multiple

comparisons consideration”: Only one of the two doses will be selected

 Assumption of normal distribution in the data is

not necessary as long as the test statistics are (asymptotically) normal

– However, the correlation of the final test statistic and interim statistic need to be estimated

April, 2008 Sanofi-Aventis 26

Some Key Points (4)

 Results available for 3 or more treatment groups

– Calculation is much simpler under H0 – Recommend using targeted power for each comparison (not need to re-calculate the power)

 Any better method for the interim dose selection?

– Should we only be interested in a “meaningful difference” at the interim for selection? – What if the two are very close at interim? – What if we change our mind (do not want to drop an arm) at the interim?

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Summary

 The “Two-Stage Winner Design” is a

combination of “adaptive design”, “multiple comparison”, and “winner selection”

 Distribution of the test statistic is not

normal but the closed forms are provided

 Good approximation by normal

distributions (skewed normal)

 Can be practically implemented easily

without complicated numerical integration

April, 2008 Sanofi-Aventis 28

Computation Demo

By Philip He, sanofi-aventis