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

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. April, 2008 Sanofi-Aventis 2

Introduction  Background  Statistical concepts of “Two Stage Winner Design”  Distribution and mathematical details  Practical solutions  Example  Some key points  Summary  Computation demo April, 2008 Sanofi-Aventis 3

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 4

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? April, 2008 Sanofi-Aventis 5

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 April, 2008 Sanofi-Aventis 6

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 0 – H 0 :  1 =  2 = 0 versus H a :  1 > 0 or  2 > 0 – We assume  j = δ j under H a for j = 1 and 2 April, 2008 Sanofi-Aventis 7

Data Assumptions and Notation (2)  Test Statistics – Interim analysis is performed at time  – Final sample means: Y n (0) , Y n (1) , and Y n (2) (1) and X n1 – Interim sample means: X n1 (2) (don’t care the control) – Final test statistics for pair j : Z (j) = √( n/2  Y (j) - Y n 2 ) ( Y n (0) ) – “Winner” test statistic W is defined as (1) > X n1 W = Z (1) , if X n1 (2) , (1) < X n1 W = Z (2) , if X n1 (2)  W is not longer normally distributed April, 2008 Sanofi-Aventis 8

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 April, 2008 Sanofi-Aventis 9

Type-I Error Inflation Figure 2A: Overall Alpha With Interim Dose Selection Different Measurements: One-Sided Alpha = 0.025 0.050 rho = 1 0.045 rho = 0.9 rho = 0.8 rho = 0.7 0.040 rho = 0.6 rho = 0.5 0.035 rho = 0.4 Type I Error rho = 0.3 rho = 0.2 0.030 rho = 0.1 rho = 0 0.025 0.020 0.015 0.0 0.2 0.4 0.6 0.8 1.0 Information Time April, 2008 Sanofi-Aventis 10

Type-I Error Inflation Figure 2B: Adjusted Alpha With Interim Dose Selection Different Measurements: One-Sided Alpha = 0.025 0.025 rho = 0 rho = 0.1 rho = 0.2 0.020 Adjusted Type I Error rho = 0.3 rho = 0.4 rho = 0.5 rho = 0.6 rho = 0.7 0.015 rho = 0.8 rho = 0.9 rho = 1 0.010 0.0 0.2 0.4 0.6 0.8 1.0 Information Time April, 2008 Sanofi-Aventis 11

Distribution of W Density function of W is f W (w) = p f 1 ( w-w 1 ) + q f 2 (w -w 2 ) Where (1) > X n1 (2) ), q = 1 - p p = Pr ( X n1 w j = √ ( n/2σ Y ² ) δ j for j = 1, 2 f 1 ( w ) = ( 1/p ) Φ ( k 0 + kw ) υ ( w ) f 2 ( w ) = ( 1/q ) Φ ( - k 0 + kw ) υ ( w ) with k 0 = λ / √ ( 1- η² ) and k = η / √ ( 1- η² ) and η = 0.5ρ√τ λ = √ ( n 1 / ( 2σ X ² )) (  X 1 -  X 2 ) Note: f i ( w ) here are “skewed normal” April, 2008 Sanofi-Aventis 12

Normal Approximation  The mean and S.D. for f j are μ 1 = ( Λ / p ), σ 1 ² = 1 – λ η μ 1 - μ 1 ² μ 2 = ( Λ / q ), σ 2 ² = 1 + λ η μ 2 – μ 2 ² Λ = η / √( 2π ) exp{-(1/2) λ² }  f j can be approximated by N ( μ j , σ j 2 ) for j = 1, 2  Therefore, f W can be “replaced” by a linear combination of two normal density functions  When δ 1 = δ 2 = 0 (assume  X 1 =  X 2 ) or under H 0 μ 1 = μ 2 = μ 0 = η√(2/π) σ 1 = σ 2 = σ 0 = 1 - (2/π)η² April, 2008 Sanofi-Aventis 13

Normal Approximation: Density Function 0.4 0.0010 0.0005 0.3 Difference in Density Density 0.0000 0.2 -0.0005 0.1 -0.0010 0.0 -2 0 2 4 6 8 -2 0 2 4 6 8 w w April, 2008 Sanofi-Aventis 14

Normal Approximation: Type-I Error (Tail probability under H 0 ) 1.0 0.0002 0.8 0.0000 Tail Probability Under H0 Pr(W*>w) - Pr(Z>w) 0.6 -0.0002 0.4 -0.0004 0.2 -0.0006 0.0 -4 -2 0 2 4 -4 -2 0 2 4 w w April, 2008 Sanofi-Aventis 15

Normal Approximation: Power (Tail probability under H a ) 1.0 Difference in Tail Prob. of W and Its Normal Approximation 0.0002 0.8 0.0000 0.6 Tail Probability -0.0002 0.4 -0.0004 0.2 -0.0006 -0.0008 0.0 1 2 3 4 5 6 1 2 3 4 5 6 w w April, 2008 Sanofi-Aventis 16

Design a Study Using Normal Approximation  Interim sample size n 1 : it can be determined by assumed treatment difference and p from p=Φ ( λ ). n 1 = 2σ X ²z p ² /(  X 1 -  X 2 )²  Sample size under overall power – For δ 1 = δ 2 : • Given p n = ( z β + z α )² (  Y / δ )²{1 + √[1 - ( n 1 ρ² ) / ( π ( z β + z α )² (  Y / δ )²)]} • Given  n = 2( z β + z α )² ( σ Y / δ )²{1 - ρ²  / 2 π } – For δ 1  δ 2 n =2 ( μ 0 +σ 0 z α – μ 1 + σ 1 z β1 )² (  Y / δ 1 )² with two constraints 1. δ 1 / δ 2 = ( μ 0 +σ 0 z α – μ 1 + σ 1 z β1 ) / ( μ 0 +σ 0 z α – μ 2 + σ 2 z β2 ), 2. β = p β 1 + q β 2 April, 2008 Sanofi-Aventis 17

Confidence Interval and p-value  Confidence interval – For δ 1 = δ 2 Δ n - √(( 2σ Y ² ) / n ) ( μ 0 + σ 0 z p ) < δ < Δ n - √(( 2σ Y ² ) / n ) ( μ 0 + σ 0 z p ) – For δ 1  δ 2 , only “conditional” confidence interval can be provided Δ j n - √ (( 2σ Y ² ) / n ) ( μ j + σ j z p ) < δ < Δ j n - √(( 2σ Y ² ) / n ) ( μ j + σ j z p ) 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 z 1 or z 2 ), then Pr ( W > w ) = Pr [( W - μ 0 ) / σ 0 > ( w - μ 0 ) / σ 0 ] ≈ 1 – Φ [( w - μ 0 ) / σ 0 ] April, 2008 Sanofi-Aventis 18

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 = ( δ 1 + δ 2 ) / 2 – Iterative method: no closed-form solution from the formula of sample size April, 2008 Sanofi-Aventis 19

Algorithm for Iterative Method  Step 1: – Choose n (0) and n 1 based on the targeted p using the “naive method” with δ (0) = ( δ 1 + δ 2 ) /2 . 2 using δ 1 , δ 2 , n (0) , and τ (0) = n 1 / n (0 ) . – Then calculate β (0) 1 and β (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) + δ 2 ) /2 or δ (1) = ( δ 1 + δ (0) ) /2 depending on 1- β (0) < 1 - β or 1- β (0) > 1 - β . 2 using n (1) and τ (1) = n 1 / n (1 ) . – Then calculate β (1) 1 and β (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. April, 2008 Sanofi-Aventis 20