Optimal Two-Stage Adaptive Enrichment Designs, using Sparse Linear Programming

Transcription

1 Optimal Two-Stage Adaptive Enrichment Designs, using Sparse Linear Programming Michael Rosenblum Johns Hopkins Bloomberg School of Public Health Joint work with Xingyuan (Ethan) Fang and Han Liu Princeton University Working paper giving full results available here: October, 05

2 Problem Motivation Goal: Testing Treatment Effects in Two Subpopulations and the Overall Population in Adaptive Enrichment Designs Example: Treating resistant HIV. Recent HIV drugs (maraviroc, raltegravir) have shown stronger benefit in those with lower phenotypic sensitivity to background therapy. We assume two, predefined, subpopulations that partition the overall population.

3 Multiple Testing Problem: Null Hypotheses Definition Define three treatment effects of interest: : Mean Treatment Effect for Subpopulation (i.e., difference between population mean of the primary outcome under treatment and under control) : Mean Treatment Effect for Subpopulation C = p + ( p ) : Mean Treatment Effect for Combined Population

4 Multiple Testing Problem: Null Hypotheses Definition Define three treatment effects of interest: : Mean Treatment Effect for Subpopulation (i.e., difference between population mean of the primary outcome under treatment and under control) : Mean Treatment Effect for Subpopulation C = p + ( p ) : Mean Treatment Effect for Combined Population Goal: construct adaptive enrichment design D and multiple testing procedure M for: H 0 : 0, H 0 : 0, H 0C : p + ( p ) 0, that strongly controls familywise Type I error rate, and is optimal in sense defined below.

5 General Two-Stage Adaptive Enrichment Design Stage Decision Stage Hyp.Test Recruit Both Popula)ons Subpopula)on Subpopula)on D... Subpopula)on : Subpopula)on : Subpopula)on : Subpopula)on : (D=) n (D=) n (D=) n (D=) n M K Subpopula)on : Subpopula)on : (D=K ) n (D=K ) n

6 Example Two-Stage Adaptive Enrichment Design Stage Decision Stage Stop trial Hyp.Test Recruit Both Popula7ons Subpop Subpop D Recruit Both Pop. Subpop Subpop Recruit Only Subpop. T > T? Subpopula7on M 4 Recruit Only Subpop. Subpopula7on

7 Example Two-Stage Adaptive Enrichment Design n = total sample size if both subpopulations enrolled in stage. Stage Decision Stage Stop trial Hyp.Test Recruit Both Popula7ons Subpop Subpop } n D 4 } Recruit Both Pop. Subpop n Subpop Recruit Only Subpop. T > T? Subpopula7on } n 4 M } Recruit Only Subpop. n Subpopula7on 4

8 Two-Stage Adaptive Enrichment Design Assume known variances and normally distributed outcomes; subpopulation cumulative sample sizes and z-statistics and are then sufficient statistics for,, C.

9 Two-Stage Adaptive Enrichment Design Assume known variances and normally distributed outcomes; subpopulation cumulative sample sizes and z-statistics and are then sufficient statistics for,, C. Z s () : z-statistic for subpopulation s at end of stage ; Z s (F ) : Final, cumulative z-statistic for subpop. s at end of stage.

10 Two-Stage Adaptive Enrichment Design Assume known variances and normally distributed outcomes; subpopulation cumulative sample sizes and z-statistics and are then sufficient statistics for,, C. Z s () : z-statistic for subpopulation s at end of stage ; Z s (F ) : Final, cumulative z-statistic for subpop. s at end of stage. Decision rule D is map from Z () = (Z (), Z () ) to possible decisions D. Multiple testing procedure M is map from Z (F ) = (Z (F ), Z (F ) ) and decision D to set of null hypotheses rejected (if any).

11 Two-Stage Adaptive Enrichment Design Assume known variances and normally distributed outcomes; subpopulation cumulative sample sizes and z-statistics and are then sufficient statistics for,, C. Z s () : z-statistic for subpopulation s at end of stage ; Z s (F ) : Final, cumulative z-statistic for subpop. s at end of stage. Decision rule D is map from Z () = (Z (), Z () ) to possible decisions D. Multiple testing procedure M is map from Z (F ) = (Z (F ), Z (F ) ) and decision D to set of null hypotheses rejected (if any). Power at alternative, to reject H 0 is Pr (, )[M(Z (F ), D(Z () )) rejects H 0 ]. User specifies: (i) loss function L(D, M;, ), e.g., total sample size; and (ii) distribution Λ on alternatives (, ).

12 Constrained Bayes Optimization Problem Problem inputs: p ; set of possible stage decisions; σ, σ ; clinically meaningful min. treatment effect min ; loss function L; distribution Λ on alternatives (, ); α, β, β, β C. Recall D = D(Z () ) and M = M(Z (F ), D(Z () )).

13 Constrained Bayes Optimization Problem Problem inputs: p ; set of possible stage decisions; σ, σ ; clinically meaningful min. treatment effect min ; loss function L; distribution Λ on alternatives (, ); α, β, β, β C. Recall D = D(Z () ) and M = M(Z (F ), D(Z () )). Constrained Bayes Opt. Problem: Find pair (D, M) minimizing: E (, )[L(D, M;, )]dλ(, ), under familywise Type I error constraints: sup Pr (, )[M rejects any true null hypothesis] α, (, ) R

14 Constrained Bayes Optimization Problem Problem inputs: p ; set of possible stage decisions; σ, σ ; clinically meaningful min. treatment effect min ; loss function L; distribution Λ on alternatives (, ); α, β, β, β C. Recall D = D(Z () ) and M = M(Z (F ), D(Z () )). Constrained Bayes Opt. Problem: Find pair (D, M) minimizing: E (, )[L(D, M;, )]dλ(, ), under familywise Type I error constraints: sup Pr (, )[M rejects any true null hypothesis] α, (, ) R and power constraints: Pr ( min,0)[m rejects H 0 ] β. Pr (0, min )[M rejects H 0 ] β. Pr ( min, min )[M rejects H 0C ] β C.

15 Our Method to Solve Optimization Problem Computational challenge: continuum of Type I error constraints. (It s not enough to satisfy only at global null (, ) = (0, 0).)

16 Our Method to Solve Optimization Problem Computational challenge: continuum of Type I error constraints. (It s not enough to satisfy only at global null (, ) = (0, 0).) Our solution: Discretize decision region R into small rectangles R; for any r R, enforce decision rule D makes same decision for any (Z (), Z () ) r. For each decision d {,..., K}, discretize rejection regions R into small rectangles R d ; for any r R d, enforce that if D = d, multiple testing procedure M rejects same set of hypotheses for any (Z (F ), Z (F ) ) r. Discretize Type I error constraints into fine grid on boundaries of null spaces.

17 Our Method to Solve Optimization Problem Computational challenge: continuum of Type I error constraints. (It s not enough to satisfy only at global null (, ) = (0, 0).) Our solution: Discretize decision region R into small rectangles R; for any r R, enforce decision rule D makes same decision for any (Z (), Z () ) r. For each decision d {,..., K}, discretize rejection regions R into small rectangles R d ; for any r R d, enforce that if D = d, multiple testing procedure M rejects same set of hypotheses for any (Z (F ), Z (F ) ) r. Discretize Type I error constraints into fine grid on boundaries of null spaces. Discretized opt. problem is not convex. However, we construct reparametrization that is sparse, linear program: max c T x s.t. Ax b. x We apply advanced optimization methods to solve this.

18 Example p = /, α = 0.05, σ = σ. L =total sample size. Prior Λ equally weighted pt. masses at (, ) equal to (0, 0), ( min, 0), (0, min ), ( min, min ).

19 Example p = /, α = 0.05, σ = σ. L =total sample size. Prior Λ equally weighted pt. masses at (, ) equal to (0, 0), ( min, 0), (0, min ), ( min, min ). Sample size n is min s.t. standard design satisfies each power constraint at β j = 0.64 for j {,, C}. Set each β j = 0.8. Stage Decision Stage Stop trial Hyp.Test M(Z (F),) Recruit Both Popula7ons Subpop Subpop } n D(Z () ) 4 Recruit Both Pop. Subpop n Subpop Recruit Only Subpop. Subpopula7on T > T? } } } n 4 M(Z (F),) M(Z (F),) Recruit Only Subpop. n Subpopula7on 4 M(Z (F),4)

20 Example p = /, α = 0.05, σ = σ. L =total sample size. Prior Λ equally weighted pt. masses at (, ) equal to (0, 0), ( min, 0), (0, min ), ( min, min ). Sample size n is min s.t. standard design satisfies each power constraint at β j = 0.64 for j {,, C}. Set each β j = 0.8.? T T > Subpopula)on Stage Decision Stage Hyp.Test Subpopula)on Subpopula)on Subpopula)on Recruit Both Popula)ons Recruit Only Subpop. Recruit Only Subpop. Recruit Both Pop. Subpopula)on Subpopula)on Stop trial D 4 M Z Z 0 0

21 Approximately Optimal Solution Decision Rule for Stage Enrollment: Z Z 0 0 Rejection Regions under Each Decision: (in terms of Z (F ), Z (F ) ) D = "" D = "" D = "" D = ""

22 Approximately Optimal Solution Decision Rule to Enroll Stage : Z Z 0 0 Rejection Regions for Decision: D = "" Z F Z F 0 0 H0 H0, H0C H0, H0C H0

23 Approximately Optimal Solution Decision Rule to Enroll Stage : Z Z 0 0 Rejection Regions for Decision: D = "" Z F Z F 0 0 H0 H0, H0C H0, H0C H0

24 Approximately Optimal Solution Decision Rule to Enroll Stage : Z Z 0 0 Rejection Regions for Decision: D = "" Z F Z F 0 0 H0, H0C H0

25 Approximately Optimal Solution Decision Rule to Enroll Stage : Z Z 0 0 Rejection Regions for Decision: D = "" Z F Z F 0 0 H0, H0C H0

26 Approximately Optimal Solution Decision Rule to Enroll Stage : Z Z 0 0 Rejection Regions for Decision: D = "" Z F Z F 0 0 H0, H0C H0

27 Power Comparison Compare to adaptive enrichment design using p-value combination approach (Bauer and Köhne, 994), with Dunnett intersection test and inverse-normal combination function. Early stopping is incorporated using O Brien-Fleming boundaries for each intersection null hypothesis. Decision rule for stage : if combined population statistic (Z () + Z () )/ > t c, enroll both subpop. else, enroll from each subpopulation s for which Z s () > t. Consider β = β = β = β C. For each power threshold β, we optimized over t, t c to minimize expected sample size under the power constraints. n = total sample size if both enrolled stage. Table: Minimum of ESS dλ, as power constraint β varied. Required Power β: 70% 74% 78% 8% Comparator 0.97n.0n infeasible infeasible Optimal 0.79n 0.84n 0.9n.0n

28 References Rosenblum, M., Fang, X., and Liu, H., Optimal, Two Stage, Adaptive Enrichment Designs for Randomized Trials Using Sparse Linear Programming (04). Johns Hopkins University, Dept. of Biostatistics Working Papers. Working Paper 7. Rosenblum, M., Liu, H., and Yen, E.-H. (04), Optimal Tests of Treatment Effects for the Overall Population and Two Subpopulations in Randomized Trials, using Sparse Linear Programming, Journal of American Statistical Association, Theory and Methods Section, Volume 09. Issue Rosenblum, M. (In Press), Adaptive Randomized Trial Designs that Cannot be Dominated by Any Standard Design at the Same Total Sample Size. Biometrika.

29 Collaborators Collaborators Ethan X. Fang Han Liu Princeton University

30 Thank you! Acknowledgement: This research was supported by the U.S. Food and Drug Administration (HHSF0400C) and the Patient- Centered Outcomes Research Institute (ME ). This work is solely the responsibility of the authors and does not represent the views of these agencies.