Stat Seminar 10/26/06. General position results on uniqueness of nonrandomized Group-sequential decision procedures in Clinical Trials Eric Slud, UMCP

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1 Stat Seminar 10/26/06 General position results on uniqueness of nonrandomized Group-sequential decision procedures in Clinical Trials Eric Slud, UMCP OUTLINE I. Two-Sample Clinical Trial Statistics A. Large-sample background B. Loss-functions & constraints on actions II. Decision Theory Formulation III. General Two-look Optimized Plans A. Reductions & results B. Computed Boundary Example C. Role of uniqueness & nonrandomized solutions IV. Random Perturbations of Loss Functions A. General result on a.s. unique minima B. Proof Sketch: a.s. unique optimal plans Joint work with Eric Leifer, NHLBI. 1

2 Two-Sample Clinical Trial Statistics Data format : (E i,ti, i,z i, for analysis at times t. i =1,...,N A (τ)) E i entry-times, N A arrival-counting, τ accrual-horizon X i failure time, C i indep. right-cens., Z i trt. gp. (X i,c i cond. indep. given Z i & strat. variable V i ) T i = X i C i (t E i ), i = I [Xi C i (t E i ]) PROBLEM: test H 0 : S X Z (t z) S X (t),z=0, 1 with multiple interim looks & experimentwise validity. TEST STATISTIC : for look at t, define Y z (s) = j M(t )= i I [Zj =z,t j s], Y (s) =Y 1 (s)+y 0 (s) at-risk K(s, Ŝ X (s)){z i Y 1 (T i s) Y (T i s) } i di [T i s] asympt. indep. incr., with estimated variance ˆV (t ) = i K 2 (s, ŜX(s)){ Y 1 (T i s)y0 (T i s) Y (Ti } i s) di [T i s] Reject if M(t ) ˆV (t ) b(t ), Accept if a(t ) 2

3 Abstracted Asymptotic Problem Most methods rely on asymptotically Gaussian timeindexed statistic-numerator n 1/2 M(t) with indep. incr. s, and variance function V (t) to be estimated in real or information time. V (t) in H 0 is functional of ( S X,S C Z, Λ A E(N A )) K 2 (S X (s)) p(1 p)λ A(t s)s C Z (s 1)S C Z (s 0) Λ A (t)(p(s C Z (s 1) + (1 p)s C Z (s 0)) df X(s) Control parameters: At each t = t k, can choose t k+1,a k+1,b k+1 PROBLEM: To optimize expected Loss or Cost over times and cutoffs while maintaining overall nominal significance level. Main Computational Method of optimizing boundaries is parametric search for parametric boundary classes, or backward induction. 3

4 Decision-Theory Ingredients Actions: look-times t k and boundaries b k = b(t ) for W/ ˆV 1/2 for now consider only upper rejection boundary Prior: π(ϑ) density for group-difference log hazard ratio parameter ϑ (within semiparametric model). Losses: costs of experimentation c 1 (t, ϑ), wrong decision c 2 (ϑ), late correct decision c 3 (t, ϑ), these loss elements introduced in Leifer (2000) thesis. Costs are economic within-trial, ethical within-trial, and economic after-trial. In 2-Look Trials, adaptive actions are: t 1 constant depending on prior & losses Followup time t 2 = t 2 (W (t 1 ),t 1 ) which if = t 1 indicates imediate Reject or Accept, and final boundary: reject iff W (t 2 ) b(t 2,t 1,W(t 1 )) 4

5 More on Decision Theory Problem (I) state-of-nature parameter ϑ R, target parameter z = γ(ϑ) (= ϑ in estimation, I[ϑ 0] in testing), and prior prob. π on Θ (II) action-space A = {(t,a):t =(t 1,...,t K ), 0 t 1 t K T, a γ(θ)} (III) observable process W =(W (x), 0 x T ) Wiener process with drift ϑ, and suff. stat. W (τ) for data up to stopping-time τ, and auxiliary randomization r.v. U Unif[0, 1] independent of W (IV) strategies consist of: increasing nonnegative stoppingtime random variables τ =(τ 1,...,τ K ) 0 τ 1 τ 2 τ K T a.s. satisfying causality restrictions on measurability, [τ j x] σ(τ i,i<j; W (u), u x ; U), and terminal action r.v. a γ(θ) meas. wrt (W (x), x t K ) and U 5

6 (V) a loss-function L(t,a,ϑ,λ) depending on looktimes t only as smooth fcn of terminal time t K, with Lagrange multipliers λ 0,λ 1 L(t,a,ϑ,λ) = C(t K,a,z)+λ 0 aδ ϑ,0 + λ 1 (1 a) δ ϑ,ϑ1 where z = I [ϑ 0], ϑ 1 > 0 and C(t, a, z) = c 1 (t, ϑ) +I [z a] c 2 (ϑ) +I [z=a] c 3 (t, ϑ) and c j (,ϑ), c 3 (t, ϑ) c 2 (ϑ). (VI) risk function to be minimized over d =(τ,a) is r(d, λ) = Θ E ϑ L(τ, α, ϑ, λ) dπ(ϑ) (1) NOTE: For terminal time τ K = s and data history (W (u), u s), there exists unique optimal terminal action I [W (s) w(s)], with w( ) = w(,λ 0,λ 1 ) uniquely determined by e θw(y) θ 2 y/2 g 1 (y,ϑ,λ 0,λ 1 ) dπ(θ) = 0 where g 1 (y) =(c 2 (θ) c 3 (y,θ)) (2I [θ 0] 1)+ λ 0 π 0 I [θ=0] λ 1 π 1 I [θ=θ1 ] 6

7 Optimizing the Decision Rule We know that at the final look-time, the best decision is to Reject when W (s) w(s), w computed uniquely from s, λ 0,λ 1. Now let t 1 be fixed and imagine W (t 1 )/ t 1 = x observed. Can express conditional expected final loss at time t 1 + s in form r 2 (t 1,x,s) R(t 1,x,s,ϑ) dπ(ϑ) = e θx t 1 θ 2 t 1 /2 {a 0 (t 1 + s, ϑ) +a 1 (t 1 + s, ϑ) 1 Φ w(t 1 + s) x t 1 s ϑ s } dπ(θ) where a 0 (t, ϑ) = c 1 (t, ϑ)+c 2 (ϑ) I [ϑ>0] + λ 1 π 1 I [θ=θ1 ] + c 3 (t, ϑ) I [ϑ 0] Backward Induction Idea: if we can uniquely optimize r 2 (t 1,x,s) in s = s(t 1,x) uniquely (a.e. x), and then choose t 1 uniquely to minimize 1 R(t1,x, s(t 1,x), ϑ) e (x ϑ t 1 ) 2 /2 dx dπ(ϑ) 2π 7

8 Optimizing, Continued then we would have specified the unique nonrandomized optimal decision rule through the constant t 1 and functions s(t 1,x), w(t 1 + s). But for these general loss functions there is no hope of explicit formulas, and even showing unique optima is complicated by the form of optimal s which will generally be 0 on the complement of some interval. This behavior is unavoidable, reflecting the necessity to stop after one look at the data at time t 1 if the observed statistic value W (t 1 ) is too extreme! We show this next is a computed example. 8

9 Optimal Boundary in 2-Look Example Data: W (t) =B(t i )+ϑt i, i =1, 2 t 1 is fixed in advance, continuation-time t 2 t 1 0 is chosen as function of W (t 1 ). Loss for stopping at τ with Rejection indicator z : c 1 (τ,ϑ) +c 3 (τ,ϑ) +z (c 2 (ϑ) c 3 (τ,ϑ)) (2I [ϑ 0] 1) Problem to find min-risk test under prior π(dϑ), with sig. level α and type II error at ϑ 1 β. Under regularity conditions on loss elements (piecewise smoothness, c 2 c 3, c 1 ) and prior π(dϑ) assigning positive mass to neighborhoods of 0, ϑ 1 > 0: can show that optimal procedures are nonrandomized (w.p.1 after smallrandom perturbation of c 1 ) and unique, rejecting for W (t 2 ) b 2 (t 1,t 2,W(t 1 )) V (t 2 ). Example. α =.025, β =.1, ϑ 1 = log(1.5), time scaled so τ fix = 1. Optimized t 1 =.42 τ fix. e ϑ = hazard ratio π({ϑ}) c 1 (t, ϑ) t t t t t c 2 (ϑ)

10 Total Trial Time normalized first-look statistic U 1

11 Second Look Critical Value normalized first-look statistic U 1

12 = -0.2 = 0.4 = 0.56 (lower bdy of cont. reg.) s s s = 0.7 =1 = s s s = 2.56 (upper bdy of cont. reg.) = 2.7 = s s s

13 Lemma on Unique Random-Function Minima Lemma 1 (extending Bulinskaya 1961 ) Let K be a fixed compact interval in R, and suppose that g 1,g 2 are fixed smooth real-valued functions on K, with inf K g 2 > 0. Let ζ( ) =ζ(,ω) be a random process on K with sample paths a.s. in C 2 (K) such that for all δ>0, the joint density of (ζ(s 1 ),ζ (s 1 ),ζ(s 2 ),ζ (s 2 )) (with respect to Lebesgue measure, on R 4 ) is uniformly bounded, uniformly over all s 1,s 2 K for which s 1 s 2 δ. Then (i) With probability 1 (ω), there do not exist distinct elements s 1,s 2 K such that the sample path of the function ρ(s) g 1 (s) +g 2 (s)ζ(s) satisfies ρ (s 1 )=ρ (s 2 )=0 along with ρ(s 1 )=ρ(s 2 ), and (ii) Under the assumptions above, with probability 1, the number of zeroes of ρ on K is finite, and ρ (s) 0 at all values of s for which ρ (s) =0. Corollary. Analogous result for minima wrt s with functions g j (t 1,x,s) holds a.e. in (t 1,x,ω). Via Implicit Function Theorem for minimizer s = s(t 1,x) find s a.e. smooth in its arguments, a.s. for argument ω. 14

14 Further Steps to Establish a.s. unique rule Want to substitute s = s(t 1,x) and integrate out standardized variable x = W (t 1 )/ t 1 to get smooth time-t 1 risk r 1 (t 1 ) except that this function s is only a.e. defined and smooth. Introduce technical assumption for small ɛ>0 that t 2 t 1 + ɛ whenever t 2 >t 1 (No real loss in generality: can likely prove it directly.) Remaining proof-step to prove r 1 (t 1 ) smooth: for V Unif[0,ɛ) indep. of other data, r 1 (t 1 ) = min (E(r 2 (t 1,W(t 1 )/ t 1, 0+), E( r 2 (t 1 + V, W (t 1 + V ) t1 + V,s(t 1 + V, W (t 1 + V ) t1 + V ))) Last expectations with respect to measure on (ϑ, W ( )) given by A P ϑ (dw )dπ(ϑ). Overall results hold a.s. (ω) for a.e. (λ 0,λ 1 ). But that seems to be enough. 15

15 References For clinical-trial large sample theory: Tsiatis (1982), Andersen & Gill (1982), Sellke & Siegmund (1983), Slud (1984) For group-sequential boundaries: Pocock (1977), O Brien & Fleming (1979), Slud & Wei (1982), Lan & DeMets (1983), Pampallona & Tsiatis (1994) plus others For adaptive modifications: Burman & Sondesson (2006, current Biometrics) For decision-theoretic formulations: Anscombe (1963), Colton (1963), Berger (1985 book), Siegmund (1985 book), Jennison (1987), Freedman & Spiegelhalter (1989), Leifer (2000, thesis) For level-crossings theorems: Bulinskaya (1962) cited in Cramer and Leadbetter (1967) Stationary and Related Stochastic Processes book 16