Sample size considerations for precision medicine

Transcription

1 Sample size considerations for precision medicine Eric B. Laber Department of Statistics, North Carolina State University November 28

2 Acknowledgments Thanks to Lan Wang NCSU/UNC A and Precision Medicine Lab PR Department of Natural Resources National Science Foundation National nstitutes of Health Joint work with Eric Rose Marie Davidian Butch Tsiatis Michael Kosorok Students in lab

3 Precision medicine The right treatment for the right patient at the right time. Mantra of personalized medicine advocates Widely recognized that best clinical care requires treatment decisions tailored to individual patient characteristics mprove patient outcomes, reduce cost and patient burden Treatment regimes Formalize clinical decision making as sequence of decision rules One rule per stage of clinical intervention Maps current patient info to recommended treatment Optimal regime maximizes the mean of some cumulative clinical outcome if applied to population of interest / 3

4 Ex. Treatment regime: mhealth for PTSD in cancer patients (P S. Smith) First stage decision rule f distress 3 then: Cancer Distress Coach (CDC) Else if PTSD symptom score 2 then: CDC Else: usual care Second stage decision rule f responder then: continue first stage treatment Else if using CDC and PSTD change 3 then: add mcoaching Else if using CDC and distress 4 then: add FaceTime CBT Else FaceTime CBT only 2 / 3

5 Ex. Trial design: mhealth for PTSD in cancer Continue patients (P S. Smith) Distress coach Yes Treatment AA Add mcoaching Treatment A Distress Coach Response? Treatment AB No Facetime CBT R R Continue Follow-up only Yes Treatment BA DC + mcoaching Treatment B Standard Care Response? Treatment BB No Facetime CBT R 3 / 3

6 Sample size calculations for precision medicine Sample size typically based on primary aim Power for simple hypotheses, e.g., compare mean outcome under two non-overlapping treatment sequences t-test Clear criterion for power calculations Test statistics are standard Estimation of treatment regime typically secondary analysis Composite hypotheses, high-dim nuisance parameters Unclear what criteria should be used Test statistics are non-standard 4 / 3

7 State-of-the-art: estimate and validate Estimate optimal regime Run SMART Run follow-up RCT Continue Est. opt. regime Distress coach Yes Add mcoaching Treatment A Distress Coach Response? Treatment AB No Facetime CBT R R Continue Follow-up only Yes Treatment BA DC + mcoaching Treatment B Standard Care π bn Treatment AA Response? Treatment BB No Facetime CBT π bn R Primary analyses Eval via CV R Control Control Standard hypothesis test 5 / 3

8 Estimate and validate: discussion Sound scientific premise but potentially inefficient Requires two distinct clinical trials Higher cost Delayed clinical impact π bn may be of low-quality Poor patient outcomes Obscure new clinical knowledge deal: run single trial that allows primary aims and estimation and evaluation of optimal treatment regime 6 / 3

9 Perception of power and precision medicine dentifying subgroups has negative connotations Data snooping Lack of reproducibility/ generalizability Lack of statistical rigor Widely held that sizing trial to estimate opt treatment regime Would require an enormous sample size Cannot guarantee frequentist operating characteristics ntractable without strong assumptions 7 / 3

10 Perception of power and precision medicine dentifying subgroups has negative connotations Data snooping Lack of reproducibility/ generalizability Lack of statistical rigor Widely held that sizing trial to estimate opt treatment regime Would require an enormous sample size Cannot guarantee frequentist operating characteristics ntractable without strong assumptions 8 / 3

11 Preview of remainder of this talk Goal: sample size procedures to ensure estimation and evaluation of opt regime with performance guarantees Develop sample size procedures for two extremes Strong distributional assumptions which (implicitly) exclude non-standard test statistics Non-parametric model and asymptotic approximations that allow for non-standard test statistics Many intermediate procedures between these extremes 9 / 3

12 Outline Estimation of optimal treatment regimes SMART sample size formulae via parametric models SMART sample size formulae under minimal assumptions Simulation experiments

13 Outline Estimation of optimal treatment regimes SMART sample size formulae via parametric models SMART sample size formulae under minimal assumptions Simulation experiments

14 Setup and notation Observe {(X,i, A,i, X 2,i, A2,i, Yi )}ni=, i.i.d. from P X Rp A {, } : first treatment : baseline subj. info. X 2 Rp A2 {, } : second treatment Y R : outcome, higher is better 2 : interim subj. info. during course of A H = X, H 2 = (X, A, X 2) Define history Treatment regime π = (π, π2 ) where πt : supp H t supp At, patient presenting with H t = ht assigned treatment πt (ht ) / 3

15 Characterizing optimal treatment regime π ) = Eπ Y Optimal regime maximizes value V (π Define Q-functions Q2 (h 2, a2 ) = E Y H 2 = h 2, A2 = a2 Q (h, a ) = E max Q2 (H 2, a2 ) H = h, A = a a2 Dynamic programming (Bellman, 957) πtopt (h t ) = arg maxat Qt (h t, at ) / 3

16 Q-learning Assume estimation via Q-learning with linear models Regression-based dynamic programming algorithm (Q) Postulate working models for Q-functions Qt (h t, at ; βt ) = h t, βt, + at h t, βt,, h t,, h t, features of ht 2 (Q) Compute βb2,n = arg minβ2 Pn {Y Q2 (H 2, A2 ; β2 )} (Q2) Compute n o2 βb,n = arg minβ Pn maxa2 Q2 (H 2, A2 ; βb2,n ) Q (H, A ; β ) (Q3) π bt,n (h t ) = arg maxat Qt (h t, at ; βbt,n ) 2 / 3

17 Sample size criteria Let B >, γ, α, η,, ζ (, ) be fixed Want to choose sample size n so that π opt ) B with power at (PWR) there exists an α-level test of H : V (π π opt ) B + η least ( γ) + o() provided V (π π opt ) } ζ + o() (OPT) P {V (b π n ) V (π 3 / 3

18 Outline Estimation of optimal treatment regimes SMART sample size formulae via parametric models SMART sample size formulae under minimal assumptions Simulation experiments

19 Approach one: impose structure Assume the generative model satisfies + a2 h 2, β2, (AN) Q2 (h 2, a2 ) = h 2, β2, (AN2) E H2, β2, H = h, A = a + a h, ξ, = h, ξ, β2, = H,2 $,2 + A H,3 $,3 + τ Z, where (AN3) H2, Z Normal(, ) ind error (AN4) H, ξ,, H, ξ,, H,2 $,2, H,3 $,3 Normal(ω, Ω ) 4 / 3

20 Approach one: impose structure Assume the generative model satisfies + a2 h 2, β2, (AN) Q2 (h 2, a2 ) = h 2, β2, (AN2) E H2, β2, H = h, A = a + a h, ξ, = h, ξ, β2, = H,2 $,2 + A H,3 $,3 + τ Z, where (AN3) H2, Z Normal(, ) ind error (AN4) H, ξ,, H, ξ,, H,2 $,2, H,3 $,3 Normal(ω, Ω ) Note* implicitly make technical assumptions including compactness of parameter space, existence of requisite moments etc. 4 / 3

21 Approach one: impose structure Assume the generative model satisfies + a2 h 2, β2, (AN) Q2 (h 2, a2 ) = h 2, β2, (AN2) E H2, β2, H = h, A = a + a h, ξ, = h, ξ, β2, = H,2 $,2 + A H,3 $,3 + τ Z, where (AN3) H2, Z Normal(, ) ind error (AN4) H, ξ,, H, ξ,, H,2 $,2, H,3 $,3 Normal(ω, Ω ) Note* implicitly make technical assumptions including compactness of parameter space, existence of requisite moments etc. 4 / 3

22 Characterizing the optimal regime Define g : R4 R as g (v ) = max ρ {,} (v + ρv2 )2 v3 + ρv4 + exp 2 2π + (v + ρv2 ) { 2Φ (v + ρv2 )} Lemma Assume (A)-(A4). Let ψ denote normal density, then Z π opt ) = ν {τ, ω, vech(ω )} = V (π τ g (v /τ ) ψ(v ; ω, Ω )d v. R4 5 / 3

23 π opt ) B Sizing for power to reject H : V (π Lemma π opt ). Assume (AN)-(AN4). Let Vbn be plug-in estimator of V (π Then, o n π opt ) n Vbn V (π Normal(, σ 2 ), where σ 2 is a smooth functional of generative model. 6 / 3

24 π opt ) B Sizing for power to reject H : V (π Lemma π opt ). Assume (AN)-(AN4). Let Vbn be plug-in estimator of V (π Then, o n π opt ) n Vbn V (π Normal(, σ 2 ), where σ 2 is a smooth functional of generative model. Corollary Assume (AN)-(AN4) and let σ bn2 be a consistent estimator of σ 2. b σn zα is an Then a test that rejects when n Vn B /b π opt ) B. Moreover, (asymptotic) α-level test of H : V (π choosing l m 2 n = (σ 2 /η 2 ) Φ ( γ) + zα satisfies (PWR). 6 / 3

25 Sizing to ensure high-quality estimation of π opt Lemma Assume (AN)-(AN4). Then under mild regularity conditions there exists K, δ > such that π opt ) Knδ Vbn V (π π opt ) + op (/ n). V (b π n ) V (π Subexpontential tail bounds and a bounded density for the treatment effect at each stage are sufficient. 7 / 3

26 Sizing to ensure high-quality estimation of π opt Lemma Assume (AN)-(AN4). Then under mild regularity conditions there exists K, δ > such that π opt ) Knδ Vbn V (π π opt ) + op (/ n). V (b π n ) V (π Corollary Assume (A)-(AN4). Then under mild regularity conditions & 2 ' Φ ( ζ)σ n= satisfies (OPT). Subexpontential tail bounds and a bounded density for the treatment effect at each stage are sufficient. 7 / 3

27 Sample size formulae via parametric models Reduce to simple formulae Linear mean models and normal error Not vacuous Can be checked empirically Can be relaxed/extended Requires elicitation and/or historical data to estimate σ 2 8 / 3

28 Outline Estimation of optimal treatment regimes SMART sample size formulae via parametric models SMART sample size formulae under minimal assumptions Simulation experiments

29 Q-learning (again) Assume estimation via Q-learning with linear models Regression-based dynamic programming algorithm (Q) Postulate working models for Q-functions Qt (h t, at ; βt ) = h t, βt, + at h t, βt,, h t,, h t, features of ht 2 (Q) Compute βb2,n = arg minβ2 Pn {Y Q2 (H 2, A2 ; β2 )} (Q2) Compute n o2 βb,n = arg minβ Pn maxa2 Q2 (H 2, A2 ; βb2,n ) Q (H, A ; β ) (Q3) π bt,n (h t ) = arg maxat Qt (h t, at ; βbt,n ) Population parameters βt obtained by replacing Pn with P; define π Q,opt (h t ) = arg maxat Qt (h t, at ; βt ) t 9 / 3

30 nference and Q-learning mpose less structure on generative model Only impose moment and tail conditions Do not assume postulated models are correct Both (PWR) and (OPT) are meaningful under misspecification π Q,opt ) nonsmooth functionals of Both β and V (π data-generating model No regular asymptotically unbiased estimators exist Standard normal approximations can perform poorly Base sample size on conservative interval estimators 2 / 3

31 (Nonpar) characterizing the optimal regime For each β = (β, β2 ) define (β) = 4Y A H, β, > A 2 H 2, β2, > + A (H 2, A2, β), where A(H 2, A2, β) mean zero aug term (Tsiatis 26) Let V Q (β) = P (β) and Vb Q (β) = Pn (β) π Q,opt ) VbnQ (βbn ) is APWE of V Q (β ) = V (π Estimators consistent even if Q-functions misspecified 2 / 3

32 Projection test Define ςn2 (β) = P { (β) P (β)}2 then o n bq n V (β) V Q (β) Normal(, ) Zn (β) = ςbn (β) Let Ξn,ϑ be a ( ϑ ) % confidence region for β 2 then a test that rejects when zϑ2 ςbn (β) Q b inf Vn (β) B β Ξn,ϑ n has type error no more than α = ϑ + ϑ2 + o() under π Q,opt ) B H : V (π 2 See L. et al. 24; Robins / 3

33 Power calculation Power of projection test P inf β Ξn,ϑ [Zn (β) + Nn (β)] zϑ2 nη P inf zϑ2 Zn (β) + min Nn (β), β Ξn,ϑ ςbn (β) where Nn (β) = Q n V (β) B /b ςn (β) No closed form for power of projection test Assume pilot data of size n available Use bootstrap oversampling to estimate power Solve for sample size that attains nominal power 23 / 3

34 Power calculation cont d Bootstrap resample size n from n (b) Pn,n bootstrap empirical distn f Wn = f (P, Pn ) then Wn,n = f (Pn, Pn,n ) (b) (b) Bootstrap sample size for (PWR) is smallest integral n such that PB " inf β Ξ(b) ( (b) (b) Zn,n (β) + min Nn,n (β), n,n,ϑ nη (b) ςbn,n (β) )# zϑ2 exceeds nominal power ( γ) 24 / 3

35 Consistency of bootstrap oversampling Theorem Let ϑ (, ) be fixed, κ, K >, s R be arbitrary, and ηn = η + s/ n. Then, under moment conditions ( lim P n,n sup PB v K P " inf (b),n,ϑ β Ξn (b) Z(b) n,n (β) + min Nn,n (β), nηn Zn (β) + min Nn (β), v β Ξn,ϑ ςbn (β) inf ( nηn (b) )# ) v ςbn,n (β)! >κ =. 25 / 3

36 Sample size formulae with minimal assumptions Projection test + bootstrap oversampling Consistent estimation of power curve (PWR) Analogous procedure for (OPT) (not shown) Did not assume parametric or correctly specified model ncreased complexity Nuisance parameters difficult to elicit pilot/hist data 26 / 3

37 Outline Estimation of optimal treatment regimes SMART sample size formulae via parametric models SMART sample size formulae under minimal assumptions Simulation experiments

38 Simulation experiment Overview of generative model From L. et al. (24), focus on non-standard cases Linear at second stage Normal errors Correct and incorrectly specified Q-functions Consider pilot data of size n = 5 Additional details Level α =.5 Desired power ( γ) % = 9% Probability of being near-optimal ( ζ) =.9 Sample size selected so that both (PWR) and (OPT) hold 27 / 3

39 Simulation experiments cont d Compare parametric and nonparametric sample size procedure with sample size required for comparing fixed regimes (t-test) is difference between mean outcome under optimal regime and best fixed (non-adaptive) regime in η units = represents worst case tailoring has no benefit Consider two methods for selecting σ Known Estimated from pilot/historical data (n = 5) 28 / 3

40 Method for σ (PWR) (OPT) Correctly specified Misspecified Results: parametric sample size formulae n fixed En SDn / 3

41 Method for σ (PWR) (OPT) Correctly specified Misspecified Results: parametric sample size formulae n fixed En SDn / 3

42 Method for σ (PWR) (OPT) Correctly specified Misspecified Results: parametric sample size formulae n fixed En SDn / 3

43 Method for σ (PWR) (OPT) Correctly specified Misspecified Results: parametric sample size formulae n fixed En SDn / 3

44 Method for σ (PWR) (OPT) Correctly specified Misspecified Results: parametric sample size formulae n fixed En SDn / 3

45 (PWR) (OPT) n fixed En SDn n = NA P. Correct P. ncorrect Results: nonparametric sample size formulae / 3

46 (PWR) (OPT) n fixed En SDn n = NA P. Correct P. ncorrect Results: nonparametric sample size formulae / 3

47 (PWR) (OPT) n fixed En SDn n = NA P. Correct P. ncorrect Results: nonparametric sample size formulae / 3

48 Discussion Sample size procedures possible for precision medicine Not excessively conservative* Attain nominal frequentist operating characteristics Range of assumptions on generative model Canonical information requirement in sample size procedures Pilot/historical data Elicitation Modeling assumptions Presented first steps, many interesting open problems! 3 / 3

49 Thank you. laber-labs.com