SEQUENTIAL MULTIPLE ASSIGNMENT RANDOMIZATION TRIALS WITH ENRICHMENT (SMARTER) DESIGN

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1 SEQUENTIAL MULTIPLE ASSIGNMENT RANDOMIZATION TRIALS WITH ENRICHMENT (SMARTER) DESIGN Ying Liu Division of Biostatistics, Medical College of Wisconsin Yuanjia Wang Department of Biostatistics & Psychiatry, Columbia University Donglin Zeng Department of Biostatistics, University of North Carolina Duke Industry Statistics Symposium, Sep 7th, 2017

2 INTRODUCTION TO DTR AND SMART Topic of this talk: Design SMART Enrichment Trial A new design to save time and cost for SMART trial. A way to incorporate big observational data in to randomized clinical trial design:a future foresight. Reference: Liu, Ying, Yuanjia Wang, and Donglin Zeng. "Sequential multiple assignment randomization trials with enrichment design." Biometrics 73, no. 2 (2017):

3 INTRODUCTION TO DTR AND SMART MOTIVATION: MULTIPLE STAGE DECISIONS Dynamic Treatment Regimes (DTRs) are sequential decision rules, tailored at each stage by patients time-varying features and intermediate outcomes in previous stages (Lavori & Dawson 1998, Murphy et al. 2001, Lei et al. 2014). Optimizing DTR is to address the personalized medicine quest to diliver the best treatment to the right patient at the right time.

4 INTRODUCTION TO DTR AND SMART CLINICAL TRIAL DESIGN FOR INFERRING DTRS SMART: Sequential Multiple Assignment Randomized Trial (Lavori & Dawson 2000, 2004; Murphy 2005) Patients are sequentially randomized at each critical decision point. Enables efficient causal comparisons among different DTRs.

5 INTRODUCTION TO DTR AND SMART SMART EXAMPLE: ADHD TRIAL Figure: SMART Design of Adaptive Pharmacological Behavioral Treatments for Children with ADHD Trial (Pelham 2002)

6 INTRODUCTION TO DTR AND SMART RESEARCH QUESTIONS TO BE ANSWERED FROM A SMART SMART Design Powered for Comparing Effects of fixed DTRs Comparing BMOD+Intensify vs BMOD+add MED; Comparing BMOD+add MED vs MED+add BMOD; SMART variance (Murphy 2005) ( ) I(A1 = d 1 (S 1 ), A 2 = d 2 (S 1, A 1 )) Var s (Y µ(d 1, d 2 )). p(a 1 S 1 )p(a 2 S 1, A 1 )

7 INTRODUCTION TO DTR AND SMART HOWEVER, PRACTICAL CONCERNS REMAIN... SMART requires all participants to stay through multi-stage randomization and be compliant. High cost and long period. With drop-out and non-compliance, need a large sample size to achieve sufficient power for comparing DTRs. Clinical Antipsychotic Trials of Intervention and Effectiveness (CATIE, Stroup et al. 2003): 705 out of 1460 (48%) stayed for the full 18 months In ExTENd (Lei et al., 2012), there was a drop-out rate of 17% during the first-stage treatment (52 out of 302), and an additional 13% during the second stage (41 out of 302).

8 SMART WITH ENRICHMENT (SMARTER) NEW DESIGN: SMART-ENRICHMENT TRIAL (SMARTER) Main Idea: At the kth stage, (k>1), augment the original SMART with new patients randomized among the k th stage treatment options without requiring randomization of previous stage treatments. A Two stage example: Group 1. Compliant and complete the SMART trial in both stages. Group 2. Dropouts before the second stage randomization after complete the first stage. Group 3. Newly recruited enrichment sample at the second stage. They only receive randomization at the second stage, and they receive one of the stage 1 treatment by observation.

9 SMART WITH ENRICHMENT (SMARTER) NEW DESIGN: SMART-ENRICHMENT TRIAL (SMARTER) Figure: Diagram of SMART-EnRichment Trial (SMARTER)

10 SMART WITH ENRICHMENT (SMARTER) RATIONALE BEHIND SMARTER At Stage 2, the continuing participants from SMART and the enrichment participants provide unbiased prediction of Stage 2 treatment effect given history at Stage 1, due to RANDOMIZATION. This prediction provides imputed and unbiased future outcomes for the participants who drop out before Stage 2 we recover the drop-out participants. At Stage 1, the imputed and observed outcomes from SMART can be used to infer unbiased treatment effects, again due to RANDOMIZATION. Therefore, SMARTER protects against bias due to sequential randomization; SMARTER improves efficiency due to enrichment.

11 DATA COLLECTED IN SMARTER DESIGNS Notation. Stage 1: {S 1, A 1 }; Stage 2: S 2 = {(S 1, A 1 ), A 2 }; Final outcome: Y. The goal is to evaluate the expected final outcome for any given treatment strategy: a 1 = d 1 (S 1 ), a 2 = d 2 (S 2 ). Data from the SMART sample: S 1i, A 1i, Z i A 2i, Z i Y i, i = 1,..., n. (Z i : Stage 2 continuation status) Data from the enrichment sample: S 1j, A 1j, A 2j, Y j, j = 1,..., m. Note that the distributions of (S 1, A 1 ) may be different between the SMART group and the enrichment group!

12 KEY ASSUMPTIONS IN SMARTER (C.1) Stable unit treatment value assumption (SUTVA): treatment applied to one unit does not effect the outcome for another unit (C.2) Non-informative dropout: the dropout is independent of {Y (a 1, a 2 )} given (S, A 1 ) (C.3) No selection bias: the conditional distribution of Y given (A 1, S, A 2 ) in the enrichment group is the same as that in the original SMART population. (C.4) The first stage treatments of A 1 for the enrichment group is identical to the treatment A 1 in SMART population.

13 Under conditions (C.1)-(C.4), SMARTer can provide an unbiased estimation for the average potential outcome E(Y (d 1, d 2 )) under the DTR(d 1, d 2 ).

14 INFERENCE FROM SMARTER First, we estimate the predicted outcome using the Stage 2 data for Group 2 patients using Group 1 and 3. Ŷ (a 1, a 2, s) = n i=1 Z iy i I(A 1i = a 1, A 2i = a 2, S 1i = s) + m j=1 Y ji(a 1j = a 1, A 2j = a 2, S 1j = s) n i=1 Z ii(a 1i = a 1, A 2i = a 2, S 1i = s) + m j=1 I(A. 1j = a 1, A 2j = a 2, S 1j = s)

15 INFERENCE FROM SMARTER For any given treatment regimen (d 1, d 2 ), the nonparametric estimator of its value using SMARTER is a weighted average of the outcomes from the SMART participants who were assigned to treatment (d 1, d 2 ): SMART Subjects Outcome Weights Group 1:Z i = 1 Y i I(A 1i =d 1 (S 1i ),A 2i =d 2 (S 1i,A 1i )) p(a 1i S 1i )p(a 2i S 1i,A 1i ) Group 2:Z i = 0 Ŷ (A 1i, d 2 (S 1i, A 1i ), S 1i ) I(A 1i =d 1 (S 1i )) p(a 1i S 1i )

16 VARIANCE COMPUTATION FOR SMARTER For any given treatment regime (d 1, d 2 ), the variance of the value estimator depends on the continuation ratio α = n 1 /n, enrichment ratio β = m/n : ( Var smart Z I(A 1 = d 1 (S 1 ), A 2 = d 2 (S 1, A 1 )) p(a 1 S 1 )p(a 2 S 1, A 1 ) { } 1 α(a 1, S 1 ) (Y µ(d 1, d 2 )) + α(a 1, S 1 ) + βr(a 1, S 1 ) (Y E[Y A 1, A 2, S 1 ]) + (1 Z ) I(A 1 = d 1 (S 1 )) E[Y µ(d 1, d 2 ) A 1, A 2 = d 2 (S 1, A 2 ), S 1 ] p(a 1 S 1 ) ( (1 α(a1, S 1 )) (Y E[Y A 1, A 2, S 1 ]) +βvar enrichment α(a 1, S 1 ) + βr(a 1, S 1 ) I(A ) 1 = d 1 (S 1 ), A 2 = d 2 (S 1, A 1 )). p(a 1 S 1 )p(a 2 S 1, A 1 ) )

17 VARIANCE COMPUTATION FOR SMARTER When continuation ratio α = 1, enrichment ratio β = 0, reduces to SMART variance (Murphy 2005) ( ) I(A1 = d 1 (S 1 ), A 2 = d 2 (S 1, A 1 )) Var s (Y µ(d 1, d 2 )). p(a 1 S 1 )p(a 2 S 1, A 1 )

18 EFFICIENCY OF SMARTER COMPARED TO SMART Simplifications: (1) pure randomization: p 1 (S) = p 1, p 2 (S) = p 2 ; (2) drop-out completely at random : P(Z = 1 A 1, S) = α; (3) same S distribution in the enrichment and SMART populations Relative efficiency of SMARTER to SMART: ρ = Var SMART Var enrichment 1 + γ, 1 (1 α)(1 p 2 ) + γ α(1+β)2 +β(1 α) 2 (α+β) 2 γ is the ratio of the within-strata variance versus the between-strata variance. The relative efficiency depends on randomization probabilities, within- and between-strata (S) variability, drop out rate, enrichment rate.

19 RELATIVE EFFICIENCY IN SOME SIMPLE CASES ρ > 1 implies the proposed SMARTER is more efficient than a SMART without enrichment and no dropout For α = 0, all subjects drop out of the first stage. Thus, ρ (1 + γ)/(p 2 + γ/β) so β 1 leads to efficiency gain. SMARTER always more efficient if β > γ/(1 + γ p 2 ). For any 0 α < 1, if α(1 + β) 2 + β(1 α) 2 (α + β) 2, ρ > 1 implies efficiency gain. Particularly, the latter condition holds if we choose β 1.

20 CONTOUR PLOTS OF RES Figure: Relative efficiencies of SMARTER compared to SMART; γ = 2 (ratio of within and between stratum variance); β α

21 SAMPLE SIZE CONSIDERATION SMART without dropout: 8(z 0.05/2 + z 0.2 ) 2 σ 2 ( d 2 ) ( µ) 2 ; SMART with α attrition rate: 8(z 0.05/2 + z 0.2 ) 2 σ 2 ( d 2 ) α( µ) 2 ; SMARTER: 8(z 0.05/2 + z β ) 2 σ2 ( µ) 2 ρ.

22 SAMPLE SIZE CONSIDERATION Table: Sample sizes of SMARTER to achieve the same efficiency as SMART with 100 subjects α SMARTER β = 0.5 n m n m n m n m n m n m γ = γ = γ = β = 1 γ = γ = γ = β = 2 γ = γ = γ = SMART-mis NA : Sample sizes for SMARTER are to achieve same efficiency as a SMART trial with 100 patients and in an ideal case of no dropout. n is the sample size for the SMART group, m is the sample size for the enrichment group; β = m/n is the ratio of sample size between enrichment and SMART group; 1 α is dropout rate; γ is ratio of within- and between-stratum variance. : SMART-mis is the sample size for a SMART accounting for the dropout rate of 1 α in the second stage in the design, i.e., 100/α.

23 ESTIMATING OPTIMAL DTRS USING SMARTER Use Group 1 and 3 to train a model ˆf 1 to optimize A 2. Inputs are (S 1, A 1, S 2 ) and outputs are actual observed outcome Y. Use Group 1 and 2 to train a model to optimize A 1. Inputs are S 1 and outputs is the predicted optimal outcome from ˆf 1.

24 A SIMULATION STUDY FOR ESTIMATING OPTIMAL DTRS Simulation setting R 1 = 1 + A 1 S 1 + N(0, 2); R 2 = A 2 R 1 + N(0, 2). S 1 N(0, 1) plus 4 additional noise baseline covariates. In SMART group, A 1 and A 2 are purely randomized. In the enrichment group of the same size, A 1 is observational and depends on R 1 and R 2 ; only A 2 is purely randomized. We vary the drop-out rates of subjects in SMART component.

25 SIMULATIONS FOR EXPLORING OPTIMAL DTRS Emipirical Value Super(Analysis 1) SMART(group1) dropout proportion at stage 1 Figure: Estimates of the value functions using the complete SMART subjects (yellow) and the SMARTER (blue)

26 SIMULATIONS FOR EXPLORING OPTIMAL DTRS Results: Scenario 2 Emipirical Value Super(Analysis 1) SMART(group1) dropout proportion at stage 1 Figure: Mean Value Functions in Scenario 2

27 DISCUSSION DISCUSSION SMARTER advantages: SMARTER supplements SMART to improve efficiency to salvage potential high drop-out in SMART. By enrichment, it ensures sufficient sample size at each stage.

28 DISCUSSION DISCUSSION A more radical departure from SMART: All subjects in SMART sample have dropped out after the first stage treatment. SMARTER essentially synthesizes two independent trials: Trial 1 on Stage 1 and Trial 2 on Stage 2. Key information: Stage 1 treatment and tailoring variables for Trial 2 participants are available. Key assumptions: the two trial populations should be the same; tailoring variables S 1 and S 2 are measured in Trial 1, and S 2 measured in Trial 2

29 DISCUSSION DISCUSSION Concerns of SMARTER: Quality of the first stage (naturalistic) treatment delivery in the enrichment sample: When the first stage treatment includes some common treatment. SMARTER can be used to improve efficiency, if one option of A 1 can be found in naturalistic treatment delivery, despite the other treatments are novel and cannot be found in observational data sets. Recruiting similar enrichment population with the SMART population: A future forsight for the big medical data set to improve its precision.