sequential multiple assignment randomization t … · assignment randomization trials with...

SEQUENTIAL MULTIPLE ASSIGNMENT

RANDOMIZATION TRIALS WITH ENRICHMENT

(SMARTER) DESIGN

Ying LiuDivision of Biostatistics, Medical College of Wisconsin

Yuanjia WangDepartment of Biostatistics & Psychiatry,

Columbia UniversityDonglin Zeng

Department of Biostatistics, University of North Carolina

Duke Industry Statistics Symposium , Sep 7th, 2017

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 torandomized clinical trial design:a future foresight.

Reference:Liu, Ying, Yuanjia Wang, and Donglin Zeng. "Sequential multipleassignment randomization trials with enrichment design." Biometrics73, no. 2 (2017): 378-390.


MOTIVATION: MULTIPLE STAGE DECISIONS

Dynamic Treatment Regimes (DTRs) are sequential decisionrules, tailored at each stage by patients’ time-varying featuresand intermediate outcomes in previous stages (Lavori &Dawson 1998, Murphy et al. 2001, Lei et al. 2014).Optimizing DTR is to address the personalized medicine questto diliver the best treatment to the right patient at the right time.


CLINICAL TRIAL DESIGN FOR INFERRING DTRS

SMART: Sequential Multiple Assignment Randomized Trial(Lavori & Dawson 2000, 2004; Murphy 2005)

Patients are sequentially randomized at each criticaldecision point.

Enables efficient causal comparisons among differentDTRs.


SMART EXAMPLE: ADHD TRIAL

Figure: SMART Design of Adaptive Pharmacological BehavioralTreatments for Children with ADHD Trial (Pelham 2002)


RESEARCH QUESTIONS TO BE ANSWERED FROM A

SMART

SMART Design Powered for Comparing Effects of fixedDTRs

Comparing BMOD+Intensify vs BMOD+add MED;Comparing BMOD+add MED vs MED+add BMOD;

SMART variance (Murphy 2005)

Vars

(I(A1 = d1(S1),A2 = d2(S1,A1))

p(A1|S1)p(A2|S1,A1)(Y − µ(d1,d2))

).


HOWEVER, PRACTICAL CONCERNS REMAIN...

SMART requires all participants to stay through multi-stagerandomization and be compliant. High cost and longperiod.With drop-out and non-compliance, need a large samplesize to achieve sufficient power for comparing DTRs.

Clinical Antipsychotic Trials of Intervention andEffectiveness (CATIE, Stroup et al. 2003): 705 out of 1460(48%) stayed for the full 18 monthsIn ExTENd (Lei et al., 2012), there was a drop-out rate of17% during the first-stage treatment (52 out of 302), and anadditional 13% during the second stage (41 out of 302).

SMART WITH ENRICHMENT (SMARTER)

NEW DESIGN: SMART-ENRICHMENT TRIAL

(SMARTER)

Main Idea: At the k th stage, (k>1), augment the originalSMART with new patients randomized among the k th stagetreatment options without requiring randomization of previousstage treatments.A Two stage example:

Group 1. Compliant and complete the SMART trial in bothstages.

Group 2. Dropouts before the second stage randomizationafter complete the first stage.

Group 3. Newly recruited enrichment sample at the secondstage. They only receive randomization at the secondstage, and they receive one of the stage 1 treatment byobservation.


NEW DESIGN: SMART-ENRICHMENT TRIAL

(SMARTER)

Figure: Diagram of SMART-EnRichment Trial (SMARTER)


RATIONALE BEHIND SMARTER

At Stage 2, the continuing participants from SMART andthe enrichment participants provide unbiased prediction ofStage 2 treatment effect given history at Stage 1, due toRANDOMIZATION.

This prediction provides imputed and unbiased futureoutcomes for the participants who drop out before Stage2–we “recover” the drop-out participants.

At Stage 1, the imputed and observed outcomes fromSMART can be used to infer unbiased treatment effects,again due to RANDOMIZATION.

Therefore, SMARTER protects against bias due tosequential randomization; SMARTER improves efficiencydue to enrichment.

EFFICIENCY OF SMARTER

DATA COLLECTED IN SMARTER DESIGNS

Notation. Stage 1: {S1, A1}; Stage 2: S2 = {(S1,A1), A2};Final outcome: Y .

The goal is to evaluate the expected final outcome for anygiven treatment strategy: a1 = d1(S1),a2 = d2(S2).

Data from the SMART sample:S1i ,A1i ,ZiA2i ,ZiYi , i = 1, ...,n. (Zi : Stage 2 continuationstatus)

Data from the enrichment sample:S1j ,A1j ,A2j ,Yj , j = 1, ...,m.

Note that the distributions of (S1,A1) may be differentbetween the SMART group and the enrichment group!


KEY ASSUMPTIONS IN SMARTER

(C.1) Stable unit treatment value assumption (SUTVA):treatment applied to one unit does not effect the outcomefor another unit

(C.2) Non-informative dropout: the dropout is independentof {Y (a1,a2)} given (S,A1)

(C.3) No selection bias: the conditional distribution of Ygiven (A1,S,A2) in the enrichment group is the same asthat in the original SMART population.

(C.4) The first stage treatments of A1 for the enrichmentgroup is identical to the treatment A1 in SMART population.


Under conditions (C.1)-(C.4),SMARTer can provide an unbiased estimation for the averagepotential outcome E(Y (d1,d2)) under the DTR(d1,d2).


INFERENCE FROM SMARTER

First, we estimate the predicted outcome using the Stage 2data for Group 2 patients using Group 1 and 3.

Y (a1,a2, s) =∑ni=1 ZiYi I(A1i = a1,A2i = a2,S1i = s) +

∑mj=1 Yj I(A1j = a1,A2j = a2,S1j = s)∑n

i=1 Zi I(A1i = a1,A2i = a2,S1i = s) +∑m

j=1 I(A1j = a1,A2j = a2,S1j = s).


INFERENCE FROM SMARTER

For any given treatment regimen (d1,d2), thenonparametric estimator of its value using SMARTER is aweighted average of the outcomes from the SMARTparticipants who were assigned to treatment (d1,d2):

SMART Subjects Outcome Weights

Group 1:Zi = 1 YiI(A1i =d1(S1i ),A2i =d2(S1i ,A1i ))

p(A1i |S1i )p(A2i |S1i ,A1i )

Group 2:Zi = 0 Y (A1i ,d2(S1i ,A1i),S1i)I(A1i =d1(S1i ))

p(A1i |S1i )


VARIANCE COMPUTATION FOR SMARTER

For any given treatment regime (d1,d2), the variance of thevalue estimator depends on the continuation ratioα = n1/n, enrichment ratio β = m/n :

Varsmart

(Z

I(A1 = d1(S1),A2 = d2(S1,A1))

p(A1|S1)p(A2|S1,A1)

×{(Y − µ(d1,d2)) +

1− α(A1,S1)

α(A1,S1) + βr(A1,S1)(Y − E [Y |A1,A2,S1])

}+ (1− Z )

I(A1 = d1(S1))

p(A1|S1)E [Y − µ(d1,d2)|A1,A2 = d2(S1,A2),S1]

)

+βVarenrichment

((1− α(A1,S1)) (Y − E [Y |A1,A2,S1])

α(A1,S1) + βr(A1,S1)

× I(A1 = d1(S1),A2 = d2(S1,A1))

p(A1|S1)p(A2|S1,A1)

).


VARIANCE COMPUTATION FOR SMARTER

When continuation ratio α = 1, enrichment ratio β = 0,reduces to SMART variance (Murphy 2005)

Vars

(I(A1 = d1(S1),A2 = d2(S1,A1))

p(A1|S1)p(A2|S1,A1)(Y − µ(d1,d2))

).


EFFICIENCY OF SMARTER COMPARED TO SMART

Simplifications:(1) pure randomization: p1(S) = p1,p2(S) = p2; (2)drop-out completely at random : P(Z = 1|A1,S) = α; (3)same S distribution in the enrichment and SMARTpopulations

Relative efficiency of SMARTER to SMART:

ρ =VarSMART

Varenrichment≈ 1 + γ

1− (1− α)(1− p2) + γ α(1+β)2+β(1−α)2

(α+β)2

,

γ is the ratio of the within-strata variance versus thebetween-strata variance.

The relative efficiency depends on randomizationprobabilities, within- and between-strata (S) variability,drop out rate, enrichment rate.


RELATIVE EFFICIENCY IN SOME SIMPLE CASES

ρ > 1 implies the proposed SMARTER is more efficient than aSMART without enrichment and no dropout

For α = 0, all subjects drop out of the first stage. Thus,ρ ≈ (1 + γ)/(p2 + γ/β) so β ≥ 1 leads to efficiency gain.SMARTER always more efficient if

β > γ/(1 + γ − p2).

For any 0 ≤ α < 1, if α(1 + β)2 + β(1− α)2 ≤ (α+ β)2,ρ > 1 implies efficiency gain. Particularly, the lattercondition holds if we choose β ≥ 1.


CONTOUR PLOTS OF RES

Figure: Relative efficiencies of SMARTER compared to SMART;γ = 2 (ratio of within and between stratum variance);

α

β

0.1 0.2

0.3 0.4

0.5

0.6

0.7 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

1


SAMPLE SIZE CONSIDERATION

SMART without dropout: 8(z0.05/2 + z0.2)2 σ2(d2)

(∆µ)2 ;

SMART with α attrition rate: 8(z0.05/2 + z0.2)2 σ2(d2)α(∆µ)2 ;

SMARTER: 8(z0.05/2 + zβ)2 σ2

(∆µ)2ρ.


SAMPLE SIZE CONSIDERATION

Table: Sample sizes of SMARTER to achieve the same efficiency asSMART with 100 subjects

α 0 .2 .4 .5 .6 .8

SMARTER∗ β = 0.5n m n m n m n m n m n m

γ = 0.5 100 50 92 46 91 46 92 46 93 46 96 48γ = 1 125 62 109 54 102 51 100 50 99 50 99 49γ = 2 150 75 125 62 112 56 108 54 105 53 102 51

β = 1γ = 0.5 67 67 73 73 80 80 83 83 87 87 93 93γ = 1 75 75 80 80 85 85 88 88 90 90 95 95γ = 2 83 83 87 87 90 90 92 92 93 93 97 97

β = 2γ = 0.5 50 100 61 122 72 143 77 153 82 163 91 182γ = 1 50 100 62 124 73 145 78 155 82 165 91 183γ = 2 50 100 62 125 73 147 78 157 83 166 92 184

SMART-mis† NA 500 250 200 167 125∗: 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.


ESTIMATING OPTIMAL DTRS USING SMARTER

Use Group 1 and 3 to train a model f1 to optimize A2.Inputs are (S1,A1,S2) and outputs are actual observedoutcome Y .

Use Group 1 and 2 to train a model to optimize A1. Inputsare S1 and outputs is the predicted optimal outcome fromf1.


A SIMULATION STUDY FOR ESTIMATING OPTIMAL

DTRS

Simulation setting

R1 = 1 + A1 ∗ S1 + N(0,2); R2 = A2 ∗ R1 + N(0,2).

S1 ∼ N(0,1) plus 4 additional noise baseline covariates.

In SMART group, A1 and A2 are purely randomized.

In the enrichment group of the same size, A1 isobservational and depends on R1 and R2; only A2 is purelyrandomized.

We vary the drop-out rates of subjects in SMARTcomponent.


SIMULATIONS FOR EXPLORING OPTIMAL DTRS

0 0.25 0.5 0.75 1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

dropout proportion at stage 1

Em

ipir

ical V

alu

e

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Super(Analysis 1)

SMART(group1)

Figure: Estimates of the value functions using the complete SMARTsubjects (yellow) and the SMARTER (blue)


SIMULATIONS FOR EXPLORING OPTIMAL DTRS

Results: Scenario 2

0 0.25 0.5 0.75 1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

dropout proportion at stage 1

Em

ipir

ical V

alu

e

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Super(Analysis 1)

SMART(group1)

Figure: Mean Value Functions in Scenario 2

DISCUSSION

DISCUSSION

SMARTER advantages:

SMARTER supplements SMART to improve efficiency tosalvage potential high drop-out in SMART.

By enrichment, it ensures sufficient sample size at eachstage.

DISCUSSION

DISCUSSION

A more radical departure from SMART:

All subjects in SMART sample have dropped out after thefirst 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 variablesfor Trial 2 participants are available.

Key assumptions: the two trial populations should be the same;tailoring variables S1 and S2 are measured in Trial 1, and S2

measured in Trial 2

DISCUSSION

DISCUSSION

Concerns of SMARTER:Quality of the first stage (naturalistic) treatment delivery inthe enrichment sample:

When the first stage treatment includes some commontreatment.SMARTER can be used to improve efficiency, if one optionof A1 can be found in naturalistic treatment delivery, despitethe other treatments are novel and cannot be found inobservational data sets.

Recruiting similar enrichment population with the SMARTpopulation: A future forsight for the big medical data set toimprove its precision.

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