design and analysis of sequential multiple assignment...

Design and Analysis of Sequential Multiple AssignmentRandomized Trials (SMARTs)

Bibhas ChakrabortyDuke-NUS Medical School, National University of Singapore

[email protected]

IMS Workshop on Design of Healthcare StudiesSingapore

July 5, 2017

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Outline

1 Personalized Medicine and Dynamic Treatment Regimes

2 SMART DesignPrimary Analysis and Sample Size

3 Analysis: Estimation of Optimal DTRsQ-learning

4 Non-regular Inference

5 Discussion

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Outline





5 Discussion

Personalized Medicine and Dynamic Treatment Regimes

Personalized or Precision Medicine

Paradigm shift from “one size fits all” to individualized, patient-centric care

– Can address inherent heterogeneity across patients

– Can also address variability within patient, over time

– Can increase patient compliance, thus increasing the chance of treatment success

– Likely to reduce the overall cost of health care

Overarching Methodological Questions:

– How to decide on the optimal treatment for an individual patient?

– How to make these treatment decisions evidence-based or data-driven?

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Treatment Regimes or Regimens

One perspective in personalized medicine: given an individual patient’scharacteristics, can we identify a treatment, among the available options, that ismost likely to confer the most benefit?

A treatment regimen is a rule (or rules) that specifies how to allocate treatmentbased on an individual patient’s characteristics

If treatment is administered only once, then there is a single decision point, oftenthe time of diagnosis

In case of chronic diseases, treatments are typically administered at multipledecision points

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Dynamic Treatment Regime(n)s

A dynamic treatment regimen (DTR) is a sequence of decision rules, one perdecision point (or stage), that specify how to adapt the type, dosage and timingof treatment according to the ongoing information on an individual patient

– Each decision rule takes a patient’s treatment and covariate history as inputs, andoutputs a recommended treatment

Decision points can occur at regular intervals (e.g. yearly follow-up visits) orclinical events (e.g. remission, relapse, complication)

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Dynamic Treatment Regimens (DTRs)

DTRs offer a data-driven framework for operationalizing the adaptive clinicaldecision-making in a time-varying setting, and thereby potentially improving it

– Clinical decision support systems for treating chronic diseases

A subject’s stage-specific treatment is not known at the start of a dynamicregimen, since treatment depends on time-varying variables

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ADHD Example: One Simple DTR

“Give Low-intensity BMOD initially; if the subject responds, then continue BMOD,otherwise prescribe BMOD + MEDS”

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Treatment Regimen vs. Realized Treatment Experience

Subjects following the same DTR can have different realized treatmentexperiences:

– Subject 1 experiences “Low-intensity BMOD, followed by response, followed byLow-intensity BMOD”

– Subject 2 experiences “Low-intensity BMOD, followed by non-response, followedby BMOD + MEDS”

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Notation and Data Structure

K stages (or decision points) on a single patient:

O1,A1, . . . ,OK ,AK ,OK+1

Oj : Observation (pre-treatment) at the j-th stageAj : Treatment (action) at the j-th stage, Aj ∈ Aj

Hj : History at the j-th stage,Hj = {O1,A1, . . . ,Oj−1,Aj−1,Oj}Y : Primary Outcome (assume larger is better)

A DTR is a sequence of decision rules:

d ≡ (d1, . . . , dK) with dj(hj) ∈ Aj

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The Big Scientific Questions in DTR Research

What would be the mean outcome if the population were to follow a particularpre-conceived DTR?

How do the mean outcomes compare among two or more DTRs?

What is the optimal DTR in terms of the mean outcome?

– What individual information (tailoring or prescriptive variables) do we use to makethese decisions?

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The Big Statistical Questions

1 What is the right kind of data for comparing two or more DTRs, or estimatingoptimal DTRs? What is the appropriate study design?

– Sequential Multiple Assignment Randomized Trial (SMART)

2 How can we compare pre-conceived, embedded DTRs?– primary analysis of SMART data

3 How can we estimate the “optimal” DTR for a given patient?

– secondary analysis of SMART data– e.g. Q-learning, a stagewise regression-based approach

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Outline





5 Discussion

SMART Design

DTR Discovery: Experimental Data

As is well-known, experimental data are generally better than observational data(if you can afford)

Standard RCTs assess the effectiveness of a single dose-level of a singletreatment, as compared to another

Estimating the sequence of treatments that optimizes response in a longitudinalsetting requires studying the elements in the sequence

Can this be accomplished by a series of single-stage RCTs?

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SMART Design

Example: Treating MDD

Suppose we wish to compare both front-line and second-line treatment of majordepressive disorder (MDD):

– Front-line options: citalopram (Cit) or cognitive behavioral therapy (CBT)

– Second-line options: treatment switch to Cit, CBT, or Lithium (Li)

– All responders to first-line therapy will continue with maintenance and follow-up

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SMART Design

Example: Treating MDD

CBT

Cit

R

Maintenance dose +telephone monitoring

RCBT

Li

RCit

Li

Telephone monitoring

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SMART Design

Front-line Treatment of MDD

CBT

Cit

R

Suppose we observe 60% response with Cit, andonly 50% with CBT

Conclude: Cit is the best front-line therapy

Now run another one-stage trial amongst Citnon-responders

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SMART Design

Second-line Treatment of MDD

Cit


RCBT

Li

60% respond

We now observe 40% response to CBT and 20% to Li

Conclude: CBT is the best second-line therapy

Final treatment sequence: “Cit, followed by CBT for non-responders”

– Under this regimen, we expect to see 76% of patients respond

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SMART Design

Delayed Effect

What if initial treatment with CBT increases treatment adherence⇒ subsequenttherapies more successful?

CBT

Cit

R


RCBT

Li

RCit

Li

Telephone monitoring

60% respond

50% respond

40% respond

20% respond

60% respond

30% respond

Optimal DTR: “CBT, followed by Cit for non-responders”; 80% responseexpected

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SMART Design

SMART!

Two single-stage trials would not have detected the best overall strategy fortreatment

Instead, we should have used a two-stage trial

Such two-stage trials are known as SMARTs (Lavori and Dawson, 2004;Murphy, 2005), i.e.

Sequential Multiple Assignment Randomized Trials

Susan Murphy won the MacArthur Foundation “Genius Grant” in 2013 forinventing the SMART design:

http://www.macfound.org/fellows/898/

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SMART Design

SMARTs, in general

Multi-stage trials with a goal to inform the development of DTRs

Same subjects participate throughout (they are followed through stages oftreatment)

Each stage corresponds to a treatment decision

At each stage the patient is randomized to one of the available treatment options

Treatment options at randomization may be restricted on ethical grounds,depending on intermediate outcome and/or treatment history

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SMART Design

SMART: The Gains

Ability to detect

– delayed therapeutic effects (treatment interactions)

– diagnostic effects

More generalizable than standard RCT?

Better recruitment and retention potential than standard RCT?– By virtue of the option to alter a non-functioning treatment while in the trial

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SMART Design

SMART: The Costs

Unfortunately there is no free lunch!!

More expensive than a single-stage RCT

Longer follow-up

Need more participants only if the trial is powered to compare whole sequences oftreatments – but there exist less expensive alternatives to power SMARTs!

More complex methods for planning and analysis: requires an experiencedstatistician, or one with time to learn new methods

May require additional work to fund: still relatively new, unfamiliar

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SMART Design

SMART vs. Crossover Trial Designs

Operationally, SMARTs look similar to crossover trials

However, conceptually they are very different

– Unlike SMART, Crossover trials aim to evaluate stand-alone treatments, not DTRs

– Unlike a crossover trial, treatment allocation in a SMART is typically adaptive to asubject’s intermediate outcome

– Crossover trials attempt to “wash out” the “carry-over” effect while SMARTsattempt to capture it

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SMART Design

SMART vs. Usual Adaptive Trial Designs

SMARTs are different from usual adaptive trials

– Within-subject vs. between-subject adaptation

In an adaptive trial, the randomization probabilities can change during the courseof the trial depending on the relative success of already-recruited patients oneach of the treatments

– Adaptive trials typically aim to reduce the number of trial participants exposed tothe inferior treatment

– Adaptive trials are most useful for examining questions where the outcome followsfairly quickly after treatment and/or the recruitment process is (relatively) slow

– SMARTs are fixed trial designs – while randomization probabilities may depend oncovariates, those probabilities remain constant over the course of the trial

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SMART Design

From SMART to SMART-AR

Integration of the two concepts (SMART and adaptive design) is a topic ofcurrent research

– SMARTs can be made more ethically appealing by incorporating adaptiverandomization

– In certain modern contexts (e.g., implementation research and mHealth), SMARTwith Adaptive Randomization (SMART-AR)1 has been developed recently

– In general, how best to do this is not known yet

1Cheung YK, Chakraborty B, and Davidson K (2015). Sequential multiple assignment randomized trial(SMART) with adaptive randomization for quality improvement in depression treatment program.Biometrics, 71(2): 450-459.

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SMART Design

Does anyone actually use SMARTs?

ADHD Trial (PI: Pelham; see Lei et al., 2012, for design details)

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SMART Design

Other Examples of SMART Studies

Schizophrenia: CATIE (Schneider et al., 2001)

Depression: STAR*D (Rush et al., 2003)

Prostate Cancer: Trials at MD Anderson Cancer Center (e.g., Thall et al., 2000)

Leukemia: CALGB Protocol 8923 (e.g., Stone et al., 1995; Wahed and Tsiatis,2004)

Smoking Cessation: Project Quit (Strecher et al., 2008; Chakraborty et al.,2010)

Alcohol Dependence: Oslin et al. (see, e.g., Lei et al., 2012)

Many recent examples available at the Methodology Center, PennStateUniversity website:

http://methodology.psu.edu/ra/smart/projects

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SMART Design Primary Analysis and Sample Size

SMART Design Principles

Primary and Secondary Hypotheses

Choose scientifically important primary hypotheses that also aid in developingDTRs

– Power trial to address these hypotheses

Choose secondary hypotheses that further develop the DTR, and userandomization to eliminate confounding

– Trial is not necessarily powered to address these hypotheses

– Still better than post hoc observational analyses

– Underpowered randomizations can be viewed as pilot studies for future full-blowncomparisons

From a funding perspective, it may be more feasible to design a standardtwo-arm trial as primary goal, and add the secondary randomizations and DTRestimation as a secondary goal

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SMART Design: Embedded DTRs

Embedded DTR-1

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Embedded DTR-2

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Embedded DTR-3

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Embedded DTR-4

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Primary Hypothesis and Sample Size: Scenario 1

Hypothesize that averaging over the secondary treatments, the initial treatmentBMOD is as good as the initial treatment MEDS

– Sample size formula is same as that for a two group comparison

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Hypothesize that among non-responders an augmentation (BMOD+MEDS) is asgood as an intensification of treatment

– Sample size formula is same as that for a two group comparison ofnon-responders (overall sample size depends on the presumed non-response rate)

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Hypothesize that the “red” DTR (embedded DTR-2) is as good as the “green” DTR(embedded DTR-3)

– Sample size formula involves a two group comparison of “weighted” means(overall sample size depends on the presumed non-response rate)

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A Bit of Formalism...

Objective: compare two embedded regimens in the SMART with continuousoutcome

Embedded regimens are:

d1 =(

BMOD,BMODR(BMOD + MEDS)1−R)

d2 =(

BMOD,BMODR(BMOD + +)1−R)

d3 =(

MEDS,MEDSR(BMOD + MEDS)1−R)

d4 =(

MEDS,MEDSR(MEDS + +)1−R)

(where R = 1 denotes responder and R = 0 denotes non-responder)

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Data Structure

(O1i,A1i,O2i,A2i,Yi) for i = 1, · · · ,N, where

– O1 = ∅

– A1 ∈ {BMOD,MEDS}

– O2 = R is the response indicator (1/0)

– A2 ∈ {BMOD,BMOD ++,MEDS,MEDS ++,BMOD + MEDS}

– Y is the continuous end-of-trial primary outcome

– N is the trial size

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Regimen Mean: The Key Estimand

For the embedded regimen d1 = (BMOD,BMODR(BMOD + MEDS)1−R), themean outcome is

µd1 = Ed1(Y)

= E[ I{A1 = BMOD,A2 = BMODR(BMOD + MEDS)1−R}

π1(A1)π2(A2|A1,R)Y]

= E(Wd1 Y), say

– π1 and π2 are the randomization probabilities

– Inverse probability weighting (IPW) approach

– Change of probability distribution (measure) by Radon-Nikodym derivative

Similarly, one can write expressions for other regimens d2, d3, d4

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Estimation of Regimen Mean

The regimen mean can be estimated by the IPW estimator (Robins et al., 2000):

Yd1 =

N∑i=1

Wd1i Yi

/ N∑i=1

Wd1i

For large samples (Murphy, 2005):

Var(Yd1) =1

N2

N∑i=1

(Wd1i )2(Yi − Yd1)

2

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Sample Size Requirements

Assume continuous outcome

Key Design Parameters:

Effect Size = ∆µσ (Cohen’s d)

Type I Error Rate = α = 0.05Desired Power = 1− β = 0.8

Initial Response Rate = γ = 0.5

Trial Size:

Effect Scenario 1 Scenario 2 Scenario 3Size0.3 N1 = 350 N2 = N1

(1−γ) = 700 N3 = N1 × (2− γ) = 5250.5 N1 = 128 N2 = N1

(1−γ) = 256 N3 = N1 × (2− γ) = 1920.8 N1 = 52 N2 = N1

(1−γ) = 104 N3 = N1 × (2− γ) = 78

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SMART Design: Other Comparisons

Comparing embedded DTR-1 (d1) with embedded DTR-2 (d2)

d1 and d2 share the initial treatment of BMOD

Patients who respond to BMOD are ‘consistent’ with both d1 and d2, andcontribute information on the overall response to both regimens

Specialized methods are needed to account for “re-using” their information

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SMART Design: Other Comparisons

Alternatively, we may wish to:compare d1 vs. d2 vs. d3 vs. d4 to see which of the four regimens leads to the bestexpected outcome

The above question requires more specialized methods to account for bothmultiple comparisons as well as the re-use of data for patients who are consistentwith more than one regimen

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Outline





5 Discussion

Analysis: Estimation of Optimal DTRs

Secondary Analysis of SMART Data

Goal: To find optimal treatment sequence for each individual patient by deeplytailoring on their time-varying covariates and intermediate outcomes

– This is to develop the evidence-based decision support system for clinicians (theycan apply these regimens while treating future patients)

Methodologically challenging – custom-made analytic tools necessary

– One popular approach is Q-learning, a stage-wise regression-based method

– We developed an R package called qLearn (Xin et al., 2012) that conductsQ-learning (Freely available at CRAN):

http://cran.r-project.org/web/packages/qLearn/

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Analysis: Estimation of Optimal DTRs Q-learning

Q-learning: Introduction

Q-learning (Watkins, 1989)

– A popular method from Reinforcement (Machine) Learning

– A generalization of least squares regression to multistage decision problems(Murphy, 2005)

– Adapted to and implemented in the DTR context with several variations (Zhao et al.,2009; Chakraborty et al., 2010; Schulte et al., 2012; Song et al., 2014)

– Easy to understand and implement, and forms a good basis for more complexapproaches

The intuition comes from dynamic programming (Bellman, 1957) in case themultivariate distribution of the data is known

– Q-learning is an approximate dynamic programming approach

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Dynamic Programming (DP): The Background

Two Stages

Move backward in time to take care of the delayed effects

Define the “Quality of treatment”, Q-functions:

Q2(h2, a2) = E[Y∣∣∣H2 = h2,A2 = a2

]Q1(h1, a1) = E

[max

a2Q2(H2, a2)︸︷︷︸

delayed effect

∣∣∣H1 = h1,A1 = a1

]

Optimal DTR:

dj(hj) = arg maxaj

Qj(hj, aj), j = 1, 2

What if the true Q-functions are unknown?

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From DP to Q-learning

DP requires modeling the joint multivariate distribution (likelihood) oftime-varying covariates and outcome

This is hard!– The knowledge needed to model the joint distribution is often unavailable

– Model mis-specification can lead to incorrect conclusions

Turn to semi-parametric methods

In particular, Q-learning estimates the true Q-functions from data usingregression models

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Q-learning: Typical Implementation (K = 2)

Linear regression models for Q-functions:

Qj(Hj,Aj;βj, ψj) = βTj Hj + (ψT

j Hj)Aj, j = 1, 2,

At stage 2, regress Y on (H2, H2A2) to obtain (β2, ψ2)

Construct stage-1 Pseudo-outcome:

Y1i = maxa2

Q2(H2i, a2; β2, ψ2), i = 1, . . . , n

At stage 1, regress Y1 on (H1, H1A1) to obtain (β1, ψ1)

Estimated Optimal DTR:

dj(hj) = arg maxaj

Qj(hj, aj; βj, ψj) = sign(ψTj hj)

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Q-learning: Remarks

Q-learning is appealing because it is easy to implement in standard software, andeasy to explain to clinical collaborators who are familiar with regression

The approach has several limitations, e.g.:– Not robust to model mis-specification

– Only limited results are available for discrete outcomes

More sophisticated approaches exist, at least for some treatment/outcome types

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Outline





5 Discussion

Non-regular Inference

Inference for Optimal Regimen Parameters

dj(hj) = sign(ψTj hj)

“Regimen parameters” ψj – parameters that index the decision rules

– Reduce the number of variables on which data must be collected for futureimplementations of the DTR

– Know when there is insufficient evidence in the data to recommend one treatmentover another – choose treatment based on cost, familiarity, preference etc.

Inference for the optimal regimen parameters based on Q-learning has been atopic of active research for last 10+ years (Robins, 2004; Moodie and Richardson,2010; Chakraborty et al., 2010; 2013; Laber et al., 2014; Song et al., 2014)

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Non-regularity in Inference for ψ1 (K = 2)

Y1i = maxa2 Q2(H2i, a2; β2, ψ2) = βT2 H2i + |ψT

2 H2i|

Due to the non-differentiability of Y1i, the asymptotic distribution of ψ1 does notconverge uniformly over the parameter space – non-regular (Robins, 2004; Laberet al., 2014)

– It is problematic if p > 0, where p def= P[ψT

2 H2 = 0]

– The problem persists even when |ψT2 H2| is “small” with non-zero probability (“local

asymptotics”; Laber et al., 2011, 2014)

Practical consequence: Both Wald type CIs and standard bootstrap CIs performpoorly (Robins, 2004; Moodie and Richardson, 2010; Chakraborty et al., 2010)

In a K-stage setting, the same issues arise for all ψk, k = K − 1, . . . , 1

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m-out-of-n Bootstrap: A Feasible Solution

m-out-of-n bootstrap is a tool for remedying bootstrap inconsistency due tonon-smoothness (Shao, 1994; Bickel et al., 1997)

Efron’s nonparametric bootstrap with a smaller resample size, m = o(n)

Choice of m has always been difficult – resulting in a historical lack of popularityof the approach

We developed a choice of m for the regime parameters in the context ofQ-learning – adaptive to the degree of non-regularity present in the data2

2Chakraborty B, Laber EB, and Zhao Y (2013). Inference for optimal dynamic treatment regimes usingan adaptive m-out-of-n bootstrap scheme. Biometrics, 69: 714 - 723.

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Our Approach

Key idea: Since non-regularity arises when p > 0, an adaptive choice of mshould depend on an estimate of p

Consider a class of resample sizes: m = n1+α(1−p)

1+α , where α > 0 is a tuningparameter

Estimate p by “pre-test” of ψT2 H2 = 0 for fixed H2 over the training data set:

p =1n

n∑i=1

I

{n(ψT

2 H2,i)2

HT2,iΣ2H2,i

≤ χ21,1−ν

}

Plug in p for p in the above formula for m to get: m = n1+α(1−p)

1+α

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Implementation

α can be chosen in a data-driven way via double-bootstrapping (Davison andHinkley, 1997)

R package qLearn: http://cran.r-project.org/web/packages/qLearn/

Constructing one CI via double bootstrap takes about 3 minutes on a machinewith dual core 2.53 GHz processor and 4GB RAM

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Inference for ψ10: Simulation Design

A Simple Class of Generative Models

O1,A1,A2 ∈ {−1, 1} with probability 0.5

O2 ∈ {−1, 1} with P[O2 = 1|O1,A1] =exp(δ1O1 + δ2A1)

1 + exp(δ1O1 + δ2A1)

Y|· ∼ N(γ1 + γ2O1 + γ3A1 + γ4O1A1 + γ5A2 + γ6O2A2 + γ7A1A2, 1)

Analysis Model:

Q2 = β20 + β21O1 + β22A1 + β23O1A1 + (ψ20 + ψ21O2 + ψ22A1)︸︷︷︸ψT

2 S2

A2

Q1 = β10 + β11O1 + (ψ10 + ψ11O1)A1

The size of the stage-2 treatment effect ψT2 S2 determines the extent of nonregularity, e.g.

p = P[ψT2 S2 = 0]

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Example Generative Models3

Example γT δT Type p1 (0, 0, 0, 0, 0, 0, 0) (0.5, 0.5) NR 12 (0, 0, 0, 0, 0.01, 0, 0) (0.5, 0.5) NNR 03 (0, 0,−0.5, 0, 0.5, 0, 0.5) (0.5, 0.5) NR 0.54 (0, 0,−0.5, 0, 0.5, 0, 0.49) (0.5, 0.5) NNR 05 (0, 0,−0.5, 0, 1.0, 0.5, 0.5) (1.0, 0.0) NR 0.256 (0, 0,−0.5, 0, 0.25, 0.5, 0.5) (0.1, 0.1) R 07 (0, 0,−0.25, 0, 0.75, 0.5, 0.5) (0.1, 0.1) R 08 (0, 0, 0, 0, 0.25, 0, 0.25) (0, 0) NR 0.59 (0, 0, 0, 0, 0.25, 0, 0.24) (0, 0) NNR 0

3Ex. 1 – 6 taken from Chakraborty et al. (2010), and Ex. 7 – 9 taken from Laber et al. (2014)58 / 66



Focus on the 95% nominal CI for the stage-1 treatment effect parameter ψ10

Compare Monte Carlo estimates of coverage and mean width of– n-out-of-n bootstrap (usual)– m-out-of-n bootstrap

1000 simulated data sets, each of size n = 300

1000 bootstrap replications to construct CIs

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Coverage and Mean Width of the 95% nominal CI for ψ10

Table : Coverage Rates (color-coded as under-coverage, nominal coverage)

Ex. 1NR

Ex. 2NNR

Ex. 3NR

Ex. 4NNR

Ex. 5NR

Ex. 6R

Ex. 7R

Ex. 8NR

Ex. 9NNR

n-out-of-n 0.936 0.932 0.928 0.921 0.933 0.931 0.944 0.925 0.922m-out-of-n 0.964 0.964 0.953 0.950 0.939 0.947 0.944 0.955 0.960

Table : Mean Width of CIs

Ex. 1NR

Ex. 2NNR

Ex. 3NR

Ex. 4NNR

Ex. 5NR

Ex. 6R

Ex. 7R

Ex. 8NR

Ex. 9NNR

n-out-of-n 0.269 0.269 0.300 0.300 0.320 0.309 0.314 0.299 0.299m-out-of-n 0.331 0.331 0.321 0.323 0.330 0.336 0.322 0.328 0.328

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m-out-of-n Bootstrap: Remarks

At least in case of SMARTs, non-regular settings (i.e., small treatment effects)are more likely to occur than one would think – due to clinical equipoise(Freedman, 1987)

– Hence any method of inference in the DTR context should deal with non-regularityseriously

m-out-of-n bootstrap offers a feasible solution to the inference problem– The procedure is consistent, and successfully adapts to the degree of non-regularity

present in the data

– It is conceptually simple, likely to be palatable to practitioners

– It has already been implemented in the R package qLearn, to facilitate widedissemination

Applying the m-out-of-n bootstrap procedure to settings with more stages andmore treatment choices per stage is conceptually straightforward, but can beoperationally messy

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Outline





5 Discussion

Discussion

Take-home Messages

SMART is a cutting-edge clinical trial design that formalizes the sequentialdecision making in clinical practice

– Very relevant for chronic conditions

– Does not necessarily require a lot of sample size (depends on the primary question)

A variety of methods exist for (secondary) analysis of SMART data– Q-learning is probably the simplest, and already implemented in the R package

qLearn

– More non-parametric methods (e.g. outcome-weighted learning) have beendeveloped more recently

Inference is tricky, but methods exist to handle the associated non-regularity

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Shameless Self-Promotion!!

Shoot your questions, comments, criticisms, request for slides to:[email protected]

Discussion

Why move through stages? Why not run an “all-at-once”multivariable regression?

Berkson’s Paradox or Collider-stratification Bias: There may be non-causal association(s) evenwith randomized data, leading to biased stage-1 effects (Berkson, 1946; Greenland, 2003;

Murphy, 2005)66 / 66

design and analysis of sequential multiple assignment...

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