SAMSI Tutorial on Dynamic Treatment Regimes by Anastasios Tsiatis

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SAMSI Tutorial on Dynamic Treatment Regimes by
Anastasios Tsiatis

• Dr. Gong Tang
• Wentao Feng
• Sachiko Miyahara

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Introduction

• Goal of Physicians
• To give treatment to patients over
• time that will result in as favorable a
• clinical outcome as possible.

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Introduction (Cont.)

• When a new patient comes to a physicians
  office, the physician needs to make many
  decisions such as
• - Treatment Choice
• - Dose
• - When to switch
• gt Complex and often difficult to know

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Goal of This Presentation

• To find the distribution of the responses,
• based on different treatment regimes,
• using observed data from
• a controlled intervention study
• an observational study

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Notations

• For time point j 0 to k,
• Lj covariate information collected
• between time tj-1 and tj
• Aj treatment assigned at time tj
• Y Outcome

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Notations (Cont.)
L0
L1
L2
Lk

Ak
Y

A0
A1
A2
t0
t1
t2
tk
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Notations (cont.)

• (L0, Lj)
• The history of time dependent covariates
• (A0, Aj)
• The history of time dependent treatment
  decisions

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Treatment Regimes

• What is a treatment regime?
• an algorithm which dictates how each patient in
  the population treated possibly based on
  intervening covariate information.
• In formula g(tj, ) aj
• where and

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Treatment Regimes Example

• Example HIV Study
• Let L1j CD4 counts
• aj 1 to give antiretroviral therapy
• 0 to not to give the therapy
• The treatment regime
• g(tj, ) I(CD4j lt 200)

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Methods

• What are the methods to estimate the
• distribution of Y for various g from the observed
• data?
• - G-computation algorithm
• - Inverse Probability Weighting
• Need to consider
• 1. Concept of Potential Outcomes
• 2. Three assumptions

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Potential Outcomes

• Denoted as Y(g)
• Y( ) is the potential outcome of a randomly
  selected individual in our population if he/she
  hypothetically received treatment a0 at time t0,
  a1 at time t1ak at time tk
• L( ) is also referred to as potential outcome
• Also called Counterfactuals

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Potential Outcomes

• The set of all potential outcomes denoted by
• W L0(g), L1(g) Lk(g), Y(g)
• where
• L0(g) L0
• L1(g) L1(g(t0, L0)
• Lk(g) Lkg(t0, L0), , g(tk-1, (g))
• Y(g) Yg(t0, L0), , g(tk-1, (g))

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Three Assumptions

• 1. Consistency Assumptions
• 2. Sequential randomization assumption
• 3. Identification assumption

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1. Consistency Assumptions

• Assume
• Y Y( )
• Lk L( )
• In words, we assume that the potential outcome
  corresponds to observed outcome.

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2. Sequential Randomization Assumption

• No Unmeasured Confounder Assumption
• Assume
• W __ Aj ( , ) for all j 0,k
• In words, conditioning on the history of time
  dependent treatments and covariate information up
  to time tj, the treatment Aj is independent of
  the set of potential outcomes

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3. Identification Assumption

• Assume if every covariate-treatment history up to
  time tj that has a positive probability of
  observed, then there must be a positive
  probability that the corresponding treatment will
  be observed
• Example violated assumption case
• Lj shows an adverse event, so that no aj is
  given, then this assumption is violated.

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Purpose
To derive the distribution of potential outcomes
From observed data
For example, if the potential outcome Y is
survival time, we may be interested in estimating
or mean
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Inverse probability weighting
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Inverse probability weighting (continued)
The probability that a patient received regime
is
So,
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Proof of consistency of inverse probability
weighted estimator
Consistency assumption

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G-computation algorithm
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Estimating procedure

• Solve the estimating equation

• to get the estimated parameters for the
  conditional distributions,
• Then integrate out Ls to get the marginal
  distribution of

• Compare the distribution of for
  different gs.