Experiments and Dynamic Treatment Regimes

S.A. MurphyUniv. of Michigan

Yale: November, 2005

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• Joint work with– Derek Bingham (Simon Fraser)– Linda Collins (PennState)

• And informed by discussions with– Vijay Nair (U. Michigan)– Bibhas Chakraborty (U. Michigan)– Vic Strecher (U. Michigan)

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Outline

• Dynamic Treatment Regimes

• Challenges in Experimentation

• Defining Effects and Aliasing

• Examples

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Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing with ongoing subject need. Mimic Clinical Practice.

Dynamic Treatment Regimes

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k Decisions on one individual

Observation made prior to jth decision point

Treatment at jth decision point

Primary outcome Y is a specified summary of decisions and observations

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A dynamic treatment regime is a vector of decision rules, one per decision

where each decision rule

inputs the available information

and outputs a recommended treatment decision.

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Challenges in Experimentation

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Dynamic Treatment Regimes (review)

Constructing decision rules is a multi-stage decision problem in which the system dynamics are unknown.

Analysis methods for observational data dominate statistical literature (Murphy, Robins, Moodie & Richardson, Tsiatis)

Better data provided by sequential multiple assignment randomized trials: randomize at each decision point— à la full factorial.

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Reality

Unknown UnknownCauses Causes

X1 T1 X2 T2 Y

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Challenges in ExperimentationDynamic Treatment Regimes are multi-component treatments:

many possible components

• decision options for improving patients are often different from decision options for non-improving patients,

• multiple components employed simultaneously

• medications, adjunctive treatments, delivery mechanisms, behavioral contingencies, staff training, monitoring schedule…….

• Future: series of screening/refining, randomized trials prior to confirmatory trial --- à la Fisher/Box

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Screening experiments (review)1) Goal is to eliminate inactive factors (e.g. components) and

inactive effects.

2) Each factor at 2 levels

3) Screen marginal causal effects

4) Design experiment using working assumptions concerning the negligibility of certain effects. (Think ANOVA)

5) Designs and analyses permit one to determine aliasing (caused by false working assumptions)

6) Minimize formal assumptions

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Screening experiments

When the goal is to construct/optimize a dynamic treatment regime can we

design screening experiments using working assumptions concerning the marginal causal effects

andprovide an analysis method that permits the determination of the aliasing??

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Defining the Effects

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Defining the stage 2 effects

Two decisions (two stages):

Define effects involving T2 in an ANOVA decomposition of

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Defining the stage 1 effects (T1)

Unknown UnknownCauses Causes

X1 T1 R T2 Y

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Defining the stage 1 effects

Unknown UnknownCauses Causes

X1 T1 R T2 Y

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Defining the stage 1 effects

Define

Define effects involving only T1 in an ANOVA decomposition of

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Why marginal, why uniform?Define effects involving only T1 in an ANOVA

decomposition of

1) The defined effects are causal.

2) The defined effects are consistent with tradition in experimental design for screening.

– The main effect for one treatment factor is defined by marginalizing over the remaining treatment factors using an uniform distribution.

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Why marginal, why uniform?2) The defined effects are marginal consistent with

tradition in experimental design for screening.– The main effect for one treatment factor is defined by

marginalizing over the remaining factors using an uniform distribution.

When there is no R, the main effect for treatment T1 is

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An Aside: Ideally you’d like to replace

by

(X2 is a vector of intermediate outcomes)

in defining the effects of T1.

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Use an ANOVA-like decomposition:

Representing the effects

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where

Causal effects:

Nuisance parameters: and

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General FormulaNew ANOVA

Z1 matrix of stage 1 treatment columns, Z2 is the matrix of stage 2 treatment columns, Y is a vector

Classical ANOVA

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Aliasing{Z1, Z2} is determined by the experimental design

The defining words (associated with an fractional factorial experimental design) identify common columns in the collection {Z1, Z2}

ANOVA

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Aliasing

ANOVA

Consider designs with a shared column in both Z1 and Z2 only if the column in Z1 can be safely assumed to have a zero η coefficient or if the column in Z2 can be safely assumed to have a zero β, α coefficient. The defining words provide the aliasing in this case.

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Simple Examples

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Five Factors:

M1, E, C, T, A2(only for R=1), M2(only for R=0), each with 2 levels

(26= 64 simple dynamic treatment regimes)

The budget permits 16 cells --16 simple dynamic treatment regimes.

Simple Example

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Design: 1=M2M1ECT=A2M1ECT M1 E C T A2=M2

- - - - +

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- + + + -

+ - - - -

+ - - + +

+ - + - +

+ - + + -

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+ + + - -

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Assumptions

A2C, A2T, M2E, M2T and CE along with the main effects in stage 1 and 2 are of primary interest.

• Working Assumption: All remaining causal effects are likely negligible.

• Formal Assumption: Consider designs for which a shared column in Z1 and Z2 occurs only if the column in Z1 can be safely assumed to have a zero η coefficient (concerns interactions of stage 1 factors with R) or if the column in Z2 can be safely assumed to have a zero β/α coefficient (stage 2 effects).

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Design 1

• No formal assumptions. 1=M1ECT

• The design column for A2/M2 is crossed with stage 1 design.

• EC is aliased with M1T. The interaction EC is of primary interest and the working assumption was that M1T is negligible.

• A2C is aliased with A2M1ET. The interaction A2C is of primary interest and the working assumption was that A2M1ET is negligible.

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Design 1

• I=M1ECT

• Screening model:

• The estimator estimates the sum of the effects of CE and M1T.

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Design 2

• Formal assumption: No three way and higher order stage 2 causal effects & no four way and higher order effects involving R and stage 1 factors.

1=M2M1ECT=A2M1ECT

• M2T and A2T are aliased with M1CE; the interaction M2T (A2T) is of primary interest and the working assumption was that M1CE is negligible.

• M2M1T is negligible so CE is not aliased.

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Design 2

• Screening model:

• The estimator estimates the sum of the effects of M2T and of M1CE

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Design 3

• Formal assumption: No four way and higher order causal effects & no three way and higher order effects involving R and first stage factors.

I=M2M1ECT=A2M1ECT

•CE aliased with both A2M1T and with M2M1T

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Design 3

Screening model:

The estimator estimates the sum of the effects of CE and A2M1T

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Discussion

In classical screening experiments we

• Screen marginal causal effects

• Design experiment using working assumptions concerning the negligibility of the effects.

• Designs and analyses permit one to determine aliasing

• Minimize formal assumptions

We can do this as well when screening for multi-stage decision problems!

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Discussion

• Compare this to using observational studies to construct dynamic treatment regimes– Uncontrolled selection bias (causal misattributions)– Uncontrolled aliasing.

• Secondary analyses would assess if variables collected during treatment should enter decision rules.

• This seminar can be found at: http:// www.stat. lsa.umich.edu/~samurphy/seminars/Yale11.05.ppt

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Reality

Unknown UnknownCauses Causes

X1 T1 X2 T2 Y

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Conceptual Model

Unknown UnknownCauses Causes

X1 T1 X2= R T2 Y

The meaning of T2 depends on R