Experiments and Dynamic Treatment Regimes

S.A. Murphy

Univ. of Michigan

JSM: August, 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

• Estimating Effects

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

•High variability across patients in response to any one treatment

•Relapse is likely without either continuous or intermittent treatment for a large proportion of people.

•What works now may not work later

•Exacerbations in disorder may occur if there are no alterations in treatment

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The Big Questions

•What is the best sequencing of treatments?

•What is the best timings of alterations in treatments?

•What information do we use to make these decisions?

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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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An 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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Long Term Goal: Construct decision rules that lead to a maximal mean Y.

An example of a decision rule is:

stop treatment if

otherwise maintain on current treatment.

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Challenges in Experimentation

1) Dynamic Treatment Regimes are multi-component treatments• Multiple decision points through time• Different kinds of decisions• Decision options for improving patients are often

different from decision options for non-improving patients

• Delivery mechanisms, encouragement-to-adhere training of staff…….

2) Constructing decision rules is a multi-stage decision problem

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Challenges in Experimentation

• Dynamic Treatment Regimes are High Dimensional Multi-component Treatments

•series of screening/refining, randomized trials prior to confirmatory trial (MOST)--- à la G. Box!

• Multistage Decisions

•sequential multiple assignment randomized trials (SMART): randomize at each decision point— à la full factorial.

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Challenges in Experimentation

3) In the screening experiment, resources are scarce relative to the number of interesting treatment components/factors. Implementing many cells of a full factorial is very expensive.

Consider designs that are similar to balanced fractional factorials. To do this you must define the effects.

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Defining the Effects

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Treatment of Alcohol Dependency

T1 Intermediate Outcome T2

TDM +Responder counseling

TDM

Med B

Med ANonresponder

EM + Med B+ Psychosocial

Intensive OutpatientProgram

Responder TDM +counseling

TDM

Med B

Med A

Nonresponder

EM +Med A +Psychosocial

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Conceptual Model

Unknown UnknownCauses Causes

X1 T1 R T2 Y

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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

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 uniform?Define effects involving only T1 in an ANOVA

decomposition of

1) The defined effects are causal; they are total effects.

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 treatment factors using an uniform distribution.

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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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Why uniform?

3) If R were always equal to 1 then the proposal is equivalent to defining both stage 2 and stage 1 effects in an ANOVA decomposition of

T2 denotes the treatment options when R=1.

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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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Stage 1 effects:

so the interesting stage 1 and stage 2 effects are contained in the same decomposition.

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To define effects of treatment factors at first and second stages use the ANOVA-like decomposition:

where

To design an experiment we make assumptions concerning the negligibility of these effects.

Summary

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where

Causal effects:

Nuisance effects:

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Estimating the Effects

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Estimating the effects

Two decisions (two stages):

Four cells corresponding to (T1,T2)= (1,1), (1,-1), (-1,1), (-1,-1). For R=1, cells are unequal in size and similarly for R=0.

Proposal: Estimate stage 2 effects using cell means

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Proposal: Estimate stage 2 effects using cell means. This yields the same estimators as a weighted regression analysis in which an individual in the (i,j)th cell is weighted by

where pij is the proportion of responders (R=1) in the (i,j)th cell.

Estimating the stage 2 effects

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Proposal: Estimate stage 2 effects using weighted regression with Y as the outcome variable.

• The advantage is that the design matrix is orthogonal with respect to the weights. – The alias structure is easily determined using

standard design of experiments techniques. – The estimators of the stage 2 effects are the same

regardless of whether you include nuisance effects in the regression.

Estimating the stage 2 effects

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Proposal: Use a regression with outcome variable,

and regressors equal to the stage 1 treatment factors (here T1).

Why?

Estimating the stage 1 effects

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Proposal: Estimate stage 1 effects using the outcome

• The advantage is that the design matrix is orthogonal (if more than one first stage treatment). – The alias structure is easily determined using

standard design of experiments techniques. – The estimators of the stage 1 effects are the same

regardless of how many of these effects you choose to include in the regression.

Estimating the stage 1 effects

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Discussion

• In the screening experiment the goal is to ascertain which decisions (“factors”) need further investigation; these are not confirmatory experiments.

• Some fractional factorial experiments will result in aliasing between causal effects and the nuisance effects. Using these experiments requires assumptions based on design principles such as effect hierarchy and effect heredity.

• It is unclear what kinds of secondary analyses are possible if the experiment is a fractional factorial.

• This seminar can be found at: http:// www.stat. lsa.umich.edu/~samurphy/seminars/jsm0805.ppt

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Challenges in Experimentation

(2 decisions)

Unknown UnknownCauses Causes

X1 T1 X2 T2 X3