Hypothesis Testing and Dynamic Treatment Regimes

S.A. Murphy, L. Gunter & B. Chakraborty

ENAR

March 2007

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Outline

• Dynamic treatment regimes• Constructing and addressing questions regarding

an optimal dynamic treatment regime• Why and when non-regular?• A Solution• Simulation Results.

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Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing according to patient outcomes. Operationalize clinical practice.

k Stages for one individual

Observation available at jth stage

Action at jth stage

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k Stages

History available at jth stage

“Reward” following jth stage (rj is a known function)

Primary Outcome:

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Goal:

Construct decision rules that input information in the history at each stage and output a recommended decision; these decision rules should lead to a maximal mean Y.

The dynamic treatment regime is the sequence of decision rules:

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In the future we employ the actions determined by the decision rules:

An example of a simple decision rule is: alter treatment at time j if

otherwise maintain on current treatment; Sj is a summary of the history, Hj.

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Data for Constructing the Dynamic Treatment Regime:

Subject data from sequential, multiple assignment, randomized trials. At each stage subjects are randomized among alternative options.

Aj is a randomized action with known randomization probability.

binary actions with P[Aj=1]=P[Aj=-1]=.5

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Sequential, Multiple Assignment Randomized Studies

• CATIE (2001) Treatment of Psychosis in Schizophrenia

• STAR*D (2003) Treatment of Depression• Tummarello (1997) Treatment of Small Cell Lung

Cancer (many, for many years, in this field)• Oslin (on-going) Treatment of Alcohol

Dependence• Pellman (on-going) Treatment of ADHD

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Sequential Multiple Assignment Randomization

Initial Txt Intermediate Outcome Secondary Txt

Relapse

Responder R Prevention

Low-levelMonitoring

Switch toTx C

Tx A

Nonresponder RAugment withTx D

R

Responder Relapse

R Prevention

Low-levelMonitoring

Tx B

Switch toTx C

Nonresponder R

Augment withTx D

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Constructing and Addressing Questions Regarding an Optimal Dynamic

Treatment Regime

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Regression-based methods for constructing decision rules

•Q-Learning (Watkins, 1989) (a popular method from computer science)

•A-Learning or optimal nested structural mean model (Murphy, 2003; Robins, 2004)

•The first method is an inefficient version of the second method when each stages’ covariates include the prior stages’ covariates and the actions are centered to have conditional mean zero.

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(k=2)

Dynamic Programming

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Approximate for S', S vector summaries of the history and

A Simple Version of Q-Learning –binary actions

• Stage 2 regression: Use least squares with outcome, Y, and covariates to obtain

• Set

• Stage 1 regression: Use least squares with outcome, and covariates to obtain

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Decision Rules:

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Why non-regular?

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Non-regularity

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When do we have non-regularity?

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A Soft-Max Solution

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A Soft-Max Solution

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Distributions for Soft-Max

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Regularized Q-Learning (binary actions)

• Set

• Stage 1 regression: Use least squares with outcome,

and covariates to obtain

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Interpretation of λ

Future treatments are assigned with equal probability, λ=0

Optimal future treatment is assigned, λ=∞

Future treatment =1 is assigned with probability

Estimator of Stage 1 Treatment Effect when

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Interpretation of λ

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Proposal

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Proposal

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Proposal

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Simulation

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P[β2TS2=0]=1 β1(∞)=β1(0)=0

Test Statistic Nominal Type 1 based on Error=.05 .045

.047

.034*

.024*

(1)Nonregularity results in low Type 1 error

(2)Additional smoothing due to use of is useful.

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P[β2TS2=0]=1 β1(∞)=β1(0)=.1

Test Statistic Power based on

.15

.14

.10

.09

(1)The low Type 1 error rate translates into low power

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Test Statistic Power based on

.05

.13

.12

.12

(1) Averaging over the future is not a panacea

P[β2TS2=0]=0 β1(∞)=.125, β1(0)=0

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Test Statistic Type 1 Error=.05 based on

.57

.16

.05

.05

(1) The price is that the null hypothesis is altered.

P[β2TS2=0]=.25 β1(∞)=0, β1(0)=-.25

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Discussion

• We replace the hypothesis test concerning a non-regular parameter, β1(∞) by a hypothesis test concerning a near-by regular parameter β1(λ*).

• This is work in progress—limited theoretical results are available.

• If you let increase with the sample size you again end up with a non-regular problem (convergence to limiting distribution is locally non-uniform).

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Discussion

• Robins (2004) proposes several conservative confidence intervals for β1.

• Ideally to decide if the two stage 1 treatments are equivalent, we would evaluate whether the choice of stage 1 treatment influences the mean outcome resulting from the use of the dynamic treatment regime. We did not do this here.

• Constructing “evidence-based” regimes is of great interest in clinical research and there is much to be done by statisticians.

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

seminars/ENAR0307.ppt

Email me with questions or if you would like a copy!

samurphy@umich.edu