Hypothesis Testing and Dynamic Treatment Regimes

S.A. Murphy

Schering-Plough Workshop

May 2007

2

Collaborators

• Lacey Gunter

• A. John Rush

• Bibhas Chakraborty

3

Outline

• Dynamic treatment regimes

• Constructing a dynamic treatment regime

• Non-regularity & an adaptive solution

• Example/Simulation Results.

4

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 (usually a treatment)

5

Goal: Construct decision rules that input information available at each stage and output a recommended decision; these decision rules should lead to a maximal mean Y where Y is a function of

The dynamic treatment regime is the sequence of two decision rules:

k=2 Stages

6

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

7

Sequential Multiple Assignment Randomization

Stage 1 Intermediate Outcome Stage 2

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

8

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

9

Two Levels of STAR*D (Tx-resistent Depression)

Stage 1 Stage 2Preference Treatment Intermediate Treatment

Action Outcome Action

Mirtazapine

Switch R Remission Continue on Current Tx

Nortriptyline

TranylcypromineLithium

Augment R Non-remission R

ThyroidMirtazapine + Venlafaxine

10

Outline

• Dynamic treatment regimes

• Constructing a dynamic treatment regime

• Non-regularity & an adaptive solution

• Example/Simulation Results.

11

A natural approach: Myopic Decisions

•Evaluate each stage of treatment in isolation; the dependent variable is 1 if remission in that stage, 0 otherwise.

•In stage 1 there are two treatment actions for those who prefer a switch in treatment (Mirtazapine or Nortriptyline) and two treatment actions for those who prefer an augment (Lithium or Thyroid).

•Compare the two switches in treatment according to the remission rate achieved by end of stage 1. Do the same for the two augments.

12

Need an alternative

•This is not a good idea if we want to evaluate the sequence of treatments (e.g. adaptive treatment strategies).

• Some of the stage 1 non-remitters went on to have a remission in stage 2; these people have an dependent variable equal to 0 in the myopic analysis.

•the remission or lack of remission in stage 2 may be partially attributable to the stage 1 treatment.

•Patching together the separate analyses of the stages requires unnecessary causal assumptions.

13

Need an alternative for the stage 1 dependent variable

•What should the value of the stage 1 dependent variable be for those that do not remit and move to stage 2?

•We should not use a stage 1 dependent variable of Y=1 for those people who remit in stage 2.

•We should not use an stage 1 dependent variable of Y=0 for those people who remit in stage 2.

•The dependent variable should be something in between.

14

Regression-based methods for constructing decision rules

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

•Optimal nested structural mean model (Murphy, 2003; Robins, 2004; I like the term A-learning)

• When using linear models, 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.

15

There is a regression for each stage.

A Simple Version of Q-Learning –

• Stage 2 regression: Regress Y on to obtain

• Stage 1 regression: Regress on to obtain

16

for patients entering stage 2:

• is the estimated probability of remission in stage 2 as a function of variables that may include or be affected by stage 1 treatment.

• is the estimated probability of remission assuming the “best” treatment is provided at stage 2 (note max in formula).

• will be the dependent variable in the stage 1 regression for patients moving to stage 2

17

A Simple Version of Q-Learning –

• Stage 2 regression, (using Y as dependent variable) yields

• Stage 1 regression, (using as dependent variable) yields

18

Decision Rules:

19

Outline

• Dynamic treatment regimes

• Constructing a dynamic treatment regime

• Non-regularity & an adaptive solution

• Example/Simulation Results.

20

Non-regularity

21

Non-regularity

22

Non-regularity –

• Replace hard-max

• by soft-max

23

A Soft-Max Solution

24

Distributions for Soft-Max

25

To conduct inference concerning β1

• Set

• Stage 1 regression: Use least squares with outcome,

and covariates to obtain

26

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

27

Proposal

28

Proposal

29

Outline

• Dynamic treatment regimes

• Constructing a dynamic treatment regime

• Non-regularity & an adaptive solution

• Example/Simulation Results.

30

STAR*D

• Regression at stage 1:

•

•S1= ((1-Aug), Aug, Aug*Qids)

•X1 is a vector of variables available at or prior to stage 1, Aug is 1 if patient preference is augment and 0 otherwise

• We are interested in the β1 coefficients as these are used to form the decision rule at stage 1.

®T1S01+¯T1 S1A1

S01 = (1;X 1)

31

STAR*D

Decision Rule at stage 1:

•If patient prefers a Switch then

•if offer Mirtazapine, otherwise offer Nortriptyline.

•If patient prefers an Augment then

•if offer Lithium, otherwise offer Thyroid Hormone.

32

Stage 1 Augment Treatments

bbb

33

Decision Rules from Soft-Max Q-Learning Y=1 if remission or sufficient response to move to follow-up, Y=0 otherwise

Stage 1 Stage 2 Switch MIRT = NTP (225)

QIDS < 11 LI = THY (45) Augment QIDS ≥ 11 LI < THY (88)

TCP<VEN+MIRT(104)

= means not significant in two sided test at .05 level

< means significant in two sided test at .05 level

34

Simulation

35

P[β2TS2=0]=1 β1(∞)=β1(0)=0

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

.039

.025*

(1)Nonregularity results in low Type 1 error

(2) Adaptation due to use of is useful.

36

P[β2TS2=0]=1 β1(∞)=β1(0)=.1

Test Statistic Power based on

.15

.13

.09

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

37

Test Statistic Power based on

.05

.11

.12

(1) Averaging over the future is not a panacea

P[β2TS2=0]=0 β1(∞)=.125, β1(0)=0

38

Test Statistic Type 1 Error=.05 based on

.57

.16

.05

(1) Insufficient adaptation in “small” samples.

P[β2TS2=0]=.25 β1(∞)=0, β1(0)=-.25

39

Discussion

• We replace the test statistic based on an estimator of a non-regular parameter by an adaptive test statistic.

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

• The use of the bootstrap does not allow to increase too fast.

40

Discussion

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

• Ideally to decide if the 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.

41

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

seminars/Harvard0507.ppt

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

samurphy@umich.edu

42

STAR*D

• Regression at stage 2: α2TS2' + β2S2A2

• S2' =(1,X2, (1-Aug)*A1, Aug*A1, Aug*A1*Qids),

•(X2 is a vector of variables available at or prior to stage 2)

• S1 = 1

• Decision rule: Choose TCP if , otherwise offer Mirtazapine + Venlafaxine XR

43

Switch -.11(.07) -1.6

Augment .47(.25) 1.9

Augment*QIDS2 -.04(.02) -2.3

^̄(s:e:) z statistic

Stage 1 Coefficients

^̄11 =^̄12 =

^̄13 =