Planning Survival Analysis Studies of Dynamic Treatment

RegimesZ. Li & S.A. Murphy

UNC

October, 2009

Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing according to patient outcomes. Conceptualize treatment as a series of stages.

2 Stages for one individual

Observation available at jth stage

Action at jth stage (usually a treatment)

A dynamic treatment regime is the sequence of two decision rules: d1(X1), d2(X1,d1,X2) for selecting the actions in future.

In planning survival analysis trials, the observation X2 includes an indicator of response/non-response and whether the failure time has occurred.

Our Goal is to plan a sequential, multiple assignment, randomized trial (SMART). These are trials in which subjects are randomized among alternative options (the Aj’s are randomized) at each stage.

Simple regimes:

No X1, for example, d1= 1 (tx coded 1)

X2 = R, an indicator of early signs of non-response, d2(1,R) is 0 if R=1 (tx coded 0) otherwise stay on current tx

SMART• Precursors of the SMART design:

•CATIE (2001), STAR*D (2003), many in cancer

•SMART designs:•Treatment of Alcohol Dependence (Oslin, data analysis; NIAAA)•Treatment of ADHD (Pelham, data analysis; IES•Treatment of Drug Abusing Pregnant Women (Jones, in field; NIDA)•Treatment of Autism (Kasari, in field; Foundation)•Treatment of Alcoholism (McKay, in field; NIAAA)•Treatment of Prostate Cancer (Millikan, 2007)

ADHD (Pelham, PI)

A1=0. Begin low dosemedication

8 weeks

Assess-Adequate response?

Continue, reassess monthly; randomize if deteriorate

A2=0 Increase dose of medication

Randomassignment:

A2=1 Add BEMOD,medication dose remains stable

No

A1=1. Begin low-intensity BEMOD

8 weeks

Assess-Adequate response?

Continue, reassess monthly;randomize if deteriorate

A2=1 Add medication;BEMOD remains stable

Randomassignment:

A2=0 Increase intensity of BEMOD

Yes

No

Randomassignment:

Background

• Survival probabilities (and associated tests)– Lunceford et al. (2002) 3 weighted sample proportion estimators– Wahed and Tsiatis (2006) semiparametric efficient + implementable

estimator– Miyahara and Wahed (2009) weighted Kaplan-Meier estimator.– Feng and Wahed (2009) sample size formulae based on a Lunceford et

al. estimator– Guo and Tsiatis (2005) weighted cumulative hazard estimator

• Weighted version of the log rank test – Guo(2005) proposes weighted log rank test– Feng and Wahed (2008) weighted version of supremum log rank test

and associated sample size formulae

Notation

• Suppose we decide to size the study to compare regimes (A1, A2)= (1,1) versus (A1, A2)= (0,1)

• Randomization probabilities are p1, p2

• T11, T01 potential failure times under each regime

• T, S, C are the failure time, time to nonresponse, censoring time, respectively

• R is the nonresponse indicator, e.g. R=1S≤min(T,C)

Test Statistics

• Weighted version of the Kaplan-Meier to test

• Weighted version of the log rank test to test

Survival function Selected time point (usually end of study)

Weights are necessary to adjust for the trial design.

• Time independent weights (for regimes 11 and 01):

• Time dependent weights (potentially more efficient):

Weights

R=1S≤min(T,C)

Weighted Kaplan-Meier (WKM) Estimator

• Time dependent weights (tWKM):

-Asymptotically normal with mean and variance

• Can use the time independent weights (cWKM) as well.

(j,k)=(1,1), (0,1)

ith subject,

Weighted Log Rank Test (WLR)

• Time dependent weights (tWLR):

where (j,k)=(1,1), (0,1) and

• Asymptotically normal under a local alternative, PH assumption, with mean, and variance

Can use the time independent weights (cWLR) as well.

Sample Size Formulae

• Test based on WKM estimator:

• WLR test:

Challenges

• Variances are complex and depend on the joint distribution of the failure time T and the time to non-response, S.

• These two times are likely to be dependent.

• It may be hard to elicit information about this joint distribution in order to plan the trial.

Our Approach

• Use time independent weights in the sample size formulae (cWKM or cWLR).

• Express the variances in terms of the potential failure times under each regime, Tjk, e.g. in terms of

• Replace variances with simpler upper bounds.

The Variances

• cWKM:

• cWLR:

Upper Bounds on Variances

(Replace R by 1)

Sample Size Formulae• Test based on cWKM:

where

• cWLR:

Data Analysis

Use potentially more powerful tests than that used for sample size calculation.

Testing • Test based on tWKM • Test based on Lunceford 3 (Lunceford et al,

2002) • Test based on Wahed and Tsiatis, (2006)

implementable estimator, WT

Testing • tWLR

Simulation

• Proportional hazards for T11 and T01

• Frank Copula model for potential outcomes (Tjk, Sj)

• Weibull distributions for Tjk and Sj

• Compare with Feng and Wahed (2009) sample size formula:– Based on a weighted sample proportion estimator (the

second estimator in Lunceford et al., 2002).– Assumed independence between Tjk and Sj to simplify

variances.

Simulation Results for WKM(desired power is 80%)

Simulation Results for WKM(desired power is 80%)

Simulation Results for WLR (desired power is 80%)

Discussion• Working assumptions used to size the study are the same as in

the standard two arm study.

• Sample sizes are conservative, but the degree of conservatism depends on the percentage of subjects with R=1.

• cWLR yields smaller sample sizes than cWKM and needs less information, but power guarantees rely on proportional hazards assumption.

• These formulae can be easily generalized to more complex designs with the number of treatment options differing by both response status and prior treatment.

This seminar can be found at:

http://www.stat.lsa.umich.edu/~samurphy/

seminars/UNC.10.2009.ppt

Email Zhiguo Li or me with questions or if you would like a copy of the paper:

zhiguo@umich.edu or samurphy@umich.edu

Timing of movement between stages

The timing of the stages may be fixed or may be an outcome of treatment.

-----suppose the second stage is only for non-responders

Fixed timing: Second stage starts at 8 weeks after entry into trial.

Random timing: Second stage starts as soon as a nonresponse criterion is met.