Planning Survival Analysis Studies of Dynamic Treatment Regimes
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Transcript of Planning Survival Analysis Studies of Dynamic Treatment Regimes
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:
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.