Transcript Assessing the Total Effect of Time

Constructing Dynamic
Treatment Regimes &
STAR*D
S.A. Murphy, L. Gunter, B. Chakraborty
ICSA
June 2008
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)
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k=2 Stages
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 known function of
The dynamic treatment regime is the sequence of two
decision rules:
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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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Two Levels of STAR*D (Tx-resistant Depression)
Preference
Stage 1
Treatment
Action
Intermediate
Outcome
Stage 2
Treatment
Action
Mirtazapine
Switch
R
Remission
Continue on Current Tx
Nortriptyline
Tranylcypromine
Lithium
Augment R
Non-remission R
Thyroid
Mirtazapine + Venlafaxine
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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)
• The first method is an inefficient version of the second
method, when using linear models, each stages’ covariates
include the prior stages’ covariates and the actions are
centered to have conditional mean zero.
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A Simple Version of Q-Learning –
There is a regression for each stage.
• Stage 2 regression: Regress Y on
obtain
to
• Stage 1 regression: Regress
obtain
to
on
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for patients entering stage 2:
•
is the estimated probability of remission in stage 2
conditional on patient history (includes past treatment and
variables affected by stage 1 treatment).
•
is the estimated probability of remission assuming the “best”
treatment is provided at stage 2 (note max in formula).
•
is the dependent variable in the stage 1 regression for patients
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moving to stage 2
A Simple Version of Q-Learning –
• Stage 2 regression, (using Y as dependent variable)
yields
• Stage 1 regression, (using
yields
as dependent variable)
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Decision Rules:
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Non-regularity
11
Non-regularity
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Non-regularity–
• Replace hard-max
• by soft-max
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A Soft-Max Solution
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Distributions for Soft-Max
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To conduct inference concerning β1
• Set
• Stage 1 regression: Use least squares with outcome,
and covariates
to obtain
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Interpretation of λ
Estimator of Stage 1 Treatment Effect when
Future treatments are assigned
with equal probability, λ=0
Optimal future treatment is
assigned, λ=∞
Future treatment =1 is assigned
with probability
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Proposal
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Proposal
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STAR*D
• Regression at stage 1:
• S1'=(1, X1)
• 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.
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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.
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Stage 1 Augment Treatments
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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)
Augment
QIDS < 11
LI = THY (45) TCP<VEN+MIRT(104)
QIDS ≥ 11
LI < THY (88)
= means not significant in two sided test at .05 level
< means significant in two sided test at .05 level
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Simulation
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β1(∞)=β1(0)=0
P[β2TS2=0]=1
Test Statistic
based on
Nominal Type 1
Error=.05
.045
.039
.025*
(1) Nonregularity results in low Type 1 error
(2) Adaptation due to use of
is useful.
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P[β2TS2=0]=1
Test Statistic
based on
β1(∞)=β1(0)=.1
Power
.15
.13
.09
(1) The low Type 1 error rate translates into low power
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P[β2TS2=0]=0
Test Statistic
based on
β1(∞)=.125, β1(0)=0
Power
.05
.11
.12
(1) Averaging over the future is not a panacea
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P[β2TS2=0]=.25
Test Statistic
based on
β1(∞)=0, β1(0)=-.25
Type 1 Error=.05
.57
.16
.05
(1) Insufficient adaptation in “small” samples.
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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
too fast.
to increase
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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.
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This seminar can be found at:
http://www.stat.lsa.umich.edu/~samurphy/
seminars/ICSA0708.ppt
Email me with questions or if you would like a
copy!
[email protected]
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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
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Stage 1 Coefficients
^
¯(s:e:)
Augment
¯^11 = -.11(.07)
¯^12 = .47(.25)
Augment*QIDS2
¯^13 = -.04(.02)
Switch
z st at ist ic
-1.6
1.9
-2.3
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