Screening Experiments for
Developing
Dynamic Treatment Regimes
S.A. Murphy
At ICSPRAR
January, 2008
Dynamic Treatment Regimes
Are individually tailored treatments, with treatments
changing with ongoing subject assessments.
The development of a dynamic treatment regime is a
multi-stage decision problem
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Two stages of treatment for each
individual
Observation available at jth stage
Treatment action (vector) at jth stage
Primary outcome Y is a specified summary of actions
and observations
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A two stage dynamic treatment regime is a
vector of decision rules, one per stage
used to select the treatment actions,
Next: using experiments to construct decision
rules.
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Experiments
Currently the state of the art is to randomize the subject
at each of multiple stages –usually one factor per
stage.
Dynamic treatment regimes are multi-component
treatments: many possible factors (e.g. 5-6)
Future: series of screening/refining, randomized trials
prior to confirmatory trial --- à la Fisher/Box
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Screening experiments (review)
1) Goal is to eliminate inactive factors (e.g. components) and
inactive effects.
2) Each factor at 2 levels
3) Screen main effects and some interactions
4) Design experiment using working assumptions concerning
the negligibility of certain higher order factorial effects.
5) Designs and analyses permit one to determine aliasing
(caused by false working assumptions)
6) Minimize assumptions
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Simple Example
• Stage 1 Factors: T1={A, B, C, D}, each with 2
levels
• Stage 1 outcome:
• Stage 2 Factors: T2= {F2--only if R=1, G2—only if
R=0}, each with 2 levels
• Primary Outcome: Y continuous
(26= 64 simple dynamic treatment regimes)
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Screening experiments
The fact that R, an outcome of stage 1 factors,
determines the stage 2 factors + the fact that there
are 2 stages of treatment for each subject is makes
things interesting!
Can we design screening experiments using working
assumptions concerning higher order effects &
determine the aliasing & provide an analysis
method?
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Reality
Unknown
Causes
T1
R
T2
Y
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Defining the stage 2 effects via
Potential Outcomes
Simple case: two stages of treatment with one factor at
stage 1, a early measure of response, R, at the end
of stage 1 and two factors at stage 2 (one used if
R=1, the other used if R=0) and a primary outcome
Y. The potential outcomes are
Define effects involving T2 in a saturated linear model
for
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The stage 2 effects and data
Suppose the factors T1 and T2 are randomized. Assume
consistency (Robins, 1997) then
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Defining the stage 2 effects
Define (factorial) effects involving t2 in a saturated
linear model
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Defining the stage 2 effects
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Defining the stage 1 effects (T1)
Unknown
Causes
T1
R
T2
Y
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Defining the stage 1 effects
Unknown
Causes
T1
R
T2
Y
Lesson: Do not condition on R to define stage 1 effects
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Defining the stage 1 effects
Define effects involving only T1 in a saturated linear
model
The above is equal to
when {T1, T2} are randomized and T2 has a discrete
uniform distribution on {-1,1}.
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Defining the stage 1 effects
In general when {T1, T2} are randomized
G-computation Formula
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Why marginal, why uniform?
Define effects involving only T1 via
1) The defined effects are causal.
2) The defined effects are consistent with tradition in
experimental design for screening.
–
The main effect for one treatment factor is defined by
marginalizing over the remaining treatment factors
using a discrete uniform distribution.
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Representing the effects
Surprisingly both stage 1 and 2 effects can be
represented in one (nonstandard) linear model:
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where
Causal effects:
Nuisance parameters:
and
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General Formula
More than one factor per stage
Nonparametric model (for both R=0, 1)
Z1 matrix of stage 1 factor columns, Z2 is the matrix of
stage 2 factor columns, Y is a vector, p is the vector
of response rates
Classical saturated factorial effects model
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Aliasing (Identification of Effects)
When a fractional factorial design produces the data,
the effect parameters in
are not all identifiable. As in classical screening
experiments the defining words pinpoint which
effects can be identified.
The above is not immediately obvious.
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Aliasing (Identification of Effects)
The rows of {Z1, Z2} are determined by the
experimental design; each column in Z1 is associated
with a stage 1 effect and similarly each column in Z2
is associated with a stage 2 effect.
Fractional factorial designs lead to common columns
in {Z1, Z2}; these columns are given by the defining
words.
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Aliasing (Identification of Effects)
Lemma: Suppose E[R|T1]
. Suppose the
defining words indicate a shared column in both Z1
and Z2. If either the column in Z1 can be assumed to
have a zero η coefficient or the column in Z2 can be
assumed to have a zero β, α coefficient, then the
defining words provide the aliasing.
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Simple Example for Two Stages
Six Factors:
Stage 1: T1={A, B, C, D}, each with 2 levels
Stage 2: T2= {F2--only for stage 1 responders,
G2--only for stage 1 nonresponders}, each with 2
levels
(26= 64 simple dynamic treatment regimes)
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Two Stage Design: I=ABCDF2=ABCDG2
A
B
C
D
F2=G2
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Assumptions for this Design
• We use the “1=ABCDF2=ABCDG2” design if we can
•assume that all three way and higher order stage 2
effects are negligible (ABG2, ABF2, ABCG2, ABCF2,
….)—these are all β and α parameters.
•assume all four way and higher order nuisance
effects involving R and stage 1 factors are negligible
(R-p)ABCD, (R-p)ABC, (R-p)ABD …--- these are
all η parameters.
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Aliasing for this Design
•
Stage 1 three way interactions and stage 2 two way
interactions are aliased (e.g. ABC=DF2, ABC=DG2, etc.)
•
The Stage 1 four way interaction and stage 2 main effects are
aliased (e.g. ABCD=F2, ABCD=G2).
1=ABCDF2=ABCDG2
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Statistical Analysis for this design
Recall that
• Many columns in Z1, Z2 are identical (hence the
aliasing of effects). Eliminate all multiple copies of
columns and label remaining columns as stage 1 (or
stage 2) main and two-way interaction effects.
• Replace response rates in p by observed response
rates.
• Fit model.
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Interesting Result in Simulations
•
In simulations assumptions are violated.
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Response rates (probability of R=1) across 16 cells range from
.55 to .73
•
Results are surprisingly robust to violations of assumptions.
•The maximal value of the correlation between 32 estimators
of effects was .12 and average absolute correlation value is .03
•Why? Binary R variables can not vary that much. If
response rate is constant, then the effect estimators are
uncorrelated as in classical experimental design.
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Discussion
• Two competitors to this approach:
– Using nonrandomized comparisons to construct
dynamic treatment regimes
• Uncontrolled selection bias (causal misattributions)
• Uncontrolled aliasing.
– Expert opinion
• Secondary analyses would assess if other
variables collected during treatment should
enter decision rules.
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• Joint work with
– Derek Bingham (Simon Fraser)
• And informed by discussions with
– Vijay Nair (U. Michigan)
– Bibhas Chakraborty (U. Michigan)
– Vic Strecher (U. Michigan)
• This seminar can be found at: http:// www.stat.
lsa.umich.edu/~samurphy/seminars/ICPRAR01.08.ppt
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