Hypothesis Testing and Adaptive Treatment Strategies

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Hypothesis Testing and Adaptive Treatment
Strategies

• S.A. Murphy
• SCT
• May 2007

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Collaborators

• Lacey Gunter
• A. John Rush
• Bibhas Chakraborty

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Outline

• Adaptive treatment strategies
• Constructing and addressing questions regarding
  an optimal adaptive treatment strategy
• A solution to non-regularity?
• Example using STARD.

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Adaptive treatment strategies 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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k2 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 function of
The adaptive treatment strategy is the
sequence of two decision rules
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Data for Constructing the Adaptive Treatment
Strategy 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 PAj1PAj-1.5
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• 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 3 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 3.
  Do the same for the two augments.

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• 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 3 non-remitters went on to
  have a remission in stage 4 these people have an
  dependent variable equal to 0 in the myopic
  analysis.
• the remission or lack of remission in stage 4 may
  be partially attributable to the stage 3
  treatment.
• Patching together the separate analyses of the
  stages requires unnecessary causal assumptions.

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• Need an alternative for the stage 3 dependent
  variable
• What should the value of the stage 3 dependent
  variable be for those that move to stage 4?
• We should not use a stage 3 dependent variable of
  Y1 for those people who remit in stage 4.
• We should not use an stage 3 dependent variable
  of Y0 for those people who remit in stage 4.
• The dependent variable should be something in
  between.

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• Regression-based methods for constructing
  decision rules
• Q-Learning (Watkins, 1989) (a popular method from
  computer science)
• A-Learning or optimal nested structural mean
  model (Murphy, 2003 Robins, 2004)
  
  
• The first method is an inefficient version of the
  second method when each stages covariates
  include the prior stages covariates and the
  actions are coded to have conditional mean zero.

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A Simple Version of Q-Learning
There is a regression for each stage.

• Stage 4 regression Regress Y on
  to obtain
• Stage 3 regression Regress on
  to obtain

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for patients entering stage 4

• is the estimated probability of remission in
  stage 4 as a function of variables that may
  include or be affected by stage 3 treatment.
• is the estimated probability of remission
  assuming the best treatment is provided at
  stage 4 (note max in formula).
• will be the dependent variable in the stage 3
  regression for patients moving to stage 4

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A Simple Version of Q-Learning

• Stage 4 regression, (using Y as dependent
  variable) yields
• Stage 3 regression, (using as dependent
  variable) yields

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Decision Rules
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Non-regularity
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Non-regularity
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Non-regularity

• Replace hard-max
• by soft-max

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STARD Stages 3 4

• Regression at stage 3 a3TS3' ß3TS3A3
• S3' (1, X3)
• X3 is a vector of variables available at or prior
  to stage 3
• S3 ((1-Aug), Aug, AugQids2)
• We are interested in the ß3 coefficients as these
  are used to form the decision rule at stage 3.

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STARD Stages 3 4

• Decision Rule at stage 3
• 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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STARD Stages 3 4

• Regression at stage 4 a4TS4' ß4S4A4
• S4' (1,X4, (1-Aug)A3, AugA3, AugA3Qids2),
• (X4 is a vector of variables available at or
  prior to stage 4, Aug is 1 if patient preference
  is augment and 0 otherwise)
• S4 1
• Decision rule Choose TCP if ,
  otherwise offer Mirtazapine Venlafaxine XR

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Stage 3 Coefficients
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means not significant in two sided test at .05
level lt means significant in two sided test at
.05 level
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Discussion

• We replace the hypothesis test concerning a
  non-regular parameter, ß3 by a hypothesis test
  concerning a near-by regular parameter.
• These multi-stage regression methods need to be
  generalized to survival analysis.
• This is work in progress!

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Discussion

• Robins (2004) proposes several conservative
  confidence intervals for ß3.
• Ideally to decide if the two stage 3 treatments
  are equivalent, we would evaluate whether the
  choice of stage 3 treatment influences the mean
  outcome resulting from the use of the adaptive
  treatment strategy. We did not do this here.
• Constructing evidence-based strategies 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/SCT0507.ppt
• Email me with questions or if you would like a
  copy!
• samurphy_at_umich.edu

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