Bayesian Subgroup Analysis

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Bayesian Subgroup Analysis

• Gene Pennello, Ph.D. Division of Biostatistics,
  CDRH, FDA
• Disclaimer No official support or endorsement of
  this presentation by the Food Drug
  Administration is intended or should be inferred.
• FIW 2006 September 28, 2006

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Outline

• Frequentist Approaches
• Bayesian Hierarchical Model Approach
• Bayesian Critical Boundaries
• Directional Error Rate
• Power
• Summary

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Frequentist Approaches

• Strong control of FWE
• Weak control of FWE
• Gatekeeper test subgroups (controlling FWE) only
  if overall effect is significant
• Confirmatory Study confirm with a new study in
  which only patients in the subgroup are enrolled.

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Concerns with Frequentist Approaches

• Limited power of FWE procedures
• Powerlessness of gatekeeper if overall effect is
  insignificant
• Discourages multiple hypothesis testing, thereby
  impeding progress.
• Confirmation of findings, one at a time, impedes
  progress.

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• No aphorism is more frequently repeated in
  connection with field trials, than that we must
  ask Nature few questions, or, ideally, one
  question at a time. The writer is convinced that
  this view is wholly mistaken. Nature, he
  suggests, will best respond to a logical and
  carefully thought out questionnaire

Fisher RA, 1926, The arrangement of field
experiments, Journal of the Ministry of
Agriculture, 33, 503-513.
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A Bayesian Approach

• Adjust subgroup inference for its consistency
  with related results.
• Choices Build prior on subgroup relationships.
• Invoke relatedness by modeling a priori
  exchangeability of effects.

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Prior Exchangeability Model

• Subgroups Labels do not inform on magnitude or
  direction of main subgroup effects.
• Treatments Labels do not inform for main
  treatment effects.
• Subgroup by Treatment Interactions Labels do not
  inform for treatment effects within subgroups.

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Prior Exchangeability Model

• Exchangeability modeled with random effects
  models.
• Key Result Result for a subgroup is related to
  results in other subgroups because effects are
  iid draws from random effect distribution.

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Bayesian Two-Way Normal Random Effects Model
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Bayesian Two-Way Normal Random Effects Model
Note In prior distribution, Pr(zero effect)
0 That is, only directional (Type III) errors
can be made here.
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Known Variances Inference

• Subgroup Problem
• Posterior

Note In prior distribution, Pr(zero effect)
0 That is, only directional (Type III) errors
can be made here.
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Bayes Decision Rule
Declare difference gt 0 if
Let
Note In prior distribution, Pr(zero effect)
0 That is, only directional (Type III) errors
can be made here.
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Bayes Critical z Value
Linear dependence on standardized marginal
treatment effect ? with ? interaction (?
) ?with ? subgroups b.
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Bayes Critical z Value
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Bayes Critical z Value
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Full Bayes Critical t Boundaries
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Directional Error Control

• Directional FDR controlled at A under 0-1-A loss
  function for correct decision, incorrect
  decision, and no decision (Lewis and Thayer,
  2004).
• Weak control of FW directional error rate,
  loosely speaking, because of dependence on F
  ratio for interaction.

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Comparisons of Sample Size to Achieve Same Power

• ULSD 5 level unadjusted z test
  Bonf Bonferonni 5 level z test HM EB
  hierarchical model test

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EX. Beta-blocker for Hypertension

• Losartan versus atenolol randomized trial
• Endpoint composite of Stroke/ MI/ CV Death
• N9193 losartan (4605), atenolol (4588)
• Events losartan (508), atenolol (588)
• 80 European Caucasians 55-80 years old.

http//www.fda.gov/cder/foi/label/2003/020386s032l
bl.pdf
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EX. Beta-blocker for Hypertension

• Cox Analysis
• N logHR SE HR (95 CI) p val
• Overall 9193 .87 ( .77, .98) 0.021
• Race SubgroupsNon-Black 8660 -.19 .06 .83 (
  .73, .94) 0.003Black 533 .51 .24 1.67
  (1.04,2.66) 0.033
• Is Finding Among Blacks Real or a Directional
  Error?

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EX. Beta-blocker for Hypertension

• Bayesian HM Analysis
• logHR se/sd HR (95CI) p val
  Prgt0non-black frequentist -.19 .06
  0.83 ( .73 .94) 0.003 0.001Bayesian -.18 .06
  0.84 ( .74, .95) 0.003
• blackfrequentist .51 .24 1.67 (1.04, 2.67)
  0.033 0.983 Bayesian .38 .27 1.47
  (0.87, 2.44) 0.914Bayesian analysis cast
  doubt on finding, but is predicated on not
  expecting a smaller effect in blacks a priori.

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Suggested Strategy

• Planned subgroup analysis
• Bayesian adjustment using above HM or similar
  model
• Pennello,1997, JASASimon, 2002, Stat. Med.
  Dixon and Simon, 1991, Biometrics

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Suggested Strategy

• Unplanned subgroup analysis
• Ask for confirmatory trial of subgroup.
• Posterior for treatment effect in the subgroup
  given by HM is prior for confirmatory trial.
• Prior information could reduce size of
  confirmatory trial.

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Summary

• Bayesian approach presented here considers trial
  as a whole, adjusts for consistency in finding
  over subgroups.
• Error rate is not rigidly pre-assigned Can vary
  from conservative to liberal depending on
  interaction F ratio and marginal treatment
  effect.
• Power gain can be substantial.Control for
  directional error rate is made only when
  warranted.

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References

• Dixon DO and Simon R (1991), Bayesian subset
  analysis, Biometrics, 47, 871-881.
• Lewis C and Thayer DT (2004), A loss function
  related to the FDR for random effects multiple
  comparisons, Journal of Statistical Planning and
  Inference 125, 49-58.
• Pennello GA (1997), The k-ratio multiple
  comparisons Bayes rule for the balanced two-way
  design, J. Amer. Stat. Assoc., 92, 675-684
• Simon R (2002), Bayesian subset analysis
  appliation to studying treatment-by-gender
  interactions, Statist. Med., 21, 2909-2916.
• Sleight P (2000), Subgroup analyses in clinical
  trials fun to look at but dont believe them!,
  Curr Control Trials Cardiovasc Med, 1, 25-27.

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Other Notable References

• Berry DA, 1990, Subgroup Analysis
  (correspondence) Biometrics, 46, 1227-1230.
• Gonen M, Westfall P, Johnson WO (2003), Bayesian
  multiple testing for two-sample multivariate
  endpoints, Biometrics, 59, 76-82.
• Westfall PH, Johnson WO, and Utts JM (1997), A
  Bayesian perspective on the Bonferroni
  adjustment, 84, 419-427