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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. Outline. - PowerPoint PPT Presentation
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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.
ChoicesBuild 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 subgroupsbecause effects are iid draws fromrandom effect distribution.
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Bayesian Two-Way Normal Random Effects Model
2~ ( , / ), 1, , 1,ijijy N n i a j b
ij i j ij 2 2 2~ (0, ), ~ (0, ), ~ (0, )i j ijN N N
2 2 2 2Jeffreys prior on ( , , , , )
2 2 2/ ~ ( ), ( 1)fs f f ab r
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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
Note: In prior distribution, Pr(zero effect) = 0
That is, only directional (Type III) errors can be made here.
Subgroup Problem:
Posterior
12, 1 2j j j
212( , ( ( 1) ) / )),A C C A C dN S d S d S b S b
212, | , ~j y
12 12, 121 2 , C jd y y d d d 2 2 2 2 21 1 / , / , A A A A A CS br
2 2 2 2 21 1/ , / , C C C C CS r
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Bayes Decision Rule
212,Pr 0 | , 1 / 2j y
Declare difference > 0 if
Let12, 12, 12, 1 2/ 2 / , j j j j jz d r d y y
12, 12, 12, 1 2/ 2 / , z d br d y y
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Bayes Critical z Value
212,Pr 0 | , 0.975j y if
1/ 2
12,1 / 212,
11A A
jC CC
zz S Sbz
bS b SS b
12,z
Linear dependence on standardized marginal treatment effect
↑ with ↓ interaction (↑ )↓with ↑ # subgroups b.
/A CS S
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Bayes Critical z Value
C
1 / 212, j
C
zz
S
Full Interaction Case: A CS S
Critical z value
↑ with ↓ true F ratio measuring heterogeneity of interaction effects.
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Bayes Critical z Value
A
1 / 212,
A
zz
S
No Interaction Case: 0CS
Critical z value
Power can be > than for unadjusted 5% level z test for subgroup if true F ratio measuring heterogeneity of treatment effects is large.
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Full Bayes Critical t Boundaries
12,t
12, jt
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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
0 /ULSD HMr r b
1 / 2 1
1 / 2 1
0 / bBonf ULSD
z zr r
z z
1 / 2 1
1 / 2 1
0 / bBonf HM
z zr r b
z z
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EX. Beta-blocker for Hypertension
Losartan versus atenolol randomized trial
Endpoint: composite of Stroke/ MI/ CV Death
N=9193 losartan (4605), atenolol (4588)
# Events losartan (508), atenolol (588)
80% European Caucasians 55-80 years old.
http://www.fda.gov/cder/foi/label/2003/020386s032lbl.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 AnalysislogHR se/sd HR (95%CI) p val Pr>0
non-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) 0.914
Bayesian 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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SummaryBayesian approach presented here considers trial as a whole, adjusts for
consistency in finding over subgroups.
Error rate is not rigidly pre-assignedCan 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 don’t 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