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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