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STAT 422 & GS01 0013: Fall 2007Bayesian Data Analysis

• Instructors:– Gary Rosner (glrosner@mdanderson.org)– Luis Nieto-Barajas (lnietoba@mdanderson.org)

• Room: FC 2.3031• Time: Wed. & Fri. 10:30 AM – 12:00 PM• Grade:

– 3 homework sets, 1 exam, and a student presentation.• Course Text:

– An Introduction to Bayesian Analysis: Theory andMethods

Jayanta K. Ghosh, Mohan Delampady, & Tapas SamantaNew York:Springer, 2006.

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What Do Authors Mean When TheyWrite “p = 0.05”?

• Null hypothesis is probably true• Null hypothesis is probably false• Observed result is unlikely• Experimental therapy is probably

effective• None of above

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2nd-Best AnswerRepaired

a) Null hypothesis is probably trueb) Null hypothesis is probably falsec) Observed result is in a set that is

unlikely, assuming null hypothesisd) Experimental therapy is probably

effective

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What is Statistics?

• A collection of procedures andprinciples for gaining andprocessing information in order tomake decisions when faced withuncertainty.

–Seeing Through Statistics–Jessica Utts

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Bayesian or Frequentist

• Bayesian conditions on data– What do we know about the

parameters given the data?• Frequentist conditions on

hypotheses– How likely are these observations if

there is no difference?

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Advantages of Bayesian Inference

• Easier to incorporate externalinformation

• Follows learning paradigm• Easier to account for sources of

uncertainty• Inference more natural• Foundation for decision making in

the presence of uncertainty

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Criticisms of Bayesian Approach

• Prior specification– Different prior can lead to different

posterior inference• Large sample sizes minimize influence of

prior

• More difficult to carry out– Newer computing methods/programs

allow inference in complex problems

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Bayes Rule Allows InvertingConditional Probs• Sometimes have but want

• If know

• can get ( )( ) ( )

( )APr

BAPrBPrABPr

!=

!

Pr AB( ), Pr B( ), & Pr A( )

( )BAPr

( )ABPr

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Diagnostic Tests

• Sensitivity:

• Specificity:

• Want

!

Pr Test + Disease( )

!

PV+ = Pr Disease Test +( )

!

Pr Test " No Disease( )

!

PV" = Pr No Disease Test "( )

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Bayes Rule Helps DeterminePV+ & PV-

!

PV+ = Pr D + T +( ) =Pr T + D +( ) "Pr D +( )

Pr T +( )

!

Pr T +( ) = Pr T +" D +( ) + Pr T +" D #( )

= Pr T + D +( )Pr D +( ) + Pr T + D #( )Pr D #( )

– where

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Bayesian Statistical Inference• 3 main components

– Prior distribution• Initial hypothesized distribution (prior to

collecting data)– Likelihood

• Probability function associated with thedata, conditioning on parameters

– Posterior distribution• Updated distribution (from prior) after

collecting data.

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Main Bayesian Concept• Posterior Dist’n is proportional to

– Likelihood times Prior

– For example,

( )( ) ( )param.parameter Data

Data parameter

PrPr

Pr

!

"

( )

( ) ( )difftrt Nodifftrt No Data

Data difftrt No

PrPr

Pr

!

"

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• Predictive distribution– Important Bayesian concept

– For example• Probability associated with outcome

of next patient of treatment A vs B– With predicitive dist’n, can get

expected utility

( )dataCurrent nObservatioNext Pr

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Decision Making under Uncertainty

• What do we know?• How do we put it all together?

• Multiple sources of information– Multiple studies

• Randomized clinical trials• Epidemiology

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GUSTO Clinical Trial

• An International Randomized TrialComparing Four ThrombolyticStrategies for Acute MyocardialInfarction

• New England Journal of Medicine,– vol. 329:673-682, 1993

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GUSTOGlobal Utilization ofStreptokinase and TissuePlasminogen Activator forOccluded Coronary ArteriesR

A

N

D

streptokinase + i.v. heparin

streptokinase + subQ heparin

t-PA + streptokinase + i.v. heparin

t-PA + i.v. heparin

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GUSTO• Hypothesis:

– Early & sustained infarct-vesselpatency associated with bettersurvival among pts having MI

• Principal end point:– 30-day all-cause mortality

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GUSTO– Began Dec. 27, 1990– Ended Feb. 22, 1993– Enrolled 41,021 worldwide

• 15 countries; 1081 hospitals– 90% power (2-sided 0.05-level) to

detect 15% reduction in mortality (~1% difference)• E.g., 8% 6.8%

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GUSTO Results

• t-PA + i.v. heparin providessurvival benefit, compared tostreptokinase arms– 6.3% vs. 7.2%—7.4%

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Odds Ratios & 95% Confidence Intervals forMortality & Disabling Stroke Reduction

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What Else Do We Know?• Placing Trials in Context Using

Bayesian Analysis: GUSTORevisited by Reverend Bayes– James M. Brophy &– Lawrence Joseph

• Journal of the American MedicalAssociation, vol.273: 871-875, 1995

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Meta-Analysis• The statistical analysis of a large

collection of analysis results fromindividual studies for the purposeof integrating the findings.

–G.V. Glass “Primary, secondary, andmeta-analysis of research” Educ Res5:3-8, 1976

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Meta-Analysis

• Meta-analysis looks for consistencyand explanations of heterogeneity ordifferences between studies.

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GUSTO vs. Other Studies

• Two other studies evaluated t-PA& streptokinase– GISSI-2– ISIS-3

• These studies did not show sucha strong effect for t-PA

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3 Similar Studies

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How To Combine Information?

• Want to account for– Between-study differences and– Within-study heterogeneity

• Bayesian inference

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Choice of PriorDist’n

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PosteriorDistribution

SK bettertPA better

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Probability of SuperiorityDepends on Prior Belief

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But What About DifferencesBetween the Studies?

• Different tPA administration inGUSTO

• More revascularization in GUSTO• More US centers in GUSTO

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Hierarchical ModelsStudy-to-study

Center-to-center

…

…… …

…

Patient-to-patient

… ……

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Hierarchical Model: GUSTO

SK bettertPA better

Pr(tPA better) = 0.74

Pr(tPA better in next study) = 0.66

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TO BE CONTINUED…