A Bayesian self‐controlled method for drug safety ...shawn/ssimpson_nssl_poster.pdf · drug safety surveillance in large‐scale longitudinal data Introducon • Ensuring drug safety

ShawnE.Simpson DavidMadigan

ABayesianself‐controlledmethodfordrugsafetysurveillanceinlarge‐scalelongitudinaldata

Introduc?on•  Ensuringdrugsafetybeginswithextensivepre‐approval

clinicaltrials

•  Thisprocesscon:nuesa"erapprovalwhendrugsareinwidespreaduse:post‐marke?ngsurveillance

•  Drugstakenbymorepeople,forlongerperiodsof:me,andindifferentwaysthaninpre‐approvaltrials

•  Mayiden:fyadversehealthoutcomesassociatedwithdrugexposurethatwerenotpreviouslydetected

1997 2004

Sta:s:calObjec:ves•  Iden:fydrug‐condi:onpairsthatmaybelinked

•  Finddruginterac:onslinkedwithcondi:ons•  Es:matethestrengthoftheseassocia:ons

•  FundamentalDifficul:es-  Largesize:Millionsofpeople,10000’sofcondi:ons

-  Highdimension:10000’sofdrugs,millionsofinterac:ons

CurrentSystem:FDAAERS

LongitudinalHealthcareDatabases•  Sen?nelIni?a?ve‐FDAplanstoestablishanac've

surveillancesystemusingdatafromhealthcareinforma:onholders

•  Observa?onalMedicalOutcomesPartnership(OMOP)‐Researchingmethodsforanalyzinghealthcaredatabasestoevaluatesafetyprofilesofdrugsonthemarket

Self‐ControlledCaseSeries

!" !" !" !"

MI! VIOXX!

!" !" !" !"

MI!

!" !" !" !"

#$%&'(")"

#$%&'("*" VIOXX! VIOXX! VIOXX!

+,-"

+.."

•  Personiobservedforτidays;(i,d)istheirdthday•  yid=#ofeventsobservedon(i,d)•  xid=1ifexposedtodrugon(i,d),0otherwise

•  Methoddevelopedtoes:materela:veincidenceofAEstoassessvaccinesafety[Farrington,1995]

•  Onedrug,oneadverseevent(AE)

•  Eventsariseaccordingtoanon‐homogeneousPoissonprocess,exposuremodulatestheeventrate

•  Intensityon(i,d)=

Method mAPscore22PRR 0.2251486

22OR 0.2280057

23BCPNN 0.209197

22EBGM 0.2173618

23CHI‐SQ 0.2144175

22PRR05 0.2046662

22ROR05 0.2046221

12BCPNN05 0.1832317

12EB05 0.1860902

SCCS(1AE,1drug) 0.2216072

BayesianSCCS,Normalprior,precision1(1AE,1drug)

0.26065568

BayesianLogisticRegression,Normalprior,precision1(1AE,multipledrugs)

0.2665139

Case‐Control 0.186743

•  Advantages-  Automated-  Be`ertemporaldata

•  Disadvantages-  Li`lebaselinedata-  NoOTCinforma:on

Yes No

Yes a b

No c d

AEj

Drugi

Total:N

2.  Condi:onalindependenceofeventsandexposures

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A Bayesian self-controlled method for drug safety surveillance in large-scale longitudinal data

Self-controlled case series method

Single drug

SCCS: Poisson event rate on each day

Assume that events arise according to a non-homogeneous Poissonprocess, where drug exposure modulates the baseline event rate

Poisson intensity on day (i , d) = e !i + "xid

wheree !i = fixed individual intensity ratee " = multiplicative e!ect of drug exposure

yid | xid ! Poisson( e !i+"xid )

The likelihood contribution of (i , d) is

P( yid | xid ) =e!e{!i +"xid}(e !i+"xid )yid

yid !



Single drug

SCCS: Likelihood and Assumptions

Joint probability of events for i over all observed days

Li = P( yi1, . . . , yi!i | xi1, . . . , xi!i ) = P( yi | xi ) =!i!

d=1

P( yid | xid )

= exp{!"

d

e "i+#xid} "!i!

d=1

(e "i+#xid )yid

yid !

Assumptions

1 Conditionally independent events

yid ## yid! | xi for d $= d !

2 Events are conditionally independent of exposures

yid ## xid! | xid for d $= d !

Condi:ontoremoveφi

•  CoulduseMLtogetes:mates,butdrugeffectβisofinterestandtheφi’sarenuisanceparameters

•  Condi:ononsufficientsta:s:cni=Σyid

•  Condi:onallikelihoodfori

•  Maximize togetβCMLEconsistent,asympto:callyNormal[CameronandTrivedi,1998]



Single drug

SCCS: Condition to remove !i

Could use ML to get estimates, but the drug e!ect ! is of interestand the "i ’s are nuisance parameters.

Condition on su"cient statistic ni =P

d yid to remove "i

ni | xi ! Poisson(!

d

e !i+"xid )

Conditional likelihood for i

Lci = P( yi | xi, ni ) =

P( yi | xi )

P( ni | xi )"

#i"

d=1

#e "xid

$d! e "xid!

%yid

Maximize lc =$

log Lci to get !̂CMLE

#$ consistent, asymptotically normal [Cameron and Trivedi, 1998]

If i has no events (yi = 0) then Lci = 1 $ only need cases (ni % 1)



Single drug




d yid to remove "i

ni | xi ! Poisson(!

d

e !i+"xid )



P( yi | xi )

P( ni | xi )"

#i"

d=1

#e "xid

$d! e "xid!

%yid

Maximize lc =$




•  Ifihasnoevents(yi=0)thenLiC=1,soweonlyneedcases(i.e.ni≥1)intheanalysis

•  SCCSdoeswithin‐personcomparisonofeventrateduringexposuretoeventratewhileunexposed(‘self‐controlled’)

BayesianExtensionofSCCS

•  Longitudinaldatabaseshave10000’sofpoten:aldrugs•  Intensitymodel:e(maineffects)+(2‐wayinterac:ons)

highdimensionalitywithmillionsofpredictors

•  StandardMLleadstooverfilng;needtoregularize

•  Ourapproach–putaprioronβparameterstoshrinkthees:matestowardzero,smoothoutes:ma:on,andreduceoverfilng



Single drug

SCCS: Poisson event rate on each day

Assume that events arise according to a non-homogeneous Poissonprocess, where drug exposure modulates the baseline event rate

Poisson intensity on day (i , d) = e !i + "xid

wheree !i = fixed individual intensity ratee " = multiplicative e!ect of drug exposure

yid | xid ! Poisson( e !i+"xid )

The likelihood contribution of (i , d) is

P( yid | xid ) =e!e{!i +"xid}(e !i+"xid )yid

yid !

Mul:pleDrugsandInterac:ons•  WeextendthemodeltooneAEandmul?pledrugs

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MI!

MI!

#$%&'(")"

#$%&'("*"

!" !"

!" !"

QUETIAPINE!

!" !"!" !"

OLANZAPINE!

VIOXX! VIOXX!

VIOXX! VIOXX!

+,-"

.*/"

!"!"!" !"

OLANZAPINE!CELECOXIB!

QUETIAPINE!

•  Intensityon(i,d)=



Multiple drugs

SCCS: Multiple drugs

We extend the model to one adverse event and multiple drugs:

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MI!!" !"

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MI!

MI!

#$%&'(")"

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QUETIAPINE!

!" !"!" !"

OLANZAPINE!

VIOXX! VIOXX!

VIOXX! VIOXX!

+,-"

.*/"

!"!"!" !"


QUETIAPINE!

Intensity on (i , d): e !i + !Txid = e !i + "1 xid1 + ··· + "p xidp

xidj =

!1 exposed to drug j on (i , d)0 otherwise

xid = ( xid1, . . . , xidp )T

! = ( !1, . . . , !p )T

Intensity with drug interactions, time-varying covariates

e {!i + !Txid +P

r !=s #rs xidr xids + "Tzid}

•  xidj=1ifexposedtodrugj,0otherwise•  Intensitywithdruginterac:ons,:me‐varyingcovariates:



Multiple drugs



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MI!!" !"

!" !"!" !"

MI!

MI!

#$%&'(")"

#$%&'("*"

!" !"

!" !"

QUETIAPINE!

!" !"!" !"

OLANZAPINE!

VIOXX! VIOXX!

VIOXX! VIOXX!

+,-"

.*/"

!"!"!" !"


QUETIAPINE!


xidj =


xid = ( xid1, . . . , xidp )T

! = ( !1, . . . , !p )T


e {!i + !Txid +P




Multiple drugs



!"!" !"

!"

MI!!" !"

!" !"!" !"

MI!

MI!

#$%&'(")"

#$%&'("*"

!" !"

!" !"

QUETIAPINE!

!" !"!" !"

OLANZAPINE!

VIOXX! VIOXX!

VIOXX! VIOXX!

+,-"

.*/"

!"!"!" !"


QUETIAPINE!


xidj =


xid = ( xid1, . . . , xidp )T

! = ( !1, . . . , !p )T


e {!i + !Txid +P




Multiple drugs



!"!" !"

!"

MI!!" !"

!" !"!" !"

MI!

MI!

#$%&'(")"

#$%&'("*"

!" !"

!" !"

QUETIAPINE!

!" !"!" !"

OLANZAPINE!

VIOXX! VIOXX!

VIOXX! VIOXX!

+,-"

.*/"

!"!"!" !"


QUETIAPINE!


xidj =


xid = ( xid1, . . . , xidp )T

! = ( !1, . . . , !p )T


e {!i + !Txid +P


1.  Normalprior(ridgeregression)

2.  Laplacianprior(lasso)

3.4 Shrinkage Methods 71

TABLE 3.4. Estimators of !j in the case of orthonormal columns of X. M and "are constants chosen by the corresponding techniques; sign denotes the sign of itsargument (±1), and x+ denotes “positive part” of x. Below the table, estimatorsare shown by broken red lines. The 45! line in gray shows the unrestricted estimatefor reference.

Estimator Formula

Best subset (size M) !̂j · I[rank(|!̂j | ! M)

Ridge !̂j/(1 + ")

Lasso sign(!̂j)(|!̂j |" ")+

(0,0) (0,0) (0,0)

|!̂(M)|

"

Best Subset Ridge Lasso

!^ !^2. .!

1

! 2

!1 !

FIGURE 3.11. Estimation picture for the lasso (left) and ridge regression(right). Shown are contours of the error and constraint functions. The solid blueareas are the constraint regions |!1| + |!2| ! t and !2

1 + !22 ! t2, respectively,

while the red ellipses are the contours of the least squares error function.

3.4 Shrinkage Methods 71

TABLE 3.4. Estimators of !j in the case of orthonormal columns of X. M and "are constants chosen by the corresponding techniques; sign denotes the sign of itsargument (±1), and x+ denotes “positive part” of x. Below the table, estimatorsare shown by broken red lines. The 45! line in gray shows the unrestricted estimatefor reference.

Estimator Formula

Best subset (size M) !̂j · I[rank(|!̂j | ! M)

Ridge !̂j/(1 + ")

Lasso sign(!̂j)(|!̂j |" ")+

(0,0) (0,0) (0,0)

|!̂(M)|

"

Best Subset Ridge Lasso

!^ !^2. .!

1

! 2

!1 !

FIGURE 3.11. Estimation picture for the lasso (left) and ridge regression(right). Shown are contours of the error and constraint functions. The solid blueareas are the constraint regions |!1| + |!2| ! t and !2

1 + !22 ! t2, respectively,

while the red ellipses are the contours of the least squares error function.


Bayesian Extension of SCCS

Overfitting

Priors on model parameters !

1 Normal prior (ridge regression)

!j ! N(0, "2!)

max likelihood subject top!

j=1

!2j " s

2 Laplacian prior (lasso)

!j ! Laplace(0, 1/#)


j=1

|!j | " s

Posterior modes via cyclic coordinate descent [Genkin et al, 2007].Handles millions of predictors in logistic case (BBR).

Log likelihood is concave # convex optimization problem. Each stepincreases log posterior, “climb to top of the hill”

maxliksubjectto



Overfitting



!j ! N(0, "2!)


j=1

!2j " s




j=1

|!j | " s





Overfitting



!j ! N(0, "2!)


j=1

!2j " s




j=1

|!j | " s



maxliksubjectto



Overfitting



!j ! N(0, "2!)


j=1

!2j " s




j=1

|!j | " s



References

Results:OMOPMethodsEvalua?on•  Methodsevalua:on:

-  Chose10drugs,10condi:onsofinterest-  9drug‐condi:onpairswithatrueassocia'on-  Pairsdeterminedtohavenolinkserveasnega'vecontrols

•  Evalua:onisbasedonmeanaverageprecision(mAP)score:measuresthedegreetowhichamethodmaximizes‘trueposi:ves’whileminimizing‘falseposi:ves’

MSLRdatabase(1.5Mpeople)

FurtherWork

•  CameronandTrivedi(1998)RegressionAnalysisofCountData.CambridgeUniversityPress.

•  Farrington(1995)“Rela:veincidencees:ma:onfromcaseseriesforvaccinesafetyevalua:on,”Biometrics,Vol.51,No.1,pg.228‐235.

•  Genkinetal.(2007)“Large‐scaleBayesianlogis:cregressionfortextcategoriza:on,”Technometrics,Vol.49,No.3,pg.291‐304.

•  CurrentapproachtosurveillanceisbasedontheFDA’sAdverseEventRepor?ngSystem(AERS)

•  Anyonecanvoluntarilysubmitareportonadverseevents(AEs)thatmayberelatedtodrugexposures

•  Difficul:eswithAERS

-  Messy–spellingerrors,etc.-  Bias–underrepor:ng,duplicatereports,media-  Unreliabletemporalinforma:on

•  Mul:pledrugsandAEsmaybelistedononereport

•  15000drugs×16000AEs=240milliontables

•  MostAEsdonotoccurwithmostdrugs;smallcountsina

•  FDAuses2×2summaries,appliesBayesianshrinkagemethodstodealwithvariabilityduetosmallcounts

•  Limita:ons

-  Noadjustmentforconfoundingdrugs-  Ignoresinterac:ons-  Maynotu:lizetemporalinforma:on

xi1 xi2 xi3 xiτ...

...

yi1 yi2 yi3 yiτ...•  Relaxindependenceassump:onstoallowdependencebetweenevents

•  Alloweventstoinfluencefutureexposures

β1x1+…+βjxj+…+βkxk+…+βpxpeN(μ[1],σ[1]2) N(μ[D],σ[D]2)...

drugclass[1] drugclass[D]...

β1x1+...+βpxpe

β1x1+...+βpxpe

...

...

ym

y1~

~

...

N(μ1(c),σ(1)2) N(μp(c),σ(c)2)...

cond

itio

ncl

ass

(c)

•  Hierarchicalmodelingofdrugsintodrugclasses

•  Hierarchicalmodelingofcondi:onsintoclasses

•  Convexop:miza:on:Posteriormodesviacycliccoordinatedescent[Genkinetal,2007]

•  Handlesmillionsofpredictorsinlogis:ccase(BBR)



Single drug




d yid to remove "i

ni | xi ! Poisson(!

d

e !i+"xid )



P( yi | xi )

P( ni | xi )"

#i"

d=1

#e "xid

$d! e "xid!

%yid

Maximize lc =$




⌃

DataReduc:ontoCasesOnly

•  Poten:alanalysistechniques:maxSPRT,cohortmethods,casecontrol,case‐crossover,self‐controlledcaseseries…

Documents

A Bayesian self‐controlled method for drug safety ...shawn/ssimpson_nssl_poster.pdf · drug safety surveillance in large‐scale longitudinal data Introducon • Ensuring drug safety