Sufficient Covariate, Propensity Variable and Doubly Robust Estimation

0.3 MB·English

by  Hui Guo

Sufficient Covariate, Propensity Variable and
Doubly Robust Estimation
HuiGuo,PhilipDawidandGiovanniBerzuini
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3 AbstractStatistical causalinferencefromobservationalstudiesoftenrequiresad-
justment for a possibly multi-dimensionalvariable, where dimension reduction is

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crucial.Thepropensityscore,firstintroducedbyRosenbaumandRubin,isapopular
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approachtosuchreduction.We addresscausalinferencewithinDawid’sdecision-
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. theoretic framework, where it is essential to pay attention to sufficient covariates
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andtheirproperties.Weexaminetheroleofapropensityvariableinanormallinear
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a model.We investigatebothpopulation-basedandsample-basedlinearregressions,
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with adjustmentsfor a multivariate covariate and for a propensity variable. In ad-
[ dition, we study the augmented inverse probability weighted estimator, involving

a combinationof a responsemodelanda propensitymodel.In a linear regression
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v withhomoscedasticity,apropensityvariableisprovedtoprovidethesameestimated
1 causaleffectasmultivariateadjustment.Anestimatedpropensityvariablemay,but
6 need not, yield better precision than the true propensity variable. The augmented
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inverseprobabilityweightedestimatorisdoublyrobustandcanimproveprecision
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ifthepropensitymodeliscorrectlyspecified.
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HuiGuo
r
a Centre for Biostatistics, Institute of Population Health, The University of Manch-
ester, Jean McFarlane Building, Oxford Road, Manchester M13 9PL, UK, e-mail:
[email protected]
PhilipDawid
StatisticalLaboratory, UniversityofCambridge, Wilberforce Road, CambridgeCB30WB,UK,
e-mail:[email protected]
GiovanniBerzuini
Department of Brain and Behavioural Sciences, University of Pavia, Pavia, Italy, e-mail:
[email protected]
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2 HuiGuo,PhilipDawidandGiovanniBerzuini
1 Introduction
Causal effects can be identified from well-designed experiments, such as ran-
domisedcontrolledtrials(RCT),becausetreatmentassignmentisentirelyunrelated
to subjects’ characteristics, both observedand unobserved.Suppose there are two
treatment arms in an RCT: treatment group and control group. Then the average
causaleffect(ACE)can simplybe estimatedasthe outcomedifferenceofthe two
groupsfrom the observeddata. However,randomisedexperiments,althoughideal
andtobeconductedwheneverpossible,arenotalwaysfeasible.Forinstance,toin-
vestigatewhethersmokingcauseslungcancer,wecannotrandomlyforceagroupof
subjectstotakecigarettes.Moreover,itmaytakeyearsorlongerfordevelopmentof
thisdisease.Instead,aretrospectivecase-controlstudymayhavetobeconsidered.
The task of drawing causal conclusion, however,becomesproblematic since sim-
ilarity of subjectsfrom the two groupswill rarelyhold, e.g., lifestyles of smokers
mightbedifferentfromthoseofnon-smokers.Thus,weareunableto“comparelike
withlike”–theclassicproblemofconfoundinginobservationalstudies,whichmay
requireadjustingforasuitablesetofvariables(suchasage,sex,healthstatus,diet).
Otherwise, the relationship between treatmentand response will be distorted, and
leadtobiasedinferences.Ingeneral,linearregressions,matchingorsubclassifica-
tionareusedforadjustmentpurpose.Iftherearemultipleconfounders,especially
formatchingandsubclassification,identifyingtwoindividualswithverysimilarval-
uesofallconfounderssimultaneouslywouldbecumbersomeorimpossible.Thus,it
wouldbesensibletoreplacealltheconfoundersbyascalarvariable.Thepropensity
score[22]isapopulardimensionreductionapproachinavarietyofresearchfields.
2 Framework
The aim of statistical causal inference is to understand and estimate a “causal ef-
fect”,andtoidentifyscientificandinprincipletestableconditionsunderwhichthe
causaleffectcanbeidentifiedfromobservationalstudies.Thephilosophicalnature
of“causality”isreflectedin thediversityofitsstatistical formalisations,asexem-
plifiedbythreeframeworks:
1. Rubin’spotentialresponseframework[24,25,26](alsoknownasRubin’scausal
model)basedoncounterfactualtheory;
2. Pearl’scausalframework[16,17]richlydevelopedfromgraphicalmodels;
3. Dawid’sdecision-theoreticframework[6,7]basedondecisiontheoryandprob-
abilisticconditionalindependence.
InDawid’sframework,causalrelationsaremodelledentirelybyconditionalproba-
bilitydistributions.Weadoptitthroughoutthischaptertoaddresscausalinference;
theassumptionsrequiredare,atleastinprinciple,testable.
LetX,T andY denote,respectively,a(typicallymultivariate)confounder,treat-
ment, and response (or outcome). For simplicity, Y is a scalar and X a multi-
SufficientCovariate,PropensityVariableandDoublyRobustEstimation 3
dimensionalvariable.WeassumethatT isbinary:1(treatmentarm)and0(control
arm). Within Dawid’s framework, a non-stochastic regime indicator variable F ,
T
takingvalues0/,0 and 1,is introducedto denotethetreatmentassignmentmecha-
nismoperating.Thisdividestheworldintothreedistinctregimes,asfollows:
1. F =0/:theobservational(idle)regime.Inthisregime,thevalueofthetreatment
T
ispassivelyobservedandtreatmentassignmentisdeterminedbyNature.
2. F =1:the interventionaltreatmentregime,i.e.,treatmentT issetto 1byma-
T
nipulation.
3. F =0:theinterventionalcontrolregime,i.e.,treatmentT issetto0bymanip-
T
ulation.
Forexample,inanobservationalstudyofcustodialsanctions,ourinterestisinthe
effect of custodial sanction, as comparedto probation (noncustodialsanction), on
theprobabilityofre-offence.ThenF =0/ denotestheactualobservationalregime
T
underwhichdatawerecollected;F =1isthe(hypothetical)interventionalregime
T
thatalwaysimposesimprisonment;andF =0isthe(hypothetical)interventional
T
regimethatalwaysimposesprobation.Throughout,weassumefullcomplianceand
no dropouts, i.e., each individual actually takes whichever treatment they are as-
signed to. Then we have a joint distribution P of all relevant variables in each
f
regimeF = f (f =0,1,0/).
T
Inthedecision-theoreticframework,causalassumptionsareconstruedasasser-
tionsthatcertainmarginalor conditionaldistributionsare commonto allregimes.
Suchassumptionscanbeformallyexpressedaspropertiesofconditionalindepen-
dence,wherethisisextendedtoallownon-stochasticvariablessuchasF [4,5,7].
T
For example, the “ignorable treatment assignment” assumption in Rubin’s causal
model(RCM)[22]canbeexpressedas
Y⊥⊥F |T, (1)
T
read as “Y is independent of F given T”. However, this condition will be most
T
likelyinappropriateinobservationalstudieswhererandomisationisabsent.
Causal effect is defined as the response difference by manipulating treatment,
which purely involves interventional regimes. In particular, the population-based
averagecausaleffect(ACE)ofthetreatmentisdefinedas:
ACE:=E(Y|F =1)−E(Y|F =0), (2)
T T
oralternatively,
ACE:=E (Y)−E (Y)1. (3)
1 0
Without further assumptions, by its definition ACE is not identifiable from the
observationalregime.
1Forconvenience,thevaluesoftheregimeindicatorF arepresentedassubscripts.
T
4 HuiGuo,PhilipDawidandGiovanniBerzuini
3 Identification ofACE
Supposethejointdistributionof(F ,T,Y )isknownandsatisfies(1).IsACEiden-
T
tifiablefromdatacollectedintheobservationalregime?Notethat(1)demonstrates
thatthedistributionofY givenT =t isthesame,whethert isobservedintheob-
servationalregimeF =0/,orintheinterventionalregimeF =t.Asdiscussed,this
T T
assumptionwouldnotbesatisfiedinobservationalstudies,andthus,directcompar-
isonofresponsefromthetwotreatmentgroupscannotbeinterpretedasthecausal
effectfromobservationaldata.
Definition1. The“face-valueaveragecausaleffect”(FACE)isdefinedas:
FACE:=E (Y|T =1)−E (Y|T =0). (4)
0/ 0/
ItwouldbehardlytruethatFACE=ACE,aswewouldnotexpecttheconditional
distribution ofY given T =t is the same in any regime. In fact, identification of
ACE fromobservationalstudiesrequires,on one hand,adjustingfor confounders,
ontheotherhand,interplayofdistributionalinformationbetweendifferentregimes.
Onecanmakenofurtherprogressunlesssomepropertiesaresatisfied.
3.1 Stronglysufficient covariate
RigorousconditionsmustbeinvestigatedsoastoidentifyACE.
Definition2. X isacovariateif:
Property1.
X⊥⊥F .
T
That is, the distribution of X is the same in any regime, be it observational or
interventional.Inmostcases,X areattributesdeterminedpriortothetreatment,for
example,bloodtypesandgenes.
Definition3. X isasufficientcovariatefortheeffectoftreatmentT onresponseY
if,inadditiontoProperty1,wehave
Property2.
Y⊥⊥F |(X,T).
T
Property 2 requires that the distribution of Y, given X and T, is the same in
all regimes. It can also be described as “strongly ignorable treatment assignment,
givenX”[22].Weassumethatreadersarefamiliarwiththeconceptandproperties
ofdirectedacyclicgraphs(DAGs).ThenProperties1 and2canberepresentedby
meansofaDAGasFig.1.ThedashedarrowfromX toT indicatesthatT ispartially
dependentonX,i.e.,thedistributionofT dependsonX intheobservationalregime,
butnotintheinterventionalregimewhereF =t.
T
SufficientCovariate,PropensityVariableandDoublyRobustEstimation 5
Fig.1 Sufficientcovariate
X
F T Y
T
Definition4. X is a strongly sufficientcovariateif, in additionto Properties1 and
2,wehave
Property3.P (T =t|X)>0withprobabilility1,fort=0,1.
0/
Property 3 requires that, for any X =x, both treatment and control groups are
observedintheobservationalregime.
Lemma1.SupposeX isastronglysufficientcovariate.Then,consideredasajoint
distributions for (Y,X,T), P is absolutely continuouswith respect to P (denoted
t 0/
byP ≪P ),fort=0andt=1.Thatis,foreveryeventAdeterminedby(X,T,Y),
t 0/
P (A)=0 =⇒ P(A)=0. (5)
0/ t
Equivalently,ifaneventAoccurswithprobability1underthemeasureP ,thenit
0/
occurswithprobability1underthemeasureP (t=0,1).
t
Proof. Property 2, expressed equivalently as (Y,X,T)⊥⊥F |(X,T), asserts that
T
thereexistsafunctionw(X,T)suchthat
P (A|X,T)=w(X,T)
f
almostsurely(a.s.)ineachregime f =0,1,0/.LetP (A)=0.Thena.s.[P ],
0/ 0/
0=P (A|X)=w(X,1)P (T =1|X)+w(X,0)P (T =0|X).
0/ 0/ 0/
ByProperty3,fort=0,1,
w(X,t)=0 (6)
a.s.[P ].Asw(X,t)isafunctionofX,itfollowsthat(6)holdsa.s.[P]byProperty
0/ t
1.Consequently,
w(X,T)=0 a.s. [P], (7)
t
since a.s. [P], T =t and w(X,T)=w(X,t) for any boundedfunctionw. Then by
t
(7),
P(A)=E{P(A|X,T)}=E{w(X,T)}=0.
t t t t
Lemma2.ForanyintegrableZ(cid:22)2 (Y,X,T),andanyversionsoftheconditional
expectations,
E(Z|X)=E(Z|X,T) a.s.[P]. (8)
t t t
2The(cid:22)symbolisinterpretedas“afunctionof”.
6 HuiGuo,PhilipDawidandGiovanniBerzuini
Proof. Let j(X,T)beanarbitrarybutfixedversionofE(Z|X,T).Then j(X,T)=
t
j(X,t)a.s.[P],and j(X,t)servesasaversionofE(Z|X,T)under[P].So
t t t
E(Z|X)=E{j(X,T)|X}=E{j(X,t)|X}= j(X,t) a.s. [P].
t t t t
Thus j(X,t)isaversionofE(Z|X)under[P]and(8)follows.
t t
Since E(Z |X) is a function of X, then by Property 1, j(X,t) is a version of
t
E(Z|X)inanyregime.Letg(X,T)besomearbitrarybutfixedversionofE (Z|
t 0/
X,T).
Theorem1.Suppose that X is a strongly sufficient covariate. Then for any inte-
grableZ(cid:22)(Y,X,T),andwithnotationasabove,
j(X,t)=g(X,t) (9)
almostsurelyinanyregime.
Proof. ByProperty2,thereexistsafunctionh(X,T)whichisacommonversionof
E (Z |X,T) under[P ] for f =0,1,0/. Thenh(X,T)servesasa versionof E (Z |
f f 0/
X,T)under[P ],andaversionofE(Z|X,T)under[P].As j(X,T)isaversionof
0/ t t
E(Z|X,T),
t
j(X,T)=h(X,T) a.s. [P],
t
andconsequently
j(X,t)=h(X,t) a.s. [P].
t
Since j(X,t)andh(X,t)arefunctionsofX,byProperty1
j(X,t)=h(X,t) a.s. [P ] (10)
f
for f =0,1,0/.Wealsohavethatg(X,T)=h(X,T) a.s.[P ],andso,byLemma1,
0/
a.s.[P]. Then g(X,t)=h(X,t) a.s.[P], where g(X,t) and h(X,t) are both func-
t t
tionsofX.ByProperty1,
g(X,t)=h(X,t) a.s. [P ] (11)
f
for f =0,1,0/.Thus(9)holdsby(10)and(11).
3.2 Specific causal effect
LetX beacovariate.
Definition5. Thespecificcausaleffect ofT onY,relativetoX,is
SCE:=E (Y |X)−E (Y |X).
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SufficientCovariate,PropensityVariableandDoublyRobustEstimation 7
WeannotateSCE toexpressSCEasafunctionofX andwriteSCE(x)toindicate
X
that X takes specific value x. Because it is defined in the interventional regimes,
SCE has a direct causal interpretation,i.e., SCE(x) is the average causal effect in
thesubpopulationwithX =x.
Although we do not assume the existence of potential responses, when this
assumption is made we might proceed as follows. Take X to be the pair Y =
(Y(1),Y(0))ofpotentialresponses—whichisassumedtosatisfyProperty1.Then
E(Y |X)=Y(t),andconsequently
t
SCE =Y(1)−Y(0),
Y
whichisthedefinitionof“individualcausaleffect”,ICE,inRubin’scausalmodel.
Thus,althoughthe formalisationsof causality are different,SCE in Dawid’sdeci-
sion theoretic framework can be reagarded as a generalisation of ICE in Rubin’s
causalmodel.
We caneasily provethat, foranycovariateX, ACE=E(SCE ), wherethe ex-
X
pectationmaybetakeninanyregime.SincebyProperty1,
E {E(Y |X)}=E{E(Y |X)}=E(Y),
0/ t t t t
fort =0,1.Thusbysubtraction,ACE=E (SCE )foranyregime f =0,1,0/ and
f X
thereforethesubscript f canbedropped.Hence,ACEisidentifiablefromobserva-
tionaldatasolongasSCE isidentifiablefromobservationaldata.IfX isastrongly
X
sufficientcovariate,byTheorem1,E(Y |X)isidentifiablefromtheobservational
t
regime.ItfollowsthatSCE canbeestimatedfromdatapurelycollectedintheob-
servationalregime.ThenACEexpressedas
ACE=E (SCE ) (12)
0/ X
isidentifiable,fromtheobservationaljointdistributionof(X,T,Y).Formula(12)is
Pearl’s“back-doorformula”[17]becausebythepropertyofmodularity,P(X)isthe
samewithorwithoutinterventiononT andthuscanbetakenasthedistributionof
X intheobservationalregime.
3.3 Dimensionreductionofstronglysufficient covariate
SupposeX isamulti-dimensionalstronglysufficientcovariate.Theadjustmentpro-
cess might be simplified if we could replace X by some reduced variableV (cid:22)X,
with fewerdimensions—solongasV is itself a stronglysufficientcovariate.Now
sinceV isafunctionofX,Properties1and3willautomaticallyholdforV.Wethus
onlyneedtoensurethatV satisfiesProperty2:thatis,
Y⊥⊥F |(V,T). (13)
T
8 HuiGuo,PhilipDawidandGiovanniBerzuini
SincetwoarrowsinitiatefromX inFig.1,possiblereductionsmaybenaturally
considered,onthepathwaysfromX toT, andfromX toY. Indeed,the following
theoremgivestwoalternativesufficientconditionsfor(13) tohold.However,(13)
canstillholdwithouttheseconditions.
Theorem2.Suppose X is a strongly sufficient covariate andV (cid:22)X. ThenV is a
stronglysufficientcovariateifeitherofthefollowingconditionsissatisfied:
(a).Response-sufficientreduction:
Y⊥⊥X|(V,F =t), (14)
T
or
Y⊥⊥X|(V,T,F =0/), (15)
T
for t =0,1. It is indicated in (14) that, in each interventional regime, X con-
tributesnothingtowardspredictingY onceweknowV.Inotherwords,aslong
asV is observed, X need not be observed to make inference onY. While (15)
impliesthatintheobservationalregime,knowingX isofnovalueofpredicting
Y ifV andT areknown.
(b).Treatment-sufficientreduction:
T⊥⊥X|(V,F =0/). (16)
T
Thatis,intheobservationalregime,treatmentdoesnotdependonXconditioning
ontheinformationofV.
Proofsoftheabovereductionswereprovidedin[9].Analternativeproofof(b)
canbe implementedgraphically[9], whichresultsina DAG asFig.2 3 offwhich
(16)and(13)canbedirectlyread.
Fig.2 Treatmentsufficient
reduction
X
V
FT T Y
Agraphicalapproachto(a)doesnotworksinceProperty3isrequired.However,
whilenotservingasaproof,Fig.3convenientlyembodiestheconditionalindepen-
denciesProperties1,2andthetrivialpropertyV⊥⊥T|(X,F ),aswellas(13).
T
3Thehollowarrowhead,pointingfromXtoV,isusedtoemphasisethatV isafunctionofX.
SufficientCovariate,PropensityVariableandDoublyRobustEstimation 9
Fig.3 Response sufficient
reduction
X
V
FT T Y
4 Propensity analysis
Here we furtherdiscussthetreatment-sufficientreduction,whichdoesnotinvolve
theresponse.Thisbringsintheconceptofpropensityvariable:aminimaltreatment-
sufficientcovariate,forwhichweinvestigatetheunbiasednessandprecisionofthe
estimatorofACE. Alsothe asymptoticprecisionoftheestimatedACE,aswellas
thevariationoftheestimatefromtheactualdata,willbeanalysed.Inasimplenor-
mal linear model that applied for covariate adjustment, two cases are considered:
homoscedasticity and heteroscedasticity. A non-parametric approach – subclassi-
fication will also be conducted, for different covariance matrices of X of the two
treatmentarms.TheestimatedACEobtainedbyadjustingformultivariateX andby
adjustingfor a scalar propensityvariable,willthen be comparedtheoreticallyand
throughsimulations[9].
4.1 Propensityscoreand propensityvariable
Thepropensityscore(PS),firstintroducedbyRosenbaumandRubin,isabalancing
score [22]. Regardedas a useful tool to reduce bias and increase precision, it is a
verypopularapproachtocausaleffectestimation.PSmatching(orsubclassification)
method,widelyusedinvariousresearchfields,exploitsthepropertyofconditional
(within-stratum) exchangeability, whereby individuals with the same value of PS
(or belonging to a group with similar values of PS) are taken as comparable or
exchangeable. We will, however, mainly focus on the application of PS within a
linear regression. The definitions of the balancing score and PS given below are
borrowedfrom[22].
Definition6. A balancingscore b(X)isa functionof X suchthat, inthe observa-
tional regime 4, the conditional distribution of X given b(X) is the same for both
treatmentgroups.Thatis,
X⊥⊥T|(b(X),F =0/).
T
4RosenbaumandRubindonotdefinethebalancingscoreandthePSexplicitlyforobservational
studies,althoughtheydoaimtoapplythePSapproachinsuchstudies.
10 HuiGuo,PhilipDawidandGiovanniBerzuini
IthasbeenshownthatadjustingforabalancingscoreratherthanX resultsinunbi-
asedestimateofACE,withtheassumptionofstronglyignorabletreatmentassign-
ment[22].Onecantriviallychooseb(X)=X,butitismoreconstructivetofinda
balancingscoretobeamanytoonefunction.
Definition7. The propensity score, denoted by P , is the probability of being as-
signedtothetreatmentgroupgivenX intheobservationalregime:
P :=P (T =1|X).
0/
We shalluse the symbolp to denotea particularrealisationof P . By (16) and
Definitions6and7,weassertthatPSisthecoarsestbalancingscore.Forasubject
i,PSisassumedtobepositive,i.e.,0<p <1.ThosewiththesamevalueofPSare
i
equallylikelytobeallocatedtothetreatmentgroup(orequivalently,tothecontrol
group),whichprovidesobservationalstudieswiththerandomised-experiment-like
propertybasedonmeasuredX.Thisisbecausethecharacteristicsofthetwogroups
with the same or similar PS are “balanced”. Therefore, the scalar PS serves as a
proxyofmulti-dimensionalvariableX,andthus,itissufficienttoadjustforthefor-
merinsteadofthelatter.Inobservationalstudies,PSisgenerallyunknownbecause
wedonotknowexactlywhichcomponentsofXhaveimpactonT andhowthetreat-
mentisassociatedwiththem.However,wecanestimatePSfromtheobservational
data.
PSanalysisforcausalinferenceisbasedonasequenceoftwostages:
Stage1:PSEstimation.ItisestimatedbytheobservedT andX,andnormally
byalogisticregressionofT onX forbinarytreatment.NotethattheresponseY is
irrelevantatthisstage.BecausewecanestimatePSwithoutobservingY,thereisno
harminfindingan”optimal”regressionmodelofT onX byrepeatedtrials.
Stage2:AdjustingforPS.Variousadjustmentapproacheshavebeendeveloped,
e.g.,linearregression.IfweareunclearabouttheconditionaldistributionofY given
T and PS, non-parametricadjustmentsuch as matching or subclassification could
beappliedinstead.
Althoughtwoalternativesfordimensionreductionshavebeenprovided,inprac-
tice, this type of reduction may be more convenientin many cases. For example,
certain values of the response may occur rarely and only after long observation
periodsaftertreatment.Inaddition,itmaysometimesbetrickytodeterminea”cor-
rect”formforaregressionmodelofY onX,T andF .SwappingthepositionsofX
T
andT,Equation(16)canbere-expressedas
X⊥⊥T|(V,F =0/), (17)
T
which states that the observationaldistribution of X givenV is the same for both
treatmentarms.Thatistosay,V isabalancingscoreforX.
Thetreatment-sufficientcondition(b)canbeequivalentlyinterpretedasfollows.
Consider the family Q ={Q ,Q } consisting of observational distributions of X
0 1
forthetwogroupsT =0andT =1.ThenEquation(16),re-expressedas(17),says
thatV isasufficientstatistic(intheusualFisheriansense[8])forthisfamily.Inpar-