MachineLearningforHealthcareHST.956,6.S897

Lecture14:CausalInferencePart1

DavidSontag

Courseannouncements• Pleasefilloutmid-semestersurvey• Projectproposals

– Youwillreceivee-mail feedback thisweek– OfficehoursnextTuesday,10-11:30am

• Problemsets– PS1-4graded(seeStellar)– PS5outtonight,duenextTuesday,April9– Lastproblem set,PS6,released in~2weeks

• Recitationthisweekwillbeadiscussionof– Bratetal.,Postsurgicalprescriptions foropioidnaïvepatients

andassociationwithoverdoseandmisuse,BMJ2018– Bertsimasetal.,Personalized diabetesmanagement using

electronicmedical records,DiabetesCare2017

Doesgastricbypasssurgerypreventonsetofdiabetes?

• InLecture4&PS2weusedmachinelearningforearlydetectionofType2diabetes

• Healthsystemdoesn’twanttoknowhowtopredictdiabetes– theywanttoknowhowtopreventit

• Gastricbypasssurgeryisthehighest negativeweight(9thmostpredictivefeature)– Doesthismeanitwouldbeagoodintervention?

1994 2000

<4.5%4.5%–5.9%6.0%–7.4%7.5%–8.9%>9.0%

2013

• Suchpredictivemodelswidelyusedtostagepatients.Shouldweinitiatetreatment?Howaggressive?

• Whatcouldgowrongifwetrainedtopredictsurvival,andthenusedtoguidepatientcare?

Mammography(86Ksubjects)

Competitive Period Launch: Nov 18, 2016Competitive Period Close: May 9, 2017

Outof1000womenscreened,only5willhavebreastcancer

Goal:developalgorithmsforriskstratificationofscreeningmammogramsthatcanbeusedtoimprovebreastcancerdetection

Whatisthelikelihoodthispatient,withbreastcancer,willsurvive5years?

𝑿𝒀

Diagnosis Death Time

“Mary”

Treatment

Alongsurvivaltimemaybebecauseoftreatment!

• Peopleresponddifferentlytotreatment• Goal:usedatafromotherpatientsandtheirjourneystoguidefuturetreatmentdecisions

• Whatcouldgowrongifwetrainedtopredict(past)treatmentdecisions?

Whattreatmentshouldwegivethispatient?Expansion Pathology

with DNA-FISH and Protein-IF

Blue =HER2ProteinRed =HER2AmpliconGreen =Centromeric probe

NegativeforHER2Amplification HER2Amplified

Expansionpathology(imagefromAndyBeck)

“David” TreatmentA

TreatmentA“Juana”“John” TreatmentB

Bestthiscandoismatchcurrentmedicalpractice!

• Doingarandomizedcontroltrialisunethical• CouldwesimplyanswerthisquestionbycomparingPr(lungcancer|smoker)vsPr(lungcancer|nonsmoker)?

• No!Answeringsuchquestionsfromobservationaldataisdifficultbecauseofconfounding

Doessmokingcauselungcancer?

Toproperlyanswer,needtoformulateascausal questions:

Intervention, 𝑇

(e.g. medication, procedure)

Outcome, 𝑌

Patient, 𝑋

(including allconfoundingfactors)

?

Highdimensional Observationaldata

PotentialOutcomesFramework(Rubin-Neyman CausalModel)

• Eachunit(individual)𝑥' hastwopotentialoutcomes:– 𝑌((𝑥') isthepotentialoutcomehadtheunitnotbeentreated:“controloutcome”

– 𝑌+(𝑥') isthepotentialoutcomehadtheunitbeentreated:“treatedoutcome”

• Conditionalaveragetreatmenteffectforunit𝑖:𝐶𝐴𝑇𝐸 𝑥' = 𝔼23~5(23|78)[𝑌+|𝑥'] − 𝔼2=~5(2=|78)[𝑌(|𝑥']

• AverageTreatmentEffect:𝐴𝑇𝐸:= 𝔼 𝑌+ − 𝑌( = 𝔼7~5(7) 𝐶𝐴𝑇𝐸 𝑥

PotentialOutcomesFramework(Rubin-Neyman CausalModel)

• Eachunit(individual)𝑥' hastwopotentialoutcomes:– 𝑌((𝑥') isthepotentialoutcomehadtheunitnotbeentreated:“controloutcome”

– 𝑌+(𝑥') isthepotentialoutcomehadtheunitbeentreated:“treatedoutcome”

• Observedfactualoutcome:𝑦' = 𝑡'𝑌+ 𝑥' + 1 − 𝑡' 𝑌((𝑥')

• Unobservedcounterfactualoutcome:𝑦'CD = (1 − 𝑡')𝑌+ 𝑥' + 𝑡'𝑌((𝑥')

Thefundamental problemofcausal inference“Thefundamental problemof

causal inference”

Weonlyeverobserveoneofthetwooutcomes

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Example– Bloodpressureandage

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Bloodpressureandage

𝐶𝐴𝑇𝐸(𝑥)

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Bloodpressureandage

𝐴𝑇𝐸

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Bloodpressureandage

Treated

Control

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Bloodpressureandage

Treated

Control

Counterfactualtreated

Counterfactualcontrol

(age,gender,exercise,treatment)

Sugarlevelshadtheyreceived

medicationA

Sugarlevelshadtheyreceived

medicationB

Observedsugarlevels

(45,F,0,A) 6 5.5 6(45,F,1,B) 7 6.5 6.5(55,M,0,A) 7 6 7(55,M,1,B) 9 8 8(65,F,0,B) 8.5 8 8(65,F,1,A) 7.5 7 7.5(75,M,0,B) 10 9 9(75,M,1,A) 8 7 8

(ExamplefromUriShalit)

(age,gender,exercise)

Sugarlevelshadtheyreceived

medicationA

Sugarlevelshadtheyreceived

medicationB

Observedsugarlevels

(45,F,0) 6 5.5 6(45,F,1) 7 6.5 6.5(55,M,0) 7 6 7(55,M,1) 9 8 8(65,F,0) 8.5 8 8(65,F,1) 7.5 7 7.5(75,M,0) 10 9 9(75,M,1) 8 7 8

(ExamplefromUriShalit)

(age,gender,exercise)

Y0:Sugarlevelshadtheyreceived

medicationA

Y1:Sugarlevelshadtheyreceived

medicationB

Observedsugarlevels

(45,F,0) 6 5.5 6(45,F,1) 7 6.5 6.5(55,M,0) 7 6 7(55,M,1) 9 8 8(65,F,0) 8.5 8 8(65,F,1) 7.5 7 7.5(75,M,0) 10 9 9(75,M,1) 8 7 8

(ExamplefromUriShalit)

(age,gender,exercise)

Sugarlevelshadtheyreceived

medicationA

Sugarlevelshadtheyreceived

medicationB

Observedsugarlevels

(45,F,0) 6 5.5 6

(45,F,1) 7 6.5 6.5

(55,M,0) 7 6 7

(55,M,1) 9 8 8

(65,F,0) 8.5 8 8

(65,F,1) 7.5 7 7.5

(75,M,0) 10 9 9

(75,M,1) 8 7 8

mean(sugar|medication B)–mean(sugar|medicaton A)=?

mean(sugar|had theyreceived B)–mean(sugar|had theyreceived A)=?

(ExamplefromUriShalit)

(age,gender,exercise)

Sugarlevelshadtheyreceived

medicationA

Sugarlevelshadtheyreceived

medicationB

Observedsugarlevels

(45,F,0) 6 5.5 6

(45,F,1) 7 6.5 6.5

(55,M,0) 7 6 7

(55,M,1) 9 8 8

(65,F,0) 8.5 8 8

(65,F,1) 7.5 7 7.5

(75,M,0) 10 9 9

(75,M,1) 8 7 8

mean(sugar|medication B)–mean(sugar|medicaton A)=7.875- 7.125=0.75

mean(sugar|had theyreceived B)–mean(sugar|had theyreceived A)=7.125- 7.875=-0.75

(ExamplefromUriShalit)

Typicalassumption– nounmeasuredconfounders

𝑌(, 𝑌+:potentialoutcomesforcontrolandtreated𝑥:unitcovariates(features)T:treatmentassignment

Weassume:(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

Thepotentialoutcomesareindependentoftreatmentassignment,conditionedoncovariates𝑥

Typicalassumption– nounmeasuredconfounders

𝑌(, 𝑌+:potentialoutcomesforcontrolandtreated𝑥:unitcovariates(features)T:treatmentassignment

Weassume:(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

Ignorability

covariates(features)

treatment

Potentialoutcomes

𝑻𝒙

𝒀𝟏𝒀𝟎

Ignorability

(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

𝑻𝒙

𝒀𝟏𝒀𝟎

anti-hypertensivemedication

bloodpressureaftermedicationA

age,gender,weight,diet,heartrateatrest,…

bloodpressureaftermedicationB

Ignorability

(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

𝒙

𝒀𝟏𝒀𝟎bloodpressureaftermedicationA

age,gender,weight,diet,heartrateatrest,…

bloodpressureaftermedicationB

𝒉

NoIgnorability

diabetic𝑻

anti-hypertensivemedication

(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

Typicalassumption– commonsupport

Y(, 𝑌+:potentialoutcomesforcontrolandtreated𝑥:unitcovariates(features)𝑇:treatmentassignment

Weassume:𝑝 𝑇 = 𝑡 𝑋 = 𝑥 > 0∀𝑡, 𝑥

Framingthequestion

1. Wherecouldwegotofordatatoanswerthesequestions?

2. WhatshouldX,T,andYbetosatisfyignorability?3. Whatisthespecificcausalinferencequestionthat

weareinterestedin?4. Areyouworriedaboutcommonsupport?

Outlineforlecture

• Howtorecognizeacausalinferenceproblem• Potentialoutcomesframework– Averagetreatmenteffect(ATE)– Conditionalaveragetreatmenteffect(CATE)

• AlgorithmsforestimatingATEandCATE

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

AverageTreatmentEffect–theadjustmentformula

• Assumingignorability,wewillderivetheadjustment formula (Hernán &Robins2010,Pearl2009)

• Theadjustmentformulaisextremelyusefulincausalinference

• AlsocalledG-formula

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

E [Y1] =

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x]

⇤=

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x, T = 1]

⇤=

Ex⇠p(x) [E [Y1|x, T = 1]]

lawoftotalexpectation

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

E [Y1] =

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x]

⇤=

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x, T = 1]

⇤=

Ex⇠p(x) [E [Y1|x, T = 1]]

ignorability(𝑌(, 𝑌+) ⫫ 𝑇|𝑥

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

E [Y1] =

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x]

⇤=

Ex⇠p(x)

⇥EY1⇠p(Y1|x) [Y1|x, T = 1]

⇤=

Ex⇠p(x) [E [Y1|x, T = 1]] shorternotation

AverageTreatmentEffect

Theexpectedcausaleffectof𝑇 on𝑌:ATE := E [Y1 � Y0]

E [Y0] =

Ex⇠p(x)

⇥EY0⇠p(Y0|x) [Y0|x]

⇤=

Ex⇠p(x)

⇥EY0⇠p(Y0|x) [Y0|x, T = 1]

⇤=

Ex⇠p(x) [E [Y0|x, T = 0]]

Quantitieswecanestimate

fromdata

Theadjustmentformula(

E[Y1|x,T=1]

E[Y0|x,T=0](E [Y1|x, T = 1]

E [Y0|x, T = 0]

ATE = E [Y1 � Y0] =

Ex⇠p(x)[ E [Y1|x, T = 1]�E [Y0|x, T = 0] ]

Undertheassumptionofignorability,wehavethat:

Quantitieswecannotdirectly

estimatefromdata

Theadjustmentformula(

E[Y1|x,T=1]

E[Y0|x,T=0]ATE = E [Y1 � Y0] =

Ex⇠p(x)[ E [Y1|x, T = 1]�E [Y0|x, T = 0] ]

Undertheassumptionofignorability,wehavethat:

E [Y0|x, T = 1]

E [Y1|x, T = 0]

E [Y0|x]E [Y1|x]

Quantitieswecanestimate

fromdata

Theadjustmentformula(

E[Y1|x,T=1]

E[Y0|x,T=0](E [Y1|x, T = 1]

E [Y0|x, T = 0]

ATE = E [Y1 � Y0] =

Ex⇠p(x)[ E [Y1|x, T = 1]�E [Y0|x, T = 0] ]

Empiricallywehavesamplesfrom𝑝(𝑥|𝑇 = 1) or𝑝 𝑥 𝑇 = 0 .Extrapolate to 𝑝(𝑥)

Undertheassumptionofignorability,wehavethat:

Manymethods!

CovariateadjustmentPropensityscore re-weightingDoublyrobustestimatorsMatching…

Covariateadjustment

• Explicitlymodeltherelationshipbetweentreatment,confounders,andoutcome

• Alsocalled“ResponseSurfaceModeling”• UsedforbothITEandATE• Aregressionproblem

𝑥+

𝑥\

𝑥]

𝑇

… 𝑓(𝑥, 𝑇)

𝑦

Regressionmodel

OutcomeCovariates(Features)

𝑥+

𝑥\

𝑥]

𝑇

…

𝑦

NuisanceParameters

Regressionmodel

Outcome

Parameterofinterest

𝑓(𝑥, 𝑇)

Covariateadjustment(parametricg-formula)

• Explicitlymodeltherelationshipbetweentreatment,confounders,andoutcome

• Underignorability,theexpectedcausaleffectof𝑇 on𝑌:𝔼7~5 7 𝔼 𝑌+ 𝑇 = 1, 𝑥 − 𝔼 𝑌( 𝑇 = 0, 𝑥

• Fitamodel𝑓 𝑥, 𝑡 ≈ 𝔼 𝑌 𝑇 = 𝑡, 𝑥

𝐴𝑇𝐸a =1𝑛c𝑓 𝑥', 1 − 𝑓(𝑥', 0)d

'e+

Covariateadjustment(parametricg-formula)

• Explicitlymodeltherelationshipbetweentreatment,confounders,andoutcome

• Underignorability,theexpectedcausaleffectof𝑇 on𝑌:𝔼7~5 7 𝔼 𝑌+ 𝑇 = 1, 𝑥 − 𝔼 𝑌( 𝑇 = 0, 𝑥

• Fitamodel𝑓 𝑥, 𝑡 ≈ 𝔼 𝑌 𝑇 = 𝑡, 𝑥

𝐶𝐴𝑇𝐸a 𝑥' = 𝑓 𝑥', 1 − 𝑓(𝑥', 0)

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Covariateadjustment

Treated

Control

Treated

𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥

Covariateadjustment

Treated

Control

Counterfactualtreated

Counterfactualcontrol

𝒇

Exampleofhowcovariateadjustmentfailswhenthereisnooverlap

TreatedTreated

Control 𝑥 = 𝑎𝑔𝑒

𝑦 =𝑏𝑙𝑜𝑜𝑑_𝑝𝑟𝑒𝑠.

𝑌+ 𝑥

𝑌( 𝑥