SAMSI Tutorial on Causal Inference

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SAMSI Tutorial onCausal Inference

• Miguel A. Hernán
• Department of Epidemiology
• Harvard School of Public Health
• www.hsph.harvard.edu/causal

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Causal inferencea central task of science

• To estimate the causal effect of variable A on
  variable Y is a common goal of the sciences
• Physics, chemistry, biology
• Experiments and observations
• Epidemiology, economics, sociology
• Mostly observations

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This tutorial covers

• The conditions required for causal inference
• The methods for causal inference under those
  conditions
• Using a language and conceptual framework that is
  common to all sciences involved in causal
  inference from observational data

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This tutorial does not cover

• whether the conditions required for causal
  inference are met in a particular case
• That is a subject-matter issue
• Expert knowledge needed for causal inference

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Causal inference in 2.5 hours?Need to focus

• Low-dimensional data
• Fixed (non time-varying) exposures
• Nonparametric analysis (not models)
• High-dimensional data
• Time-varying exposures
• Dynamic and nondynamic regimes
• Parametric/semiparametric models
• Well discuss
• low-dimensional data only
• no sampling variability
• If X1.6 and Y1.7 then X not equal to Y
• Equivalent to assuming that we work with the
  whole population (or a huge sample size)

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Outline

• Definition of causal effect
• Estimation of causal effects in randomized
  experiments
• Estimation of causal effects in observational
  studies

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An intuitive definition of cause

• Ian took the pill on Sept 1, 2003
• Five days later, he died
• Had Ian not taken the pill on Sept 1, 2003 (all
  others things being equal)
• Five days later, he would have been alive
• Did the pill cause Ians death?

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An intuitive definition of cause

• Jim didnt take the pill on Sept 1, 2002
• Five days later, he was alive
• Had Jim taken the pill on Sept 1, 2002 (all
  others things being equal)
• Five days later, he would have been alive
• Did the pill cause Jims survival?

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Human reasoning for causal inference

• We compare (often only mentally)
• the outcome when action A is present with
• the outcome when action A is absent
• all other things being equal
• If the two outcomes differ, we say that the
  action A has a causal effect
• causative or preventive
• In epidemiology, A is commonly referred to as
  exposure or treatment

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Notation for actual data

• Y1 if patient died, 0 otherwise
• Yi1, Yj0
• A1 if patient treated, 0 otherwise
• Ai1, Aj0

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Notation for ideal data

• Ya01 ?if subject had not taken the pill, he
  would have died
• Yi, a0 0, Yj, a0 0
• Ya11 ?if subject had taken the pill, he would
  have died
• Yi, a1 1, Yj, a1 0

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Clarification

• Upper-case letters for random variables
• A, Y, Ya0 , Ya1
• Lower-case letters for possible values
  (realizations) of those variables
• a is a possible value (0 or 1) of the random
  variable A
• For our purposes, random variables are variables
  with different values for different individuals

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(Individual) Causal effect

• For Ian
• Pill has a causal effect because
• For Jim
• Pill does not have a causal effect because
• Sharp causal null hypothesis holds if, for all
  subjects,

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Potential or counterfactual outcomes

• Ya0 and Ya1
• Random variables
• Amenable to mathematical treatment, e.g.,
  statistical models
• One of them is the subject's potential outcome
  that would have been observed under an exposure
  value that the subject did not actually
  experience
• Refers to a counter to the fact situation

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Consistency

• Key assumption
• One of the counterfactual outcomes is the
  subject's actual outcome under the exposure value
  that the subject actually experienced
• Refers to an observed (factual) situation
• If Aia then Yi, a Yi, A Yi
• Under consistency, a potential outcome Ya is
  factual for some subjects and counterfactual for
  others

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Available data set
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Fundamental problem of causal inference

• Individual causal effects cannot be determined
• except under extremely strong (and generally
  unreasonable) assumptions
• because only one counterfactual outcome is
  observed
• Causal inference as a missing data problem
• Whether using a randomized experiment or an
  observational study
• Need another definition of causal effect that
  requires weaker assumptions

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First, more notation

• PrYa1
• proportion of subjects that would have developed
  the outcome Y had all subjects in the population
  of interest received exposure value a
• (Counterfactual) Risk of Ya
• Unconditional or marginal probability
• Calculated by using data from the whole
  population

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(Population) Causal effect

• In the population, exposure A has a causal effect
  on the outcome Y if
• Causal null hypothesis holds if PrYa11
  PrYa01

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Equivalent representations of the causal null
hypothesis

• PrYa11 - PrYa01 0
• PrYa11 / PrYa01 1
• (PrYa11/PrYa10)/(PrYa01/PrYa00)
  1
• Causal effect can be measured in many scales
• causal risk difference, causal risk ratio, causal
  odds ratio,
• Effect measures

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Individual versus populationcausal effects

• Individual causal effects cannot be determined
• except under quite restrictive assumptions
• Population causal effects can be determined under
• no assumptions (ideal randomized studies)
• strong assumptions (observational studies)
• Well refer to population causal effects only

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Association and causationMore notation

• PrY1Aa
• proportion of subjects that developed the outcome
  Y among those who received exposure value a in
  the population
• Risk of Y among the exposed/unexposed
• Conditional probability
• Calculated by using data from a subset of the
  population

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Association

• The exposure A and the outcome Y are associated
  if
• No association independence

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Equivalent representations of independence

• PrY1A1 - PrY1A0 0
• PrY1A1 / PrY1A0 1
• (PrY1A1/PrY0A1) / (PrY1A0/PrY0A
  0) 1
• Association can be measured in many scales
• Associational risk difference, associational risk
  ratio, associational odds ratio,
• Association measures

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Again, crucial difference Association is not
causation

• Association different risk in two disjoint
  subsets of the population determined by the
  subjects' actual exposure value
• PrY1Aa is the risk in subjects of the
  population that meet the condition having
  actually received exposure level a
• Causation different risk in the entire
  population under two exposure values
• PrYa1 is the risk in all subjects of the
  population had they received the counterfactual
  exposure level a

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An example of causal concept Confounding

• There is confounding when association is not
  causation
• Confounding cannot be defined using associational
  (statistical) language
• More about confounding later

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Generalizations of counterfactual theory

• Causal effects in a subset of the population
• Non dichotomous outcome and exposure
• Non deterministic counterfactual outcomes
• Interference
• Time-varying exposures

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Counterfactual theory can be generalized
regarding a) d)

• But no major conceptual insights are gained from
  these generalizations
• For pedagogic reasons, we will stick to the
  simplified version
• Causal effect in the entire population
• Dichotomous variables
• Deterministic counterfactuals
• No interference

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e) Time-varying exposures Counterfactual
theories

• Neyman (1923)
• Effects of point exposures in randomized
  experiments
• Rubin (1974)
• Effects of point exposures in randomized and
  observational studies
• Robins (1986)
• Effects of time-varying exposures in randomized
  and observational studies

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Outline

• Definition of causal effect
• Estimation of causal effects in randomized
  experiments
• Estimation of causal effects in observational
  studies

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Association measures

• The associational risk ratio
• PrY1A1/PrY1A0
• can be directly computed in any study
• because Y is observed in all subjects of the
  population
• PrY1A1 and PrY1A0 are observed risks

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Effect measures

• The causal risk ratio
• PrYa11 / PrYa01
• cannot be directly computed (in general)
• because Ya1 and Ya0 are unobserved in some
  subjects of the population
• PrYa11 and PrYa01 are unobserved risks

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Effect measures

• can be computed using data from ideal randomized
  experiments
• with no assumptions
• (More rigorously, effect measures can be
  consistently estimated using data from ideal
  randomized experiments)
• For now lets consider experiments with
  near-infinite sample sizes only

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What is an ideal randomized experiment?

• No loss to follow-up
• Full compliance with (adherence to) assigned
  exposure or treatment
• Double blind assignment

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In ideal randomized experiments

• PrYa11 is equal to PrY1A1
• PrYa01 is equal to PrY1A0
• Therefore the associational RR
• PrY1A1 / PrY1A0
• is equal to the causal RR
• PrYa11 / PrYa01
• Well prove this equality holds because
• experimental treatment assignment ensures
  consistency
• randomization ensures exchangeability

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Experimental treatment assignment

• One (near-infinite) population
• Divided into two groups
• Group 1 and group 2
• One group is treated and the other untreated

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Consistency

• The counterfactual outcome under treatment of the
  treated is their observed outcome
• The counterfactual outcome under no treatment of
  the untreated is their observed outcome
• The counterfactual outcomes are consistent with
  the observed outcomes
• Consistency is a consequence of experimental
  treatment assignment

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Randomization (I)

• Membership in each group (1 or 2) is randomly
  assigned
• e.g., by the flip of a coin
• First option
• Treat subjects in group 1, dont treat subjects
  in group 2
• The risk is, say, PrY1A1 0.57
• Second option
• Treat subjects in group 2, dont treat subjects
  in group 1
• What is the value of the risk PrY1A1 ?

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Randomization (II)

• When group membership is randomly assigned,
  results are the same
• whether group 1 treated, group 2 untreated
• or vice versa
• Both groups are comparable or exchangeable
• Exchangeability is a consequence of randomization

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Exchangeability

• Subjects in group 1 would have had the same risk
  as those in group 2 had they received the
  treatment of those in group 2
• The counterfactual risk in the treated equals the
  counterfactual risk in the untreated

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Formal definition of exchangeability

• Exchangeability implies lack of confounding
• Exchangeability is another causal concept that
  cannot be represented by associational
  (statistical) language

for all a
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ProofWhy PrY1Aa PrYa1?

• Two steps
• PrY1Aa PrYa1Aa
• by consistency
• PrYa1Aa PrYa1
• by exchangeability
• Consistency and exchangeability ensured in ideal
  randomized studies

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In an ideal randomized experiment

• Association is causation because
• experimental treatment assignment produces
  consistency
• randomization produces exchangeability
• We have a method for causal inference!
• No need for adjustments of any sort
• Assumption-free

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Example Does heart transplant (A) increase
5-year survival (Y)?

• Select a large population of potential recipients
  of a transplant
• Get funding and IRB/Ethical approval
• Randomly allocate them to either transplant (A1)
  or medical treatment (A0)
• 5 years later, compute the associational RR
  PrY1A1 / PrY1A0
• that equals (cons. estimates) the causal RR
  PrYa11 / PrYa01

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Potential problems ofreal randomized experiments

• Loss to follow-up
• Noncompliance
• Unblinding
• Other ethics, feasibility, cost

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Consequence of problems a), b), c)

• Although exchangeability still holds in
  randomized experiments but
• available association may not be causation
  (loss to follow-up)
• exposure is misclassified (non compliance) or
  contaminated (unblinding)
• Causal inference from real randomized studies may
  require assumptions and analytic methods similar
  to those for causal inference from observational
  studies

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Conclusion

• No clear-cut separation between randomized and
  observational studies
• Observational studies are needed
• In fact, most of human knowledge comes from
  observations, e.g., evolution theory, tectonic
  plaques theory, hot coffee may cause burns
• And so are methods for causal inference from
  observational data

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Outline

• Definition of causal effect
• Estimation of causal effects in randomized
  experiments
• Estimation of causal effects in observational
  studies

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Conditions for causal inferenceconsistency and
exchangeability

• In ideal randomized experiments association is
  causation because
• experimental treatment assignment produces
  consistency
• randomization produces exchangeability
• But ideal randomized experiments are rare
• We need observational studies
• No experimental treatment assignment
  Consistency?
• No randomization Exchangeability?

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Consistency in observational studies

• If no consistency, then counterfactuals are not
  well defined
• If counterfactuals are not well defined then
  causal effects are not well defined
• Can consistency not hold?
• Lets see some examples of exposures in
  observational studies

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SEX?

• Certain chronic disease occurs more frequently in
  women (S1) than in men (S0)
• PrY1S1 gt PrY1S0
• Does sex has a causal effect on the risk of
  disease?
• PrYs11 / PrYs01

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Quite vague question

• The causal question (in English) would be
  something like
• What would have been the risk had everybody been
  of female sex compared with
• Wait a second, what do we mean by female sex?
• carrying a pair of X chromosomes
• having been brought up as a woman
• high levels of estrogens between adolescence and
  menopause
• ...?

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Another vague question

• What is the effect of obesity on mortality?
• Compare what would happen if everybody were
  obese
• If you dont think the question is vague, try to
  describe an experiment to replicate the results
  from an observational study on obesity and
  mortality
• design a randomized experiment in which
  participants are randomly assigned to either
  obesity or non obesity
• assume unlimited resources and no ethical
  constraints

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Key question what is the appropriate
intervention?

• Many possible interventions could be used to
  assign participants to the non-obese group
• Extreme exercise, starvation, surgery, genetic
  modification
• Each may lead to a different outcome even if,
  when counterfactually applied to a given
  individual, they all would produce identical body
  weight
• The counterfactual outcome under each exposure
  level is not well defined because the value of
  the counterfactual outcome may depend on the
  intervention used to manipulate the exposure

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Lack of consistency leads to ill-defined causal
effects

• Not a consequence of
• ethical constraints
• e.g., the effect of cigarette smoking can be well
  defined even if some of the hypothetical
  interventions involved are harmful
• unfeasible interventions
• e.g., the effect of long-term diet can be well
  defined even if some of the hypothetical
  interventions involved are impractical

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Conclusion Being able to utter X does not entail
that X has a meaning

• (The wishes of my running shoes?)
• In observational studies, to give an unambiguous
  meaning to a causal question, we need to be able
  to describe a hypothetical intervention
• walk 2 hours/day 7 days a week, eat 2000
  calories/day versus walk 0.5 hours, eat 3000
  calories/day
• For some questions we have a common understanding
  of the intervention
• Effect of heart transplant

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A benefit of a formal definition of causal effects

• Some interventions sound technically unfeasible
  (or plainly crazy) because formulating certain
  causal questions is not straightforward
• A counterfactual approach to causal inference
  highlights the imprecision/ vagueness of some
  causal questions
• Effect of age, HDL-cholesterol, HIV viral load,
  socioeconomic status,
• Formulating an appropriate causal question is a
  subject-matter issue

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Conditions for causal inferenceconsistency and
exchangeability

• Well assume consistency holds in observational
  studies
• That is, well assume exposures are well defined
• What about exchangeability in observational
  studies?
• First lets review exchangeability in ideal
  randomized studies

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Exchangeability in ideal randomized experiments

• Exchangeability is guaranteed
• PrYa11 is equal to PrY1A1
• PrYa01 is equal to PrY1A0
• Therefore the associational risk ratio
• PrY1A1 / PrY1A0
• is equal to the causal risk ratio
• PrYa11 / PrYa01

for all a
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But even in ideal randomized experiments

• The equality between crude association measure
  and effect measure does not always hold
• Association risk ratio not necessarily equal to
  causal risk ratio
• Need to take into account the design of the
  experiment
• Was randomization conditional or unconditional?
• So far we have considered unconditional
  (marginal) randomization only

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Consider the following ideal randomized experiment

• 2 million subjects with heart disease
• numbers divided by 100,000 in example
• Variables
• A1 heart transplant (exposure)
• Y1 death (outcome)
• L1 critical condition (prognostic factor)
• Goal to estimate the effect of A on Y
• in the risk ratio scale

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The data summarized in a table
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The data summarized in a tree
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Two designs could have produced these data

• Design 1
• randomly select and expose 65 of the individuals
• unconditional randomization probability
• Design 2
• classify individuals as in critical (L1) or
  noncritical (L0) condition
• Randomly select 75 of L1 and 50 of L0, and
  expose the selected individuals
• randomization probabilities depend (are
  conditional) on the value of L

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Implications of each design

• Design 1
• Exchangeability is guaranteed
• Design 2
• Greater proportion of L1 in the exposed
• No exchangeability
• But Design 2 is simply the combination of two
  Design 1 studies one in L1 and another one in
  L0
• Exchangeability is guaranteed in each Design 1
  substudy (i.e., conditional on L)

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Design 2Conditional exchangeability

• Within levels of L, exposed subjects would have
  had the same risk as unexposed subjects had they
  being unexposed, and vice versa
• Counterfactual risk is the same in the exposed
  and the unexposed with the same value of L

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Formal definition of conditional exchangeability
for all a

• Conditional exchangeability is equivalent to
  randomization within levels of L
• It implies no confounding within levels of the
  variable L

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Data analysis Design 1 Exchangeability

• Counterfactual risk observed risk
• PrYa1 PrY1Aa
• Causal risk ratio assoc. risk ratio
• PrYa11 / PrYa01 (7/13)/(3/7) 1.26
• But 69 exposed versus 43 unexposed were in
  critical condition
• Exchangeability does not hold
• Data not generated under Design 1

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Data analysis Design 2 Conditional
exchangeability

• Conditional counterfactual risk conditional
  observed risk
• PrYa1L0 PrY1Aa, L0
• PrYa1L1 PrY1Aa, L1
• Counterfactual risk in the population is the
  weighted average of the stratum-specific
  counterfactual risks
• From basic probability theory marginal
  probability is the weighted average of the
  conditional probabilities

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ProofPrY1Ll, Aa PrYa1Ll

• Two steps
• PrY1Ll, Aa PrYa1Ll, Aa
• by consistency
• PrYa1Ll, Aa PrYa1Ll
• by conditional exchangeability
• Therefore

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Data analysis Design 2 Conditional
exchangeability

• PrYa11 (1/4)0.4(2/3)0.6 0.5
• PrYa01 (1/4)0.4(2/3)0.6 0.5
• PrYa11 / PrYa01 0.5/0.5 1
• Whats the name of this method?
• Standardization

I
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Standardization as a simulation

• Standardization is the equivalent of simulating
  what would happen in the study population if
• everybody had received certain exposure level a,
  and
• the distribution of the covariate L were the same
  as its distribution in the standard population

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In summary, in Design 1-2 randomized experiments

• Randomization produces exchangeability (design 1)
  or conditional exchangeability (design 2)
• In both cases, the causal effect can be easily
  calculated
• Design 1 Crude association measure
• Design 2 Standardized association measure
• the g-formula for fixed exposure

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What does this have to do with observational
studies?

• Meet Design 3
• Investigators do not intervene in the assignment
  of hearts but rather they observe which
  individuals happen to receive them
• The data they observe are

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In a Design 3 (observational) study

• Absence of randomization implies that
  exchangeability is not guaranteed
• In general,
• PrYa11 is not equal to PrY1A1
• PrYa01 is not equal to PrY1A0
• Therefore the associational RR
• PrY1A1 / PrY1A0
• is not generally equal to the causal RR
• PrYa11 / PrYa01

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In a Design 3 (observational) study

• Exchangeability is too strong an assumption!
• The exposed and the unexposed are not generally
  comparable
• e.g., individuals who receive a heart transplant
  may have a more severe disease than those who do
  not receive it
• In general,

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In a Design 3 (observational) study

• The investigators may believe that the exposed
  and the unexposed are exchangeable within levels
  of L
• had exposed patients in critical condition stayed
  unexposed, they would have had the same mortality
  risk as those in critical condition who actually
  stayed unexposed (and vice versa)
• And similarly for patients in noncritical
  condition
• That is, the investigators may be willing to
  assume conditional exchangeability

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Is this a reasonable assumption in observational
studies?

• Consider only individuals with the same
  pre-exposure prognostic factors
• Then the exposed and the unexposed may be
  exchangeable
• e.g., among individuals with an ejection fraction
  of 40, those who do and do not receive a heart
  transplant may be comparable
• e.g., among individuals with CD4 countlt100, those
  who do and do not receive antiretroviral therapy
  may be comparable
• This is sometimes reasonable
• Especially if conditioning on many pre-exposure
  covariates L

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The randomized experiment paradigm for
observational studies

• An observational study (design 3) can be viewed
  as a randomized experiment (design 2) in which
• the conditional probabilities of exposure are not
  chosen by the investigators
• but can be estimated from the data
• conditional exchangeability is not guaranteed
• but only assumed based on the investigators
  expert knowledge

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In a Design 3 (observational) study,conditional
exchangeability

• Necessary condition for causal inference
• In fact, the weakest condition for causal
  inference
• Under conditional exchangeability in all strata
  Ll, we can compute (consistently estimate) the
  causal risk ratio
• PrYa11 / PrYa01
• Using standardization (g-formula)

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In summary, in a Design 3 (observational) study

• Causal effects can be calculated
• under the assumption of conditional
  exchangeability within levels of the covariates
• We have a method for causal inference from
  observational data that it is not assumption-free
• But the need to rely on this assumption is not
  THE problem

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Can we check whether conditional exchangeability
holds?

• Nope
• This is THE problem
• The assumption of conditional exchangeability is
  untestable
• Even if there is conditional exchangeability,
  there is no way we can know it with certainty

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RememberConditional exchangeability
for all a

• Conditional exchangeability is equivalent to
  randomization within levels of L
• It implies no unmeasured confounding within
  levels of the measured variables L
• Data necessary to test this condition is, by
  definition, unavailable

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Thats why causal inference from observational
data is controversial

• Expert knowledge can be used to enhance the
  plausibility of the assumption
• measure as many relevant pre-exposure covariates
  as possible
• Then one can only hope the assumption of
  conditional exchangeability is approximately true
• (All we are saying is that there may be
  confounding due to unmeasured factors)

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Identifiability and confounding

• Design 1 exchangeability guaranteed
• effect measures computed with A,Y data
• the causal effect is identifiable given A,Y data
• no confounding
• Design 2 conditional exchangeability guaranteed
• effect measures computed with L,A,Y data
• the causal effect is identifiable given L,A,Y
  data
• no unmeasured confounding
• Design 3 conditional exchangeability assumed
• effect measures computed with data L,A,Y?
• the causal effect is not identifiable given data
  only
• no unmeasured confounding?

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Outline

• Definition of causal effect
• Estimation of causal effects in randomized
  experiments
• Estimation of causal effects in observational
  studies
• Standardization (g-formula)
• Inverse probability weighting

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Under conditional exchangeability

• One can estimate causal effects in Design 2
  (randomized) and Design 3 (observational) studies
  by using standardization
• We now describe another method to estimate causal
  effects inverse probability weighting (IPW)

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IPW plan of action

• YOU will compute the causal risk ratio using IPW
  in an observational study
• i.e., you will compute PrYa11/PrYa01
• under conditional exchangeability
• We will prove that you were right

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A simplified observational study

• 2 million subjects with heart disease
• Divided by 100,000 in numerical example
• Variables
• A1 heart transplant (exposure)
• Y1 death (outcome)
• L1 critical condition (prognostic factor)
• Goal to estimate the effect of A on Y
• on the risk ratio scale

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The data summarized in a table
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The data summarized in a tree
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Your goal

• To compute the effect of a heart transplant on
  the risk of death using the causal risk ratio
  scale
• PrYa11 / PrYa01
• Assuming conditional exchangeability within
  levels of L
• First, compute PrYa01
• Second, compute PrYa11

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Data analysis in the pseudo-population

• PrYa11 10 / 20 0.5
• PrYa01 10 / 20 0.5
• Causal risk ratio 0.5 / 0.5 1

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Which assumption are you making?
for all a

• Conditional exchangeability in the population
• exposure is randomized within levels of L
• no unmeasured confounding within levels of the
  measured variable L
• Within levels of L, the risk among the exposed if
  unexposed is the same as the risk among the
  unexposed in the population
• and vice versa

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Under conditional exchangeability

• The observational study in the original
  population is a randomized experiment within
  levels of L
• The study in the pseudo-population created by IPW
  is a randomized experiment
• Exposed and unexposed subjects are
  (unconditionally!) exchangeable
• Because they are the same individuals
• Exposure is randomized, i.e., equally probable
  across levels of the covariate L
• There is no confounding
• In the pseudo-population, causal effects can be
  estimated as in a randomized experiment
• No need for adjustments of any sort

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You did it

• You computed the causal risk ratio using inverse
  probability weighting
• Right?

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Inverse probability weights

• Each individual in the population is weighted to
  create W individuals in the pseudo-population
• The denominator of your W is (informally) the
  probability of having your observed treatment
  value given your L value
• Not equal for all individuals with same L value
  because it depends on A value as well

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Notational clarification

• fA(a) or f(a) is the probability density function
  (pdf) of the random variable A evaluated at the
  value a
• For discrete A f(a) PrAa
• We need to represent the probability that each
  subject had his/her own exposure level A
• PrAA is unclear notation at best
• f(A) is the pdf evaluated at the random argument
  A (exactly what we mean)

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IPW as a simulation

• Weighting is the equivalent of simulating what
  would happen in the study population if everybody
  had received certain exposure level a
• (Hmm That sounded vaguely familiar)
• Individuals in the original population who
  received exposure level a are weighted to
  represent all individuals (regardless of exposure
  level) in the population
• sample size of pseudo-population is equal to
  number of exposure levels times the size of
  original population

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Too much of a coincidence?

• IPW risk in the exposed was equal to the
  standardized risk in the exposed
• IPW risk in the unexposed was equal to the
  standardized risk in the unexposed
• Standardized risk ratio IPW risk ratio
• Is this true in general?
• Yes

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Proof for dichotomous Y and discrete A and L

• Just algebra
• By definition (consistency)
• By assumption
• Just algebra

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Standardization IPW

• When study population is used as the standard
• Each method compute a different component of the
  joint distribution
• IPW fAL
• Standardization fL, fYA,L
• But the methods are algebraically equivalent
• The standardized risk ratio is equal to the IPW
  risk ratio
• Both are equal to the causal risk ratio
• PrYa11 / PrYa01

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Standardization IPW only in non parametric
settings

• In real applications, sparse data dont allow to
  compute the components of the joint distribution
• if L includes 20 variables or a continuous
  variable
• in the presence of time-varying exposures
• We would need to estimate these components using
  (semi)parametric models
• IPW model to estimate fAL
• Standardization models to estimate fL, fYA,L

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Something elseThe Positivity condition

• In each level of L in the population, there must
  be exposed and unexposed individuals
• If f(l)gt0 then f(al)gt0 for all a
• conditional probabilities must be positive
• IPW/Standardization cannot be used when the
  positivity condition is not met

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Generalization of standardizationto time-varying
exposures?

• G-formula (Robins 1986)
• THE method to estimate causal effects in non
  parametric settings
• Independently discovered by computer
  scientists/artificial intelligence researchers
• Problems
• For complex longitudinal data and/or continuous
  covariates it requires huge amounts of data
• Computationally intensive
• No parameter for null hypothesis

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Generalization of IPW to time-varying exposures?

• IPW has a direct generalization
• Time-varying weights W(t)
• Informally, the inverse of the probability of
  having your observed treatment history through t
  given your L history through t
• Weights can be estimated using models and then
  standard models can be used in the
  pseudo-population
• Marginal structural models (Robins 1998)
• More by Butch Tsiatis

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Conclusions (I) Causal inference from
observational data is possible

• under the assumptions of
• Consistency
• Conditional exchangeability
• assumption of conditional randomization or of no
  unmeasured confounders
• Positivity
• using IPW/Standardization
• Other assumptions are required for both
  observational and randomized data

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Conclusions (II) Causal inference from
observational data is risky

• Because the conditional exchangeability cannot be
  guaranteed or even tested
• Expert knowledge can be used to enhance the
  plausibility of the assumption
• measure as many relevant pre-exposure covariates
  as possible
• Epidemiologists must be subject-matter experts or
  work with them
• But one can only hope the assumption is
  approximately true