Download Causal inference for survival analysis (I)

Outline
Causal inference
for survival analysis (I)
1. Definition of causal effect
† Counterfactuals
2. Estimation of causal effects:
† Inverse probability weighting
Miguel A. Hernán
Department of Epidemiology
Harvard School of Public Health
3. Causal diagrams
† Directed acyclic graphs
4. The bias of standard methods
5. Causal models
Lysebu, September 2004
† Marginal structural models
Causal inference (I)
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An intuitive definition of cause
An intuitive definition of cause
† Ian took the pill on Sept 1, 2003
† Jim didn’t take the pill on Sept 1,
2002
„ Five days later, he died
„ Five days later, he was alive
† 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 Ian’s death?
Causal inference (I)
† 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 Jim’s survival?
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Causal inference (I)
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Human reasoning
for causal inference
Notation for actual data
† We compare (often only mentally)
† Y=1 if patient died, 0 otherwise
„ the outcome when action A is present with
„ the outcome when action A is absent
„ all other things being equal
„ Yi=1, Yj=0
† A=1 if patient treated, 0 otherwise
„ Ai=1, Aj=0
† 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.
Causal inference (I)
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ID
A
Y
Ian
1
1
Jim
0
0
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1
Notation for ideal data
Clarification:
† Ya=0=1 if subject would have died, had he
not taken the pill
† Upper-case letters for random variables
„ A, Y, Ya=0 , Ya=1
† Lower-case letters for possible values
(realizations) of those variables
„ Yi, a=0= 0, Yj, a=0= 0
† Ya=1=1 if patient would have died, had he
taken the pill
„ a is a possible value (0 or 1) of the random
variable A
„ Yi, a=1= 1, Yj, a=1= 0
ID
A
Ya=0
Ya=1
Ian
1
0
1
Jim
0
0
0
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† For our purposes, random variables are
variables with different values for different
individuals
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(Individual) Causal effect
Potential or
counterfactual outcomes (I)
† For Ian:
† Ya=0 and Ya=1
† Random variables
„ Pill has a causal effect because
† For Jim:
Yi,a1  Yi,a0
„ Amenable to mathematical treatment, e.g.,
statistical models
„ Pill does not have a causal effect because
Yj,a1  Yj,a0
™ Sharp causal null hypothesis holds
if, for all subjects,
Ya1  Ya0
Causal inference (I)
„ Refers to a “counter to the fact” situation
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Potential or
counterfactual outcomes (II)
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Available data set
† One of them describes the subject's
outcome value under the exposure value
that the subject actually experienced
„ Refers to an observed (factual) situation
† A given potential outcome is factual for
some subjects and counterfactual for others
† Consistency
„ By definition, if Ai=a then Yi, a = Yi, A = Yi
Causal inference (I)
† One of them describes the subject's
outcome value that would have been
observed under a potential exposure value
that the subject did not actually experience
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ID
Ian
Jim
Ken
Leo
Mike
Nick
…
A
1
0
1
0
1
0
Y
1
0
0
1
1
0
Causal inference (I)
Ya=0
?
0
?
1
?
0
Ya=1
1
?
0
?
1
?
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2
Fundamental problem
of causal inference
First, more notation
† Individual causal effects cannot be
determined
† Pr[Ya=1]
„ 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
Causal inference (I)
† Unconditional or marginal probability
„ “Calculated” by using data from the
whole population
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Causal inference (I)
† In the population, exposure A has a
causal effect on the outcome Y if
† Pr[Ya=1=1] − Pr[Ya=0=1] = 0
† Pr[Ya=1=1] / Pr[Ya=0=1] = 1
† (Pr[Ya=1=1]/Pr[Ya=1=0])/(Pr[Ya=0=1]/Pr[Ya=0=0)
=1
† Causal effect can be measured in
many scales:
PrYa1  1  PrYa0  1
™ Causal null hypothesis holds if
Pr[Ya=1=1] = Pr[Ya=0=1]
„ causal risk difference, causal risk ratio,
causal odds ratio, …
„ Effect measures
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Causal inference (I)
Individual versus population
causal effects
Association and causation:
More notation
† Individual causal effects cannot be
determined
† Pr[Y=1|A=a]
„ except under extremely strong assumptions
† Population causal effects can be
determined under
„ no assumptions (randomized studies)
„ strong assumptions (observational studies)
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„ 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
† We’ll refer to (population) causal effects
only
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Equivalent representations of the
causal null hypothesis
(Population) Causal effect
Causal inference (I)
„ 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
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„ Calculated by using data from a subset
of the population
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Equivalent representations of
independence
Association
† Pr[Y=1|A=1] − Pr[Y=1|A=0] = 0
† Pr[Y=1|A=1] / Pr[Y=1|A=0] = 1
† (Pr[Y=1|A=1]/Pr[Y=0|A=1]) /
(Pr[Y=1|A=0]/Pr[Y=0|A=0]) = 1
† The exposure A and the outcome Y
are associated if
PrY  1|A  1  PrY  1|A  0
† No association = independence
PrY  1|A  1  PrY  1|A  0  A  Y  Y  A
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† 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
„ Pr[Y=1|A=a] is the risk in subjects of the
population that meet the condition `having
actually received exposure a’
† Causation: different risk in the entire
population under two exposure values
„ Pr[Ya=1] is the risk in all subjects of the
population had they received the counterfactual
exposure a
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An example of causal concept:
Confounding
Statistics and causation
† We need counterfactuals to talk about
causation
† Statistics (as a discipline) leaned towards
banning counterfactuals from statistical
language
† Statistics w/o counterfactuals is a language
for association, not for causation
„ Causal concepts cannot be represented using
statistics w/0 counterfactuals
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† There is confounding when
association is not causation
PrYa  1  PrY  1|A  a
† Confounding cannot be defined using
associational (statistical) language
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Counterfactual theory in statistics
Outline
† Neyman (1923)
1. Definition of causal effect
„ Effects of point exposures in
randomized experiments
† Counterfactuals
2. Estimation of causal effects
† Rubin (1974)
† Inverse probability weighting
„ Effects of point exposures in
randomized and observational
studies
3. Causal diagrams
† Directed acyclic graphs
4. The bias of standard methods
5. Causal models
† Robins (1986)
„ Total and direct effects of
time-varying exposures in
longitudinal studies
Causal inference (I)
† Marginal structural models
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Causal inference (I)
Association measures
Effect measures
† The associational risk ratio
„ Pr[Y=1|A=1]/Pr[Y=1|A=0]
can be directly computed in any study
† The causal risk ratio
„ Pr[Ya=1=1] / Pr[Ya=0=1]
cannot be directly computed in
general
because Y is observed in all subjects
of the population
because Ya=1 and Ya=0 are unobserved
in some subjects of the population
„ Pr[Ya=1=1] and Pr[Ya=0=1] are
unobserved risks
„ Pr[Y=1|A=1] and Pr[Y=1|A=0] are
observed risks
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Causal inference (I)
Effect measures
What is an ideal
randomized experiment?
† can be computed using data from
ideal randomized experiments
† No loss to follow-up
„ with no assumptions
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† Full compliance with (adherence to)
assigned exposure or treatment
† More rigorously, effect measures can
be consistently estimated using data
from ideal randomized experiments
„ For now let’s consider experiments with
near-infinite sample sizes only
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† Double blind assignment
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Randomization (I)
In ideal randomized experiments
„ Pr[Ya=1=1] is equal to Pr[Y=1|A=1]
„ Pr[Ya=0=1] is equal to Pr[Y=1|A=0]
† One (near-infinite) population
† Divided into two groups
„ Group 1 and group 2
† Therefore the associational RR
† Membership in each group is
randomly assigned
„ Pr[Y=1|A=1] / Pr[Y=1|A=0]
is equal to the causal RR
„ e.g., by the flip of a coin
„ Pr[Ya=1=1] / Pr[Ya=0=1]
† One group is treated and the other
untreated
† Let’s prove it
„ first we need to describe randomization
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Randomization (II)
Randomization (III)
† First option
† When group membership is randomly
assigned, results are the same
„ Treat subjects in group 1, don’t treat
subjects in group 2
„ The risk is, say, Pr[Y=1|A=1] = 0.57
„ whether group 1: treated, group 2:
untreated
„ or vice versa
† Both groups are comparable or
exchangeable
† Exchangeability is the consequence of
randomization
† Second option
„ Treat subjects in group 2, don’t treat
subjects in group 1
„ What is the value of the risk Pr[Y=1|A=1] ?
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Exchangeability
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Formal definition of 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
Ya  A for all a
† Exchangeability implies lack of
confounding
† Exchangeability is another causal
concept that cannot be represented by
associational (statistical) language
PrYa  1|A  1  PrYa  1|A  0  A  Ya  Ya  A
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Proof:
Why Pr[Y=1|A=a] = Pr[Ya=1]?
In an ideal randomized experiment
† Association is causation
† because randomization produces
exchangeability
† Two steps:
1. Pr[Y=1|A=a] = Pr[Ya=1|A=a]
„ by definition (consistency)
2. Pr[Ya=1|A=a] = Pr[Ya=1]
™ We have a method for causal
inference!
„ by randomization (exchangeability)
„ No need for adjustments of any sort
„ Assumption-free
† Step 2 not generally true in the
absence of randomization
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Causal inference (I)
Example: Does heart transplant (A)
increase 5-year survival (Y)?
Potential problems of
real randomized experiments
† Select a large population of potential
recipients of a transplant
† Get funding and IRB/Ethical approval
† Randomly allocate them to either transplant
(A=1) or medical treatment (A=0)
† 5 years later, compute the associational RR
a) Loss to follow-up
Pr[Y=1|A=1] / Pr[Y=1|A=0]
† that equals (cons. estimates) the causal RR
Pr[Ya=1=1] / Pr[Ya=0=1]
Causal inference (I)
b) Noncompliance
c) Unblinding
d) Other: ethics, feasibility, cost…
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Causal inference (I)
Consequence of problems a), b), c)
Conclusion
† Although exchangeability still holds in
randomized experiments but
† No clear-cut separation between
randomized and observational studies
„ “available association” may not be causation
(loss to follow-up)
„ exposure is misclassified (non compliance) or
contaminated (unblinding)
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† Observational studies are needed
† 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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„ 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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In observational studies
In observational studies, can we
assume exchangeability?
† Absence of randomization implies that
exchangeability is not guaranteed
† In general,
† Too strong an assumption!
† The exposed and the unexposed are
not generally comparable
„ Pr[Ya=1=1] is not equal to Pr[Y=1|A=1]
„ Pr[Ya=0=1] is not equal to Pr[Y=1|A=0]
„ e.g., individuals who receive a heart
transplant may have a more severe
disease than those who do not receive it
† Therefore the associational RR
„ Pr[Y=1|A=1] / Pr[Y=1|A=0]
is not generally equal to the causal RR
„ Pr[Ya=1=1] / Pr[Ya=0=1]
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Dead end?
¾
¾
† In general,
PrYa  1|A  1  PrYa  1|A  0  A 
 Ya  Ya 
 A
Causal inference (I)
In search of a weaker condition
† Consider only individuals with the same
pre-exposure prognostic factors
† Then the exposed and the unexposed may
be exchangeable
Exchangeability (a consequence of
randomization) is a condition for causal
inference
Exchangeability is not generally an
acceptable assumption in observational
studies
† A condition weaker than exchangeability is
needed for causal inference from
observational data
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„ e.g., among individuals with an ejection fraction
of 50%, those who do and do not receive a
heart transplant may be comparable
„ e.g., among individuals with CD4 count<100,
those who do and do not receive antiretroviral
therapy may be comparable
† This is often reasonable
„ Especially if conditioning on many pre-exposure
covariates L
Causal inference (I)
Available data set (with covariates)
Conditional exchangeability
ID
1
2
3
4
5
6
7
8
† Within levels of the covariates 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
L
0
0
0
0
1
1
1
1
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A
1
1
0
0
1
1
0
0
Y
1
0
1
0
1
0
1
0
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Ya=0
?
?
1
0
?
?
1
0
Ya=1
1
0
?
?
1
0
?
?
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PrYa  1|A  1, L  l  PrYa  1|A  0, L  l 
A  Ya|L  l  Ya  A|L  l
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Formal definition of
conditional exchangeability
Ya  A|L  l
OK, conditional exchangeability is a
weaker condition, so what?
† Conditional exchangeability is a necessary
condition for causal inference from
observational data
† Under conditional exchangeability in all
strata L=l, we can compute (consistently
estimate) the causal risk ratio
„ Pr[Ya=1=1] / Pr[Ya=0=1]
† The assumption of conditional
exchangeability implies
„ Pr[Y=1|L=l, A=a] = Pr[Ya=1|L=l]
for all a
† Conditional exchangeability is
equivalent to randomization within
levels of L
† It implies no unmeasured (residual)
confounding within levels of the
measured variables L
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Proof:
Pr[Y=1|L=l, A=a] = Pr[Ya=1|L=l]
† Association is causation within levels of the
covariates
† under the assumption of conditional
exchangeability
„ by definition of counterfactual variable
2. Pr[Ya=1|L=l, A=a] = Pr[Ya=1|L=l]
„ by assumption (conditional
exchangeability)
™ We have a method for causal inference
from observational data that it is not
assumption-free
† Same as for randomized studies but
within levels of L
„ But the need to rely on this assumption is not
THE problem
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Causal inference (I)
Can we check whether conditional
exchangeability holds?
That’s why causal inference from
observational data is controversial
™ No
† Expert knowledge can be used to
enhance the plausibility of the
assumption
† This is THE problem
† The assumption of conditional
exchangeability is untestable
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„ measure as many relevant pre-exposure
covariates as possible
† Then one can only hope the
assumption of conditional
exchangeability is approximately true
„ Even if there is conditional
exchangeability, there is no way we can
know it with certainty
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In an observational study
† Two steps:
1. Pr[Y=1|L=l, A=a] = Pr[Ya=1|L=l, A=a]
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„ (All we are saying is that there may be
confounding due to unmeasured factors)
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Inverse probability weighting (IPW)
† 500 HIV-infected individuals
† Variables:
† A method to compute causal effects
under conditional exchangeability
† Plan of action:
„ L=1: CD4 cell count<200 cells/microL
„ A=1: on highly active antiretroviral
therapy (HAART)
„ Y=1: AIDS
„ YOU will compute the causal risk ratio
using IPTW in an observational study
† i.e., you will compute Pr[Ya=1=1]/Pr[Ya=0=1]
† Treatment status is decided after
looking at CD4 cell count
† No loss to follow-up
under conditional exchangeability
„ We will prove that you were right
Causal inference (I)
A simplified example of
observational study
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The data summarized in a table
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The data summarized in a tree
10
40
L=0
L=1
30
Y=1
Y=0
Y=1
Y=0
20
A=1
20
30
108
252
16
A=0
40
10
24
16
24
252
108
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10
Your goal
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† To compute the effect of HAART on
the risk of AIDS on the causal risk
ratio scale
„ Pr[Ya=1=1] / Pr[Ya=0=1]
„ Assuming conditional
exchangeability within levels of L
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20
16
24
252
† First, compute Pr[Ya=0=1]
† Second, compute Pr[Ya=1=1]
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10
20
Pseudopopulation data analysis
80
60
40
160
Ya = 1
Ya = 0
a=1
160
340
a=0
320
180
240
† Pr[Ya=1=1] = 160 / (160 +340) = 0.32
† Pr[Ya=0=1] = 320 / (320 + 180) = 0.64
† Causal risk ratio = 0.32 / 0.64 = 0.5
280
120
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Which assumption are you making?
Ya  A|L  l
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
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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 pseudopopulation created by IPTW 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 pseudopopulation, causal effects can be
estimated as in a randomized experiment
„ No need for adjustments of any sort
„ and vice versa
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W = 1 / f [A|L]
You did it
† You computed the causal risk ratio
using inverse-probability-oftreatment weighting
† Right?
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1/.5 = 2
80
1/.5 = 2
60
1/.5 = 2
40
1/.5 = 2
160
1/.1 = 10
240
1/.1 = 10
280
1/.9 = 1.11
120
1/.9 = 1.11
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Inverse-probability-of-treatment
weights
W
Important difference
1
fA|L
† Each individual in the population is
weighted to create W individuals in the
pseudopopulation
† The denominator of your W is (informally)
the probability of having your observed
treatment value given your L value
† Propensity score
† The PS is the probability of being treated
given L
„ Equal for all individuals with same L value
because it does not depend on the A value
„ Not equal for all individuals with same L value
because it depends on A value as well
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Proof: Inverse probability weighted
mean equals counterfactual mean
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) = Pr[A=a]
† We need to represent the probability that
each subject had his/her own exposure
level A
„ Pr[A=A] makes no sense
„ f(A) is the pdf evaluated at the random argument
A, exactly what we mean
Causal inference (I)
W 1
PS
PS  PrA  1|L
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E IA  a
E E
E E
Y
fA|L
E
IA  a
Ya|L
fA  a|L
IA  a
Ya
fA  a|L

IA  a
|L EYa |L
fA  a|L

† By definition
(consistency)
† Just algebra
„

† By assumption
„
EEYa |L  EYa 
E[X] = E[ E[X|Z] ]
E[XZ] = E[X] E[Z] if X
and Z are independent
† Just algebra
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The positivity condition
IPW estimation
† Required for the proof
† In each level of L in the population,
there must be exposed and
unexposed individuals
† If f(l)>0 then f(a|l)>0 for all a
† In general, we refer to the causal risk
ratio estimate obtained by using
weights IPT weights W as an IPTW
estimate
„ conditional probabilities must be positive
† IPW cannot be used when the
positivity condition is not met
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† The idea is a generalization of
Horvitz-Thompson (1952) estimators
for survey sampling
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IPW as a simulation
† Weighting is the equivalent of simulating
what would happen had all individuals in
the population experience every possible
exposure level
† 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 pseudopulation is equal to
number of exposure levels times the size of
original population
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