causal inference by: miguel a. hernán and james m. robins · zeus: ya=1 = 1 6= 0 = ya=0...
TRANSCRIPT
Causal Inference
By: Miguel A. Hernan and James M. Robins
Chapter 1: A definition of causal effect
Chapter 2: Randomized experiments
Chapter 3: Observational studies
4th December, 2018
Reunio GRBIO
Causal inference
1 Hernan & Robins, Chap. 1: A definition of causal
effects
2 Hernan & Robins, Chap. 2: Randomized Experiments
3 Hernan & Robins, Chap. 3: Observational Studies
Causal Inference Reunio GRBIO 4th December, 2018 2 / 25
Hernan & Robins: Causal Inference.
Chapter 1: A definition of causal effects1.1 Individual causal effects
1.2 Average causal effects
1.3 Measures of causal effect
1.4 Random variability
1.5 Causation versus association
Purpose of Chapter 1:
“... is to introduce mathematical notation that formalizes the
causal intuition that you already possess.”
Causal Inference Reunio GRBIO 4th December, 2018 3 / 25
Hernan & Robins: Causal Inference.
Chapter 1: A definition of causal effects1.1 Individual causal effects
1.2 Average causal effects
1.3 Measures of causal effect
1.4 Random variability
1.5 Causation versus association
Purpose of Chapter 1:
“... is to introduce mathematical notation that formalizes the
causal intuition that you already possess.”
Causal Inference Reunio GRBIO 4th December, 2018 3 / 25
Chapter 1.1: Individual causal effects
Some notation
Dichotomous treatment variable: A (1: treated; 0: untreated)
Dichotomous outcome variable: Y (1: death; 0: survival)
Y a=i: Outcome under treatment a = i, i ∈ {0, 1}.
Definition
Causal effect for an individual: Treatment A has a causal effect on an
individual’s outcome Y if
Y a=1 6= Y a=0
for the individual.
Causal Inference Reunio GRBIO 4th December, 2018 4 / 25
Chapter 1.1: Individual causal effects
Some notation
Dichotomous treatment variable: A (1: treated; 0: untreated)
Dichotomous outcome variable: Y (1: death; 0: survival)
Y a=i: Outcome under treatment a = i, i ∈ {0, 1}.
Definition
Causal effect for an individual: Treatment A has a causal effect on an
individual’s outcome Y if
Y a=1 6= Y a=0
for the individual.
Causal Inference Reunio GRBIO 4th December, 2018 4 / 25
Chapter 1.1: Individual causal effects (cont.)
Examples
Zeus: Y a=1 = 1 6= 0 = Y a=0 =⇒ treatment has causal effect.
Hera: Y a=1 = Y a=0 = 0 =⇒ treatment has no causal effect.
Definition
Consistency: If Ai = a, then Y ai = Y Ai = Yi.
Important:
Y a=0 and Y a=1 are counterfactual outcomes.
Only one can be observed, i.e., only one is factual.
Hence, in general, individual effects cannot be identified.
Causal Inference Reunio GRBIO 4th December, 2018 5 / 25
Chapter 1.1: Individual causal effects (cont.)
Examples
Zeus: Y a=1 = 1 6= 0 = Y a=0 =⇒ treatment has causal effect.
Hera: Y a=1 = Y a=0 = 0 =⇒ treatment has no causal effect.
Definition
Consistency: If Ai = a, then Y ai = Y Ai = Yi.
Important:
Y a=0 and Y a=1 are counterfactual outcomes.
Only one can be observed, i.e., only one is factual.
Hence, in general, individual effects cannot be identified.
Causal Inference Reunio GRBIO 4th December, 2018 5 / 25
Chapter 1.1: Individual causal effects (cont.)
Examples
Zeus: Y a=1 = 1 6= 0 = Y a=0 =⇒ treatment has causal effect.
Hera: Y a=1 = Y a=0 = 0 =⇒ treatment has no causal effect.
Definition
Consistency: If Ai = a, then Y ai = Y Ai = Yi.
Important:
Y a=0 and Y a=1 are counterfactual outcomes.
Only one can be observed, i.e., only one is factual.
Hence, in general, individual effects cannot be identified.
Causal Inference Reunio GRBIO 4th December, 2018 5 / 25
Chapter 1.1: Individual causal effects (cont.)
Examples
Zeus: Y a=1 = 1 6= 0 = Y a=0 =⇒ treatment has causal effect.
Hera: Y a=1 = Y a=0 = 0 =⇒ treatment has no causal effect.
Definition
Consistency: If Ai = a, then Y ai = Y Ai = Yi.
Important:
Y a=0 and Y a=1 are counterfactual outcomes.
Only one can be observed, i.e., only one is factual.
Hence, in general, individual effects cannot be identified.
Causal Inference Reunio GRBIO 4th December, 2018 5 / 25
Chapter 1.1: Individual causal effects (cont.)
Examples
Zeus: Y a=1 = 1 6= 0 = Y a=0 =⇒ treatment has causal effect.
Hera: Y a=1 = Y a=0 = 0 =⇒ treatment has no causal effect.
Definition
Consistency: If Ai = a, then Y ai = Y Ai = Yi.
Important:
Y a=0 and Y a=1 are counterfactual outcomes.
Only one can be observed, i.e., only one is factual.
Hence, in general, individual effects cannot be identified.
Causal Inference Reunio GRBIO 4th December, 2018 5 / 25
Chapter 1.2: Average causal effects
An example: Zeus’s extended family
Y a=0 Y a=1 Y a=0 Y a=1
Rheia 0 1 Leto 0 1
Kronos 1 0 Ares 1 1
Demeter 0 0 Athena 1 1
Hades 0 0 Hephaestus 0 1
Hestia 0 0 Aphrodite 0 1
Poseidon 1 0 Cyclope 0 1
Hera 0 0 Persephone 1 1
Zeus 0 1 Hermes 1 0
Artemis 1 1 Hebe 1 0
Apollo 1 0 Dionysus 1 0
Causal Inference Reunio GRBIO 4th December, 2018 6 / 25
Chapter 1.2: Average causal effects
An example: Zeus’s extended family
Y a=0 Y a=1 Y a=0 Y a=1
Rheia 0 1 Leto 0 1
Kronos 1 0 Ares 1 1
Demeter 0 0 Athena 1 1
Hades 0 0 Hephaestus 0 1
Hestia 0 0 Aphrodite 0 1
Poseidon 1 0 Cyclope 0 1
Hera 0 0 Persephone 1 1
Zeus 0 1 Hermes 1 0
Artemis 1 1 Hebe 1 0
Apollo 1 0 Dionysus 1 0
Causal Inference Reunio GRBIO 4th December, 2018 6 / 25
Chapter 1.2: Average causal effects (cont.)
Definition
An average causal effect of treatment A on outcome Y is present in a
population of interest if
P(Y a=1 = 1) 6= P(Y a=0 = 1).
More generally (nondichotomous outcomes):
E(Y a=1) 6= E(Y a=0).
Example:
There is no average causal effect in Zeus’s family since
P(Y a=1 = 1) = P(Y a=0 = 1) = 10/20 = 0.5.
That does not imply the absence of individual effects.
Causal Inference Reunio GRBIO 4th December, 2018 7 / 25
Chapter 1.2: Average causal effects (cont.)
Definition
An average causal effect of treatment A on outcome Y is present in a
population of interest if
P(Y a=1 = 1) 6= P(Y a=0 = 1).
More generally (nondichotomous outcomes):
E(Y a=1) 6= E(Y a=0).
Example:
There is no average causal effect in Zeus’s family since
P(Y a=1 = 1) = P(Y a=0 = 1) = 10/20 = 0.5.
That does not imply the absence of individual effects.
Causal Inference Reunio GRBIO 4th December, 2018 7 / 25
Chapter 1.2: Average causal effects (cont.)
Definition
An average causal effect of treatment A on outcome Y is present in a
population of interest if
P(Y a=1 = 1) 6= P(Y a=0 = 1).
More generally (nondichotomous outcomes):
E(Y a=1) 6= E(Y a=0).
Example:
There is no average causal effect in Zeus’s family since
P(Y a=1 = 1) = P(Y a=0 = 1) = 10/20 = 0.5.
That does not imply the absence of individual effects.
Causal Inference Reunio GRBIO 4th December, 2018 7 / 25
Chapter 1.2: Average causal effects (cont.)
Definition
An average causal effect of treatment A on outcome Y is present in a
population of interest if
P(Y a=1 = 1) 6= P(Y a=0 = 1).
More generally (nondichotomous outcomes):
E(Y a=1) 6= E(Y a=0).
Example:
There is no average causal effect in Zeus’s family since
P(Y a=1 = 1) = P(Y a=0 = 1) = 10/20 = 0.5.
That does not imply the absence of individual effects.
Causal Inference Reunio GRBIO 4th December, 2018 7 / 25
Fine Points
Fine point 1.1: Interference between subjects
Present if outcome depends on other subjects’ treatment value.
Implies that Y ai is not well defined.
Book assumes “stable-unit-treatment-value assumption (SUTVA)”
(Rubin 1980)
Fine point 1.2: Multiple versions of treatment
Different versions of treatment could exist.
Implies that Y ai is not well defined.
Authors assume “treatment variation irrelevance throughout this
book.”
Causal Inference Reunio GRBIO 4th December, 2018 8 / 25
Fine Points
Fine point 1.1: Interference between subjects
Present if outcome depends on other subjects’ treatment value.
Implies that Y ai is not well defined.
Book assumes “stable-unit-treatment-value assumption (SUTVA)”
(Rubin 1980)
Fine point 1.2: Multiple versions of treatment
Different versions of treatment could exist.
Implies that Y ai is not well defined.
Authors assume “treatment variation irrelevance throughout this
book.”
Causal Inference Reunio GRBIO 4th December, 2018 8 / 25
Chapter 1.3: Measures of causal effect
Representations of the causal null hypothesis
P(Y a=1 = 1)− P(Y a=0 = 1) = 0 (Causal risk difference)
P(Y a=1 = 1)
P(Y a=0 = 1)= 1 (Causal risk ratio)
P(Y a=1 = 1)/P(Y a=1 = 0)
P(Y a=0 = 1)/P(Y a=0 = 0)= 1 (Causal odds ratio)
The effect measures quantify the possible causal effect on different scales.
Causal Inference Reunio GRBIO 4th December, 2018 9 / 25
Chapter 1.3: Measures of causal effect
Representations of the causal null hypothesis
P(Y a=1 = 1)− P(Y a=0 = 1) = 0 (Causal risk difference)
P(Y a=1 = 1)
P(Y a=0 = 1)= 1 (Causal risk ratio)
P(Y a=1 = 1)/P(Y a=1 = 0)
P(Y a=0 = 1)/P(Y a=0 = 0)= 1 (Causal odds ratio)
The effect measures quantify the possible causal effect on different scales.
Causal Inference Reunio GRBIO 4th December, 2018 9 / 25
Chapter 1.4: Random variability
Samples: Two sources of random error
Sampling variability:
We only dispose of P(Y a=1 = 1) and P(Y a=0 = 1). Statistical
procedures are necessary to test the causal null hypothesis.
Nondeterministic counterfactuals:
Counterfactual outcomes Y a=1 and Y a=0 may not be fixed, but
rather stochastic.
“Thus statistics is necessary in causal inference to quantify random
error from sampling variability, nondeterministic counterfactuals,
or both. However, for pedagogic reasons, we will continue to
largely ignore statistical issues until Chapter 10.”
Causal Inference Reunio GRBIO 4th December, 2018 10 / 25
Chapter 1.4: Random variability
Samples: Two sources of random error
Sampling variability:
We only dispose of P(Y a=1 = 1) and P(Y a=0 = 1). Statistical
procedures are necessary to test the causal null hypothesis.
Nondeterministic counterfactuals:
Counterfactual outcomes Y a=1 and Y a=0 may not be fixed, but
rather stochastic.
“Thus statistics is necessary in causal inference to quantify random
error from sampling variability, nondeterministic counterfactuals,
or both. However, for pedagogic reasons, we will continue to
largely ignore statistical issues until Chapter 10.”
Causal Inference Reunio GRBIO 4th December, 2018 10 / 25
Chapter 1.4: Random variability
Samples: Two sources of random error
Sampling variability:
We only dispose of P(Y a=1 = 1) and P(Y a=0 = 1). Statistical
procedures are necessary to test the causal null hypothesis.
Nondeterministic counterfactuals:
Counterfactual outcomes Y a=1 and Y a=0 may not be fixed, but
rather stochastic.
“Thus statistics is necessary in causal inference to quantify random
error from sampling variability, nondeterministic counterfactuals,
or both. However, for pedagogic reasons, we will continue to
largely ignore statistical issues until Chapter 10.”
Causal Inference Reunio GRBIO 4th December, 2018 10 / 25
Chapter 1.5: Causation versus association
A “real world” example
A Y A Y A Y
Rheia 0 0 Zeus 1 1 Aphrodite 1 1
Kronos 0 1 Artemis 0 1 Cyclope 1 1
Demeter 0 0 Apollo 0 1 Persephone 1 1
Hades 0 0 Leto 0 0 Hermes 1 0
Hestia 1 0 Ares 1 1 Hebe 1 0
Poseidon 1 0 Athena 1 1 Dionysus 1 0
Hera 1 0 Hephaestus 1 1
P(Y = 1|A = 1) = 7/13 = 0.54, P(Y = 1|A = 0) = 3/7 = 0.43.
Causal Inference Reunio GRBIO 4th December, 2018 11 / 25
Chapter 1.5: Causation versus association
A “real world” example
A Y A Y A Y
Rheia 0 0 Zeus 1 1 Aphrodite 1 1
Kronos 0 1 Artemis 0 1 Cyclope 1 1
Demeter 0 0 Apollo 0 1 Persephone 1 1
Hades 0 0 Leto 0 0 Hermes 1 0
Hestia 1 0 Ares 1 1 Hebe 1 0
Poseidon 1 0 Athena 1 1 Dionysus 1 0
Hera 1 0 Hephaestus 1 1
P(Y = 1|A = 1) = 7/13 = 0.54, P(Y = 1|A = 0) = 3/7 = 0.43.
Causal Inference Reunio GRBIO 4th December, 2018 11 / 25
Chapter 1.5: Causation vs. association (cont.)
Association measures
P(Y = 1|A = 1)− P(Y = 1|A = 0) (Associational risk difference)
P(Y = 1|A = 1)
P(Y = 1|A = 0)(Associational risk ratio)
P(Y = 1|A = 1)/P(Y = 0|A = 1)
P(Y = 1|A = 0)/P(Y = 0|A = 0)(Associational odds ratio)
If P(Y = 1|A = 1) = P(Y = 1|A = 0), then A∐
Y (A, Y independent).
Example: ARD = 0.54− 0.43 = 0.11, ARR = 0.54/0.43 = 1.26.
Causal Inference Reunio GRBIO 4th December, 2018 12 / 25
Chapter 1.5: Causation vs. association (cont.)
Association measures
P(Y = 1|A = 1)− P(Y = 1|A = 0) (Associational risk difference)
P(Y = 1|A = 1)
P(Y = 1|A = 0)(Associational risk ratio)
P(Y = 1|A = 1)/P(Y = 0|A = 1)
P(Y = 1|A = 0)/P(Y = 0|A = 0)(Associational odds ratio)
If P(Y = 1|A = 1) = P(Y = 1|A = 0), then A∐
Y (A, Y independent).
Example: ARD = 0.54− 0.43 = 0.11, ARR = 0.54/0.43 = 1.26.
Causal Inference Reunio GRBIO 4th December, 2018 12 / 25
Chapter 1.5: Causation vs. association (cont.)
Association measures
P(Y = 1|A = 1)− P(Y = 1|A = 0) (Associational risk difference)
P(Y = 1|A = 1)
P(Y = 1|A = 0)(Associational risk ratio)
P(Y = 1|A = 1)/P(Y = 0|A = 1)
P(Y = 1|A = 0)/P(Y = 0|A = 0)(Associational odds ratio)
If P(Y = 1|A = 1) = P(Y = 1|A = 0), then A∐
Y (A, Y independent).
Example: ARD = 0.54− 0.43 = 0.11, ARR = 0.54/0.43 = 1.26.
Causal Inference Reunio GRBIO 4th December, 2018 12 / 25
Chapter 1.5: Causation vs. association (cont.)
P(Y = 1|A = 1) is a conditional, P(Y a = 1) an unconditional probability.
A definition of causal effect 11
We say that treatment and outcome are dependent or associated when
Pr[ = 1| = 1] 6= Pr[ = 1| = 0]. In our population, treatment andFor a continuous outcome we
define mean independence between
treatment and outcome as:
E[ | = 1] = E[ | = 0]Independence and mean indepen-
dence are the same concept for di-
chotomous outcomes.
outcome are indeed associated because Pr[ = 1| = 1] = 713 and Pr[ =
1| = 0] = 37. The associational risk difference, risk ratio, and odds ratio
(and other measures) quantify the strength of the association when it exists.
They measure the association on different scales, and we refer to them as
association measures. These measures are also affected by random variability.
However, until Chapter 10, we will disregard statistical issues by assuming that
the population in Table 1.2 is extremely large.
For dichotomous outcomes, the risk equals the average in the population,
and we can therefore rewrite the definition of association in the population as
E [ | = 1] 6= E [ | = 0]. For continuous outcomes , we can also defineassociation as E [ | = 1] 6= E [ | = 0]. Under this definition, association isessentially the same as the statistical concept of correlation between and a
continuous .
In our population of 20 individuals, we found (i) no causal effect after com-
paring the risk of death if all 20 individuals had been treated with the risk of
death if all 20 individuals had been untreated, and (ii) an association after com-
paring the risk of death in the 13 individuals who happened to be treated with
the risk of death in the 7 individuals who happened to be untreated. Figure
1.1 depicts the causation-association difference. The population (represented
by a diamond) is divided into a white area (the treated) and a smaller grey
area (the untreated). The definition of causation implies a contrast between
the whole white diamond (all subjects treated) and the whole grey diamond
(all subjects untreated), whereas association implies a contrast between the
white (the treated) and the grey (the untreated) areas of the original diamond.
Population of interest
Treated Untreated
Causation Association
vs.vs.
EYa1 EYa0 EY|A 1 EY|A 0
Figure 1.1
We can use the notation we have developed thus far to formalize the dis-
tinction between causation and association. The risk Pr[ = 1| = ] is a
conditional probability: the risk of in the subset of the population that
meet the condition ‘having actually received treatment value ’ (i.e., = ).
In contrast the risk Pr[ = 1] is an unconditional–also known as marginal–
probability, the risk of in the entire population. Therefore, association is
defined by a different risk in two disjoint subsets of the population determined
Figure: Association-causation difference (Figure 1.1 in Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 13 / 25
Chapter 1.5: Causation vs. association (cont.)
P(Y = 1|A = 1) is a conditional, P(Y a = 1) an unconditional probability.
A definition of causal effect 11
We say that treatment and outcome are dependent or associated when
Pr[ = 1| = 1] 6= Pr[ = 1| = 0]. In our population, treatment andFor a continuous outcome we
define mean independence between
treatment and outcome as:
E[ | = 1] = E[ | = 0]Independence and mean indepen-
dence are the same concept for di-
chotomous outcomes.
outcome are indeed associated because Pr[ = 1| = 1] = 713 and Pr[ =
1| = 0] = 37. The associational risk difference, risk ratio, and odds ratio
(and other measures) quantify the strength of the association when it exists.
They measure the association on different scales, and we refer to them as
association measures. These measures are also affected by random variability.
However, until Chapter 10, we will disregard statistical issues by assuming that
the population in Table 1.2 is extremely large.
For dichotomous outcomes, the risk equals the average in the population,
and we can therefore rewrite the definition of association in the population as
E [ | = 1] 6= E [ | = 0]. For continuous outcomes , we can also defineassociation as E [ | = 1] 6= E [ | = 0]. Under this definition, association isessentially the same as the statistical concept of correlation between and a
continuous .
In our population of 20 individuals, we found (i) no causal effect after com-
paring the risk of death if all 20 individuals had been treated with the risk of
death if all 20 individuals had been untreated, and (ii) an association after com-
paring the risk of death in the 13 individuals who happened to be treated with
the risk of death in the 7 individuals who happened to be untreated. Figure
1.1 depicts the causation-association difference. The population (represented
by a diamond) is divided into a white area (the treated) and a smaller grey
area (the untreated). The definition of causation implies a contrast between
the whole white diamond (all subjects treated) and the whole grey diamond
(all subjects untreated), whereas association implies a contrast between the
white (the treated) and the grey (the untreated) areas of the original diamond.
Population of interest
Treated Untreated
Causation Association
vs.vs.
EYa1 EYa0 EY|A 1 EY|A 0
Figure 1.1
We can use the notation we have developed thus far to formalize the dis-
tinction between causation and association. The risk Pr[ = 1| = ] is a
conditional probability: the risk of in the subset of the population that
meet the condition ‘having actually received treatment value ’ (i.e., = ).
In contrast the risk Pr[ = 1] is an unconditional–also known as marginal–
probability, the risk of in the entire population. Therefore, association is
defined by a different risk in two disjoint subsets of the population determined
Figure: Association-causation difference (Figure 1.1 in Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 13 / 25
Chapter 1.5: Causation vs. association (cont.)
Concluding question:
“The question is then under which conditions real world data can
be used for causal inference.”
Causal Inference Reunio GRBIO 4th December, 2018 14 / 25
Hernan & Robins: Causal Inference.
Chapter 2: Randomized Experiments2.1 Randomization
2.2 Conditional randomization
2.3 Standardization
2.4 Inverse probability weighting
“This chapter describes why randomization results in convincing
causal inferences.”
Causal Inference Reunio GRBIO 4th December, 2018 15 / 25
Hernan & Robins: Causal Inference.
Chapter 2: Randomized Experiments2.1 Randomization
2.2 Conditional randomization
2.3 Standardization
2.4 Inverse probability weighting
“This chapter describes why randomization results in convincing
causal inferences.”
Causal Inference Reunio GRBIO 4th December, 2018 15 / 25
Chapter 2.1: Randomization
Another look at the “real world” example
A Y Y 0 Y 1 A Y Y 0 Y 1
Rheia 0 0 0 ? Leto 0 0 0 ?
Kronos 0 1 1 ? Ares 1 1 ? 1
Demeter 0 0 0 ? Athena 1 1 ? 1
Hades 0 0 0 ? Hephaestus 1 1 ? 1
Hestia 1 0 ? 0 Aphrodite 1 1 ? 1
Poseidon 1 0 ? 0 Cyclope 1 1 ? 1
Hera 1 0 ? 0 Persephone 1 1 ? 1
Zeus 1 1 ? 1 Hermes 1 0 ? 0
Artemis 0 1 1 ? Hebe 1 0 ? 0
Apollo 0 1 1 ? Dionysus 1 0 ? 0
The computation of CRR is impossible because of missing data.
Causal Inference Reunio GRBIO 4th December, 2018 16 / 25
Chapter 2.1: Randomization
Another look at the “real world” example
A Y Y 0 Y 1 A Y Y 0 Y 1
Rheia 0 0 0 ? Leto 0 0 0 ?
Kronos 0 1 1 ? Ares 1 1 ? 1
Demeter 0 0 0 ? Athena 1 1 ? 1
Hades 0 0 0 ? Hephaestus 1 1 ? 1
Hestia 1 0 ? 0 Aphrodite 1 1 ? 1
Poseidon 1 0 ? 0 Cyclope 1 1 ? 1
Hera 1 0 ? 0 Persephone 1 1 ? 1
Zeus 1 1 ? 1 Hermes 1 0 ? 0
Artemis 0 1 1 ? Hebe 1 0 ? 0
Apollo 0 1 1 ? Dionysus 1 0 ? 0
The computation of CRR is impossible because of missing data.Causal Inference Reunio GRBIO 4th December, 2018 16 / 25
Chapter 2.1: Randomization (cont.)
Exchangeability
Randomization is expected to produce exchangeability.
Exchangeability means that the outcome would be the same in both
study groups if both received or if both did not receive the treatment.
Formally: Exchangeability, Y a∐
A for a ∈ {0, 1}, holds if
P(Y a=0 = 1) = P(Y a=0 = 1|A = 0)︸ ︷︷ ︸Observable
= P(Y a=0 = 1|A = 1)︸ ︷︷ ︸Counterfactual
,
P(Y a=1 = 1) = P(Y a=1 = 1|A = 0)︸ ︷︷ ︸Counterfactual
= P(Y a=1 = 1|A = 1)︸ ︷︷ ︸Observable
.
In ideal randomized experiments, association is causation.
Causal Inference Reunio GRBIO 4th December, 2018 17 / 25
Chapter 2.1: Randomization (cont.)
Exchangeability
Randomization is expected to produce exchangeability.
Exchangeability means that the outcome would be the same in both
study groups if both received or if both did not receive the treatment.
Formally: Exchangeability, Y a∐
A for a ∈ {0, 1}, holds if
P(Y a=0 = 1) = P(Y a=0 = 1|A = 0)︸ ︷︷ ︸Observable
= P(Y a=0 = 1|A = 1)︸ ︷︷ ︸Counterfactual
,
P(Y a=1 = 1) = P(Y a=1 = 1|A = 0)︸ ︷︷ ︸Counterfactual
= P(Y a=1 = 1|A = 1)︸ ︷︷ ︸Observable
.
In ideal randomized experiments, association is causation.
Causal Inference Reunio GRBIO 4th December, 2018 17 / 25
Chapter 2.1: Randomization (cont.)
Exchangeability
Randomization is expected to produce exchangeability.
Exchangeability means that the outcome would be the same in both
study groups if both received or if both did not receive the treatment.
Formally: Exchangeability, Y a∐
A for a ∈ {0, 1}, holds if
P(Y a=0 = 1) = P(Y a=0 = 1|A = 0)︸ ︷︷ ︸Observable
= P(Y a=0 = 1|A = 1)︸ ︷︷ ︸Counterfactual
,
P(Y a=1 = 1) = P(Y a=1 = 1|A = 0)︸ ︷︷ ︸Counterfactual
= P(Y a=1 = 1|A = 1)︸ ︷︷ ︸Observable
.
In ideal randomized experiments, association is causation.
Causal Inference Reunio GRBIO 4th December, 2018 17 / 25
Chapter 2.1: Randomization (cont.)
Exchangeability
Randomization is expected to produce exchangeability.
Exchangeability means that the outcome would be the same in both
study groups if both received or if both did not receive the treatment.
Formally: Exchangeability, Y a∐
A for a ∈ {0, 1}, holds if
P(Y a=0 = 1) = P(Y a=0 = 1|A = 0)︸ ︷︷ ︸Observable
= P(Y a=0 = 1|A = 1)︸ ︷︷ ︸Counterfactual
,
P(Y a=1 = 1) = P(Y a=1 = 1|A = 0)︸ ︷︷ ︸Counterfactual
= P(Y a=1 = 1|A = 1)︸ ︷︷ ︸Observable
.
In ideal randomized experiments, association is causation.
Causal Inference Reunio GRBIO 4th December, 2018 17 / 25
Chapter 2.2: Conditional randomization
The “real world” example with a 3rd variable
L A Y L A Y
Rheia 0 0 0 Leto 1 0 0
Kronos 0 0 1 Ares 1 1 1
Demeter 0 0 0 Athena 1 1 1
Hades 0 0 0 Hephaestus 1 1 1
Hestia 0 1 0 Aphrodite 1 1 1
Poseidon 0 1 0 Cyclope 1 1 1
Hera 0 1 0 Persephone 1 1 1
Zeus 0 1 1 Hermes 1 1 0
Artemis 1 0 1 Hebe 1 1 0
Apollo 1 0 1 Dionysus 1 1 0
L is supposed to be a prognosis factor (1: critical situation; 0: otherwise).Causal Inference Reunio GRBIO 4th December, 2018 18 / 25
Chap. 2.2: Conditional randomization
Conditional randomization
Both treatment groups are not exchangeable because
P(L = 1|A = 0) = 3/7, P(L = 1|A = 1) = 9/13.
In the “real world” example, treatment assignment was conditionally
randomized with probabilities:
P(A = 1|L = 0) = 0.5, P(A = 1|L = 1) = 0.75.
So, conditional exchangeability, Y a∐
A|L, holds within the levels
of the prognosis factor L.
How can the CRR be computed in a conditionally randomized experiment?
Standardization or Inverse Probability Weighting (IPW)!
Causal Inference Reunio GRBIO 4th December, 2018 19 / 25
Chap. 2.2: Conditional randomization
Conditional randomization
Both treatment groups are not exchangeable because
P(L = 1|A = 0) = 3/7, P(L = 1|A = 1) = 9/13.
In the “real world” example, treatment assignment was conditionally
randomized with probabilities:
P(A = 1|L = 0) = 0.5, P(A = 1|L = 1) = 0.75.
So, conditional exchangeability, Y a∐
A|L, holds within the levels
of the prognosis factor L.
How can the CRR be computed in a conditionally randomized experiment?
Standardization or Inverse Probability Weighting (IPW)!
Causal Inference Reunio GRBIO 4th December, 2018 19 / 25
Chap. 2.2: Conditional randomization
Conditional randomization
Both treatment groups are not exchangeable because
P(L = 1|A = 0) = 3/7, P(L = 1|A = 1) = 9/13.
In the “real world” example, treatment assignment was conditionally
randomized with probabilities:
P(A = 1|L = 0) = 0.5, P(A = 1|L = 1) = 0.75.
So, conditional exchangeability, Y a∐
A|L, holds within the levels
of the prognosis factor L.
How can the CRR be computed in a conditionally randomized experiment?
Standardization or Inverse Probability Weighting (IPW)!
Causal Inference Reunio GRBIO 4th December, 2018 19 / 25
Chap. 2.2: Conditional randomization
Conditional randomization
Both treatment groups are not exchangeable because
P(L = 1|A = 0) = 3/7, P(L = 1|A = 1) = 9/13.
In the “real world” example, treatment assignment was conditionally
randomized with probabilities:
P(A = 1|L = 0) = 0.5, P(A = 1|L = 1) = 0.75.
So, conditional exchangeability, Y a∐
A|L, holds within the levels
of the prognosis factor L.
How can the CRR be computed in a conditionally randomized experiment?
Standardization or Inverse Probability Weighting (IPW)!
Causal Inference Reunio GRBIO 4th December, 2018 19 / 25
Computation of the causal relative risk
Standardization:
CRR =P(Y a=1 = 1)
P(Y a=0 = 1)=
∑l P(Y = 1|A = 1, L = l)P(L = l)∑l P(Y = 1|A = 0, L = l)P(L = l)
.
Inverse probability weighting:
The idea is to create a pseudo population with values Y a=1 and Y a=0
for all subjects and the subsequent computation of the CRR.
For this purpose, the observations are given weights that depend on
the values of A and L. The weights are 1/f(A|L) = 1/P(A|L)(discrete case).
Technical Point 2.3 shows that both methods are identical.
Causal Inference Reunio GRBIO 4th December, 2018 20 / 25
Computation of the causal relative risk
Standardization:
CRR =P(Y a=1 = 1)
P(Y a=0 = 1)=
∑l P(Y = 1|A = 1, L = l)P(L = l)∑l P(Y = 1|A = 0, L = l)P(L = l)
.
Inverse probability weighting:
The idea is to create a pseudo population with values Y a=1 and Y a=0
for all subjects and the subsequent computation of the CRR.
For this purpose, the observations are given weights that depend on
the values of A and L. The weights are 1/f(A|L) = 1/P(A|L)(discrete case).
Technical Point 2.3 shows that both methods are identical.
Causal Inference Reunio GRBIO 4th December, 2018 20 / 25
Computation of the causal relative risk
Standardization:
CRR =P(Y a=1 = 1)
P(Y a=0 = 1)=
∑l P(Y = 1|A = 1, L = l)P(L = l)∑l P(Y = 1|A = 0, L = l)P(L = l)
.
Inverse probability weighting:
The idea is to create a pseudo population with values Y a=1 and Y a=0
for all subjects and the subsequent computation of the CRR.
For this purpose, the observations are given weights that depend on
the values of A and L. The weights are 1/f(A|L) = 1/P(A|L)(discrete case).
Technical Point 2.3 shows that both methods are identical.
Causal Inference Reunio GRBIO 4th December, 2018 20 / 25
IP weighting: an example
Figure: Graphical representation of the “real world example” (Figure 2.1 in
Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 21 / 25
IP weighting: an example
Figure: Graphical representation of the “real world example” (Figure 2.1 in
Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 21 / 25
IP weighting: an example (cont.)
Figure: Graphical representation of the inverse probability weighting in the “real
world example”. Left panel: all subjects are untreated; right panel: all subjects
are treated (Figure 2.2 in Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 22 / 25
IP weighting: an example (cont.)
Figure: Graphical representation of the inverse probability weighting in the “real
world example”. Left panel: all subjects are untreated; right panel: all subjects
are treated (Figure 2.2 in Hernan & Robins).
Causal Inference Reunio GRBIO 4th December, 2018 22 / 25
Hernan & Robins: Causal Inference.
Chapter 3: Observational Studies3.1 The randomized experiment paradigm
3.2 Exchangeability
3.3 Positivity
3.4 Well-defined interventions
3.5 Well-defined interventions are a pre-requisite for causal inference
3.6 Causation or prediction
“This chapter reviews some conditions under which observational
studies lead to valid causal inferences.”
Causal Inference Reunio GRBIO 4th December, 2018 23 / 25
Hernan & Robins: Causal Inference.
Chapter 3: Observational Studies3.1 The randomized experiment paradigm
3.2 Exchangeability
3.3 Positivity
3.4 Well-defined interventions
3.5 Well-defined interventions are a pre-requisite for causal inference
3.6 Causation or prediction
“This chapter reviews some conditions under which observational
studies lead to valid causal inferences.”
Causal Inference Reunio GRBIO 4th December, 2018 23 / 25
Conditions for causal inference
Identifiability conditions for causal inference
Three conditions must hold so that an observational study can be
conceptualized as a conditionally randomized experiment:
1 The interventions must be very well defined.
2 The conditional probability for receiving a treatment depends only on
the observed covariates.
3 The conditional probability for receiving a treatment must be positive
(Positivity).
If these three (identifiability) conditions hold,
“. . . causal effects can be identified from observational studies by
using IP weighting or standardization.”
Causal Inference Reunio GRBIO 4th December, 2018 24 / 25
Conditions for causal inference
Identifiability conditions for causal inference
Three conditions must hold so that an observational study can be
conceptualized as a conditionally randomized experiment:
1 The interventions must be very well defined.
2 The conditional probability for receiving a treatment depends only on
the observed covariates.
3 The conditional probability for receiving a treatment must be positive
(Positivity).
If these three (identifiability) conditions hold,
“. . . causal effects can be identified from observational studies by
using IP weighting or standardization.”
Causal Inference Reunio GRBIO 4th December, 2018 24 / 25
Conditions for causal inference
Identifiability conditions for causal inference
Three conditions must hold so that an observational study can be
conceptualized as a conditionally randomized experiment:
1 The interventions must be very well defined.
2 The conditional probability for receiving a treatment depends only on
the observed covariates.
3 The conditional probability for receiving a treatment must be positive
(Positivity).
If these three (identifiability) conditions hold,
“. . . causal effects can be identified from observational studies by
using IP weighting or standardization.”
Causal Inference Reunio GRBIO 4th December, 2018 24 / 25
Conditions for causal inference
Identifiability conditions for causal inference
Three conditions must hold so that an observational study can be
conceptualized as a conditionally randomized experiment:
1 The interventions must be very well defined.
2 The conditional probability for receiving a treatment depends only on
the observed covariates.
3 The conditional probability for receiving a treatment must be positive
(Positivity).
If these three (identifiability) conditions hold,
“. . . causal effects can be identified from observational studies by
using IP weighting or standardization.”
Causal Inference Reunio GRBIO 4th December, 2018 24 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25
Conditions for causal inference (cont.)
Comments
In observational studies, the value of A likely depends on some
outcome predictors (L1, L2, . . . ).
Crucial question: Are all the predictors Li with unequal distributions
among treatment groups observed?
We cannot know the answer to the previous question. There is no
guarantee that Y a∐
A|L holds.
Positivity is imperative for inverse probability weighting.
If multiple versions of a treatment are present, the interventions are
not well defined.
Problems due to unspecified interventions cannot be resolved by
applying sophisticated statistical methods.
Causal Inference Reunio GRBIO 4th December, 2018 25 / 25