Treatment: A (1: treated, 0: untreated)
Outcome: Y (1: death, 0: survival)
Potential outcomes or counterfactual outcomes:
Y ^{a=0} : the outcome variable that would have been observed under the treatment value a=0.
Y ^{a=1} : the outcome variable that would have been observed under the treatment value a=1.
Definition of individual causal effects: the treatment A has a causal effect on an individual's outcome Y if Y ^{a=1} ≠ Y ^{a=0} for the individual.
Y ^{a=1} =1
Y^{a=1}=0
In general, individual causal effects cannot be identified since only one of potential outcomes is observed for each individual.
Definition of average causal effects: an average causal effect of treatment A on outcome Y is present if E[ Y ^{a=1} ] ≠ E[ Y ^{a=0} ]
Pr[ Y ^{a=1} =1]=10/20=0.5
Absence of an average causal effect does not imply absence of individual effects.
Pr[ Y ^{a=0} =1]=10/20=0.5
Average causal effects can sometimes be identified from data even if individual causal effects cannot.
Pr[ Y ^{a} =1]
Pr[ Y=1|A=a]
Marginal probability
Conditional probability
Randomization is so highly valued because it is expected to produce exchangeability.
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0
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Can we conclude that our study is not a randomized experiment?
Marginal randomization
Conditional randomization
Exchangeability holds
Exchangeability does not hold, but conditional exchangeability holds
How to compute the causal effect?
- Stratum-specific causal effects
- Average causal effects
Standardization
Inverse Probability Weighting
Since
Similarly,
What randomized experiment are you trying to emulate?
r=0
r
Assume these 9 baseline variables are sufficient to adjust for confounding
= {
quitters
non-quitters
Fit the linear model by weighted least squares, with individuals weighted by their estimated IP weights
To obtain a 95% confidence interval:
f(A) equals to Pr[A=1] for treated, and Pr[A=0] for untreated
Under exchangeability and positivity conditional on the variables in L,
We first need to compute the mean outcomes in the uncensored treated in each stratum l of the confounders L
Use bootstrap to obtain a 95% confidence interval
The validity of causal inferences based on observational data requires the following conditions: