Hui Hu, PhD
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
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]
Randomization is so highly valued because it is expected to produce exchangeability.
Can we conclude that our study is not a randomized experiment?
Exchangeability does not hold, but conditional exchangeability holds
How to compute the causal effect?
- Stratum-specific causal effects
- Average causal effects
Inverse Probability Weighting
What randomized experiment are you trying to emulate?
Assume these 9 baseline variables are sufficient to adjust for confounding
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: