In this example "S/s" refers to "getting the surgery" and "N/n" refers to choosing not to get the surgery. The outcome Y is how healthy individuals are after getting or not getting the surgery. Annotate each numbered step.
NCE = E(Ys|S) - E(Yn|N)
NCE = E(Ys|S) - E(Yn|N) + E(Yn|S) - E(Yn|S)
Add and subtract E(Yn|S) - E(Yn|S)
NCE = E(Ys|S) - E(Yn|S) - E(Yn|N) + E(Yn|S)
NCE = [E(Ys|S) - E(Yn|S)] - [E(Yn|S) - E(Yn|N)]
ATT
Selection Bias
ATT = NCE - SB
Naive Causal Effect = NCE
ATT = [E(Y1|D=1)-E(Y0|D=0)] - [E(Y0|D=1)-E(Y0|D=0)]
In our previous notation
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If selection bias is given by these equations:
SB = E(Yn|S) - E(Yn|N)
SB = E(Y0|D=1) - E(Y0|D=0)
Explain in ordinary language what's being subtracted from what and why the difference means something important here.
what would have happened to T group had they not been treated minus what happened to C group
If selection bias is given by these equations:
SB = E(Yn|S) - E(Yn|N)
ATT = NCE - SB
SB = E(Y0|D=1) - E(Y0|D=0)
and ATT is given by:
Describe in ordinary language what happens when HEALTHIER patients choose to have the surgery.
Describe in ordinary language what happens when SICKER patients choose to have the surgery.
If healthier patients choose surgery then we can expect that their outcomes would be better even without surgery than the control group. Thus SB>0 and the NCE effectively overstates the treatment effect
If sicker patients choose surgery then we can expect that their outcomes would be worse without surgery than control group. Thus SB<0 and the NCE effectively understates the treatment effect