Update notes

Author: cxy0714

notes/main.typ

@@ -247,15 +247,15 @@ Here $serif(Pr)(Y(1) = b | G = g)$ and $serif(Pr)( Y(0) = a | G = g)$ can be ide
     - Consistency and no interference between units: 
     $ Y = Y (D) & = D Y(1) + (1-D) Y(0) \
      D & = D(Z) $
-  + [IV relevance (version 1): $ Z cancel(perp) D | X$ almost surely. ] 
+  + IV relevance (version 1): $ Z cancel(perp) D | X$ almost surely. 
   + IV independence : $ Z perp U | X$
   + IV exclusion restriction : $Z perp Y | D, X$
   + Unconfounderness/d-separation : $ (Z, D) perp Y(d) | X, U$ for $d = 0,1$
 
 As @levis2025covariate mentioned, under these assumptions, the ATE is not point identified, homogeneity assumptions are :
   + Version 1, for binary $Z$ : Either $ EE[D | Z = 1, X , U] - EE[D | Z = 0, X , U] $ or $ EE[ Y(1) - Y(0) | X , U] $ does not depend on $U$.
-    -  #quote("Assumption 5′ rules out additive effect modification by U of the Z-D relationship or d-Y (d)  relationship within levels of X. A weaker alternative is the no unmeasured common  effect modifier assumption (Cui and Tchetgen Tchetgen, 2021, Hartwig et al., 2023), which  stipulates that no unmeasured confounder acts as a common effect modifier of both the  additive effect of the IV on the treatment and the additive treatment effect on the outcome:")
-  + Version 2, for binary $Z$, following equation holds almost surely:
+    -  #quote([Assumption 5′ rules out additive effect modification by $U$ of the $Z-D$ relationship or $d-Y (d)$  relationship within levels of $X$. A weaker alternative is the no unmeasured common  effect modifier assumption (Cui and Tchetgen Tchetgen, 2021, Hartwig et al., 2023), which  stipulates that no unmeasured confounder acts as a common effect modifier of both the  additive effect of the IV on the treatment and the additive treatment effect on the outcome:])
+  + Version 2, weaker alternative for binary $Z$, following equation holds almost surely:
    $ "Cov"(EE(D| Z= 1, X, U)- EE(D|Z=0, X, U), EE(Y(1) - Y(0) | X,U) | X ) = 0  $
   + Final version, for continuous or multiple-category $Z$, for any $z$ in the support of $Z$, following equation holds almost surely:
    $ "Cov"(EE(D| Z= z, X, U)- EE(D| X, U), EE(Y(1) - Y(0) | X,U) | X ) = 0 $
@@ -265,7 +265,7 @@ As @levis2025covariate mentioned, under these assumptions, the ATE is not point
 - The real-data applicationis combine many genetic variants as weak IVs to a strong and continuous IV to solve the "obesity paradox" in oncology.
   - #quote("Obesity is typically associated with poorer oncology outcomes. Paradoxically, however,  many observational studies have reported that non-small cell lung cancer (NSCLC) patients  with higher body mass index (BMI) experience lower mortality, a phenomenon often referred  to as the “obesity paradox” (Zhang et al., 2017).")
 
-- Using the ratio of conditional weighted average treatment effect, for multiple-category(CWATE) or conditional weighted average derivative effect (CWADE) to identify the ATE.
+- Using the ratio of conditional weighted average treatment effect, for multiple-category (CWATE) or conditional weighted average derivative effect (CWADE) to identify the ATE.
 
   - Using semiparametric theory to provide the efficient influence function and build a triply robust estimator.