Section 13 Box 11: Confidence interval estimation

# transform predicted outcomes back to original scale
ObsData$Pred.updateQ1.Y1.rescaled <- 
  (max.Y- min.Y)*ObsData$Pred.updateQ1.Y1 + min.Y
ObsData$Pred.updateQ1.Y0.rescaled <- 
  (max.Y- min.Y)*ObsData$Pred.updateQ1.Y0 + min.Y

EY1_TMLE1 <- mean(ObsData$Pred.updateQ1.Y1.rescaled, 
                  na.rm = TRUE)
EY0_TMLE1 <- mean(ObsData$Pred.updateQ1.Y0.rescaled, 
                  na.rm = TRUE)

# ATE efficient influence curve
D1 <- ObsData$A/ObsData$PS.SL*
  (ObsData$Y - ObsData$Pred.updateQ1.Y1.rescaled) + 
  ObsData$Pred.updateQ1.Y1.rescaled - EY1_TMLE1
D0 <- (1 - ObsData$A)/(1 - ObsData$PS.SL)*
  (ObsData$Y - ObsData$Pred.updateQ1.Y0.rescaled) + 
  ObsData$Pred.updateQ1.Y0.rescaled - EY0_TMLE1
EIC <- D1 - D0


# ATE variance
n <- nrow(ObsData)
varHat.IC <- var(EIC, na.rm = TRUE)/n
# ATE 95% CI
ATE.TMLE.CI <- c(ATE.TMLE - 1.96*sqrt(varHat.IC), 
                 ATE.TMLE + 1.96*sqrt(varHat.IC))
ATE.TMLE.CI

## [1] 0.9578281 4.5060665