The illustrative Dataset
We begin with exploring a toy dataset with 500 observations that we included with the package for illustration. ToyData has an outcome, four covariates, and a treatment indicator. They were generated with the following model: \[
Y_i(0) = 1 + x_{1i} + 2 x_{2i} + 4x_{3i} + \epsilon_i
\] with \(\epsilon_i \sim N( 0, 1^2 )\). The treatment model is all systematic: \[
\tau_i = 2 + 2x_{1i} + x_{2i}
\] So \(x_1\) and \(x_2\) are both predictive of treatment impact. \(x_3\) also predicts control side variation, and should therefore be useful for increasing precision. \(x_4\) is useless. \(Y_i(1) = Y_i(0) + \tau_i\), so there is no ideosyncratic variation if we control for both \(x_1\) and \(x_2\).
As we generated these data, we know the true individual treatment effects.
data( ToyData )
head( ToyData )## Y Z x1 x2 x3 x4 tau
## 1 2.2430188 0 0.7635779 0.9415463 -0.50291814 -1.4374786 4.468702
## 2 0.3997241 0 -0.5884278 0.7976774 -0.32835669 -0.7140096 1.620822
## 3 3.5865779 0 0.4881701 0.1335321 0.54933171 -0.7098844 3.109872
## 4 0.9929092 0 1.3796253 0.1786193 -0.23264879 -0.3467998 4.937870
## 5 3.0077115 1 0.1722719 0.2836023 -0.03172434 -2.2100810 2.628146
## 6 2.5547212 0 -0.2978002 0.2678954 0.46776121 -1.6495943 1.672295td = gather( ToyData, x1, x2, x3, x4, key="X", value="value" )
td = gather( td, Y, tau, key="outcome", value="value2" )
ggplot( td, aes( x=value, y=value2, col=as.factor(Z) ) ) +
facet_grid( outcome ~ X, scales="free" ) +
geom_point( alpha=0.5, size=0.5) +
geom_smooth( method="loess", se=FALSE ) +
labs( x="Covariates", y="" )
As the data is simulated, we have the true finite-sample ATE:
mean( ToyData$tau )## [1] 1.858165Before testing, we quickly look at the marginal and residual CDFs of treatment and control. We see heterogeniety at left, but after controlling for observed covariates we have no visibile ideosyncratic heterogeniety left over.
par( mfrow=c(1,2) )
ll0 = lm( Y ~ Z, data=ToyData )
plot( ecdf( resid(ll0)[ToyData$Z==1] ), pch=".", main="Marginal CDFs of \n treatment and control")
plot( ecdf( resid(ll0)[ToyData$Z==0] ), pch=".", col="red", add=TRUE )
ll1 = lm( Y ~ Z + x1 + x2 + x3 + x4, data=ToyData )
plot( ecdf( resid(ll1)[ToyData$Z==1] ), pch=".", main="Residual CDFs of \n treatment and control" )
plot( ecdf( resid(ll1)[ToyData$Z==0] ), pch=".", col="red", add=TRUE )
A simple linear model should give us our parameters from above, since it is a correct specification in this case.
M0 <- lm( Y ~ Z * (x1+x2+x3), data=ToyData )
round( coef( M0 ), digits=1 )## (Intercept) Z x1 x2 x3 Z:x1
## 1.0 2.1 1.1 2.0 4.0 2.0
## Z:x2 Z:x3
## 0.9 0.1