This document presents a demonstration of calibration methods for treatment effect predictors. We generate synthetic data with covariates, treatment assignments, and outcomes. We explore two calibration methods: the Best Linear Predictor (BLP) and causal isotonic calibration.

The BLP method provides a linear calibration gaurantees by learning an optimal linear transformation of the original predictor, so that the linearly calibrated predictor cannot be improved by applying any linear transformation (i.e., scaling and shifting).

Isotonic calibration offers non-parametric (distribution-free) calibration gaurantees by (1) an optimal monotone transformation of the original predictor and (2) providing a calibrated predictor that cannot be improved by applying any transformation (linear or nonlinear).

We begin by generating synthetic data for the demonstration. We set the seed for reproducibility and create variables for covariates, treatment assignments, potential outcomes, observed outcomes, and the conditional average treatment effect (CATE).

```
# Set random seed for reproducibility
set.seed(12345)
n <- 2000
# Generate covariate W from a uniform distribution between -1 and 1
W <- runif(n, -1, 1)
# Calculate treatment assignment probabilities using logistic function
pi <- plogis(0.5 * W)
A <- rbinom(n, size = 1, pi)
# Define outcome regression functions and CATE
mu0 <- plogis(W)
mu1 <- plogis(1 + 2 * W)
cate <- mu1 - mu0
# Generate potential outcomes based on treatment assignment
Y0 <- rbinom(n, size = 1, mu0)
Y1 <- rbinom(n, size = 1, plogis(1 + 2 * W))
# Create observed outcomes based on treatment assignment
Y <- ifelse(A == 1, Y1, Y0)
```

We first create an initial predictor of the Individual Treatment Effect (ITE), denoted as tau.hat, which is a fixed function for simplicity.

```
# use machine learning to obtain initial predictor of ITE Y_1 - Y_0
# for simplicity, we define our predictor tau.hat as a fixed function.
tau.hat <- plogis(1 + W ) - 0.45
```

We visualize the initial predictor, the Best Linear Predictor (BLP), and the true CATE as functions of the covariate.

```
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## â„¹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
```

Now, lets make a calibrate plot. This is a scatter plot of the true CATE values vs the predicted CATE values.

We apply the Best Linear Predictor (BLP) method to linearly calibrate the original predictor.

```
# unbiased surrogate outcome for CATE/ITE
pseudo_outcome <- cate + (A/pi) * (Y - mu1) - ((1-A)/(1-pi)) * (Y - mu0)
# fit best linear predictor of tau.hat of the surrogate outcome
# provides estimat of BLP of ITE/CATE
fit <- lm(pseudo_outcome ~ tau.hat, data = data.frame(tau.hat, pseudo_outcome))
intercept <- coef(fit)[1]
slope <- coef(fit)[2]
# get linear calibrated predictor, i.e. BLP given tau.hat
tau.BLP.hat <- intercept + slope * tau.hat
cor_tau <- cor(tau.BLP.hat, cate)
# Calculate the regression coefficients
fit <- lm(cate ~ tau.hat, data = data.frame(tau.hat, cate))
intercept <- coef(fit)[1]
slope <- coef(fit)[2]
tau.BLP.oracle <- intercept + tau.hat * slope
```

We visualize the linearly calibrated predictor as a function of the covariate.