In this vignette, we present an example of using nearest-neighbor Gaussian processes approach in GPCERF to estimate the exposure response function (ERF) of a continuous exposure based on simulated data. We use a synthetic data generation function from the R package CausalGPS.

We first generate a synthetic dataset with six covariates, one
continuous exposure and one outcome. We consider two types of
conditional distributions (normal and student’s t) of the exposure given
covariates. For more details of the synthetic data generation, see this
document.
We then use a function `tru_R`

to derive the actual ERF of
this population at `w=seq(0,20,0.1)`

.

```
set.seed(1)
# Generate dataset with a normally distributed exposure given covariates
data_sim_normal <- generate_synthetic_data(sample_size = 500,
outcome_sd = 10,
gps_spec = 1)
# Generate dataset with a t-distributed with 2df exposure given covariates
data_sim_t <- generate_synthetic_data(sample_size = 500,
outcome_sd = 10,
gps_spec = 2)
tru_R <- function(w, sim_data) {
design_mt <- model.matrix(~cf1 + cf2 + cf3 + cf4 + cf5 + cf6 - 1,
data = sim_data)
mean(apply(design_mt, 1, function(x) {
-10 - sum(c(2, 2, 3, -1, 2, 2) * x) -
w * (0.1 - 0.1 * x[1] + 0.1 * x[4] + 0.1 * x[5] + 0.1 * x[3] ^ 2) +
0.13 ^ 2 * w ^ 3
}))
}
plot_fun <- function(object, ...) {
tmp_data <- data.frame(w_vals = object$posterior$w,
mean_vals = object$posterior$mean,
sd_vals = object$posterior$sd)
g1 <- ggplot2::ggplot(tmp_data) +
ggplot2::geom_ribbon(ggplot2::aes(.data$w_vals,
y = .data$mean_vals,
ymin = .data$mean_vals - 1.96 * .data$sd_vals,
ymax = .data$mean_vals + 1.96 * .data$sd_vals),
fill = "#FC4E07", alpha = 0.25) +
ggplot2::geom_line(ggplot2::aes(.data$w_vals, .data$mean_vals),
color = "#FC4E07",
size = 1) +
ggplot2::theme_bw() +
ggplot2::ggtitle("Estimated CERF (nngp) with credible band (1.96sd)") +
ggplot2::xlab("Exposure level") +
ggplot2::ylab("Population average counterfactual outcome")
return(g1)
}
erf_tru_normal <- sapply(seq(0, 20, 0.1), function(w) tru_R(w, data_sim_normal))
erf_tru_t <- sapply(seq(0, 20, 0.1), function(w) tru_R(w, data_sim_t))
```

GPCERF will first convert the covariate values into a single composite score (GPS) and then use it to fit the Gaussian processes. We use the GPS estimation function in CausalGPS (see here) to get the GPS model that maps covariates into GPS.

```
gps_m_normal <- estimate_gps(cov_mt = data_sim_normal[, -(1:2)],
w_all = data_sim_normal$treat,
sl_lib = c("SL.xgboost"),
dnorm_log = FALSE)
gps_m_t <- estimate_gps(cov_mt = data_sim_t[, -(1:2)],
w_all = data_sim_t$treat,
sl_lib = c("SL.xgboost"),
dnorm_log = FALSE)
```

We then use `estimate_cerf_nngp`

to estimate the ERF of
the exposure `w`

. We estimate the ERF at
`w = seq(0,20,0.1)`

. The estimated ERF as well as its
pointwise 95% credible band is visualized with a call to
`plot`

. We also plot the actual ERF on top of the estimated
ERF.

```
w_all <- seq(0, 20, 0.1)
nngp_res_normal <- estimate_cerf_nngp(data_sim_normal,
w_all,
gps_m_normal,
params = list(alpha = c(0.1),
beta = 0.2,
g_sigma = 1,
n_neighbor = 20,
block_size = 50,
tune_app = "all"),
outcome_col = "Y",
treatment_col = "treat",
covariates_col = paste0("cf", seq(1,6)),
nthread = 1)
plot_fun(nngp_res_normal) +
geom_line(data = data.frame(w = w_all, y = erf_tru_normal),
aes(x = w, y = y, color = "True"), size = 1.5)
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
```

We can see that the estimated curve follows the true ERF closely and the 95% credible band completely covers the true ERF. But the running time is much smaller than the standard GP approach.

```
nngp_res_t <- estimate_cerf_nngp(data_sim_t,
w_all,
gps_m_t,
params = list(alpha = c(0.1),
beta = 0.2,
g_sigma = 1,
n_neighbor = 20,
block_size = 50,
tune_app = "all"),
outcome_col = "Y",
treatment_col = "treat",
covariates_col = paste0("cf", seq(1,6)),
nthread = 1)
plot_fun(nngp_res_t) +
geom_line(data = data.frame(w = w_all, y = erf_tru_t),
aes(x = w, y = y, color = "True"), size = 1.5)
```

The results look very similar to the case where the exposure is normally distributed. The only difference might be that when the exposure is t-distributed, the estimated curve tends to be less smooth.