Estimating and Evaluating the Optimal Subgroup
In polle: Policy Learning

library(polle)
library(data.table)

This vignette showcase how policy_learn() and policy_eval() can be combined to estimate and evaluate the optimal subgroup in the single-stage case. We refer to [@nordland2023policy] for the syntax and methodological context.

Setup

From here on we consider the single-stage case with a binary action set ${0,1}$. For a given threshold $\eta > 0$ we can formulate the optimal subgroup function via the conditional average treatment effect (CATE/blip) as

\begin{align} d^\eta_0(v)_ = I{B_0(v) > \eta}, \end{align} where $B_0$ is the CATE defined as \begin{align} E\left[U^{(1)} - U^{(0)} \big | V = v \right]. \end{align}

The average treatment effect in the optimal subgroup is now defined as \begin{align} E\left[U^{(1)} - U^{(0)} \big | d^\eta_0(V) = 1 \right], \end{align}

which under consistency, positivity and randomization is identified as

\begin{align} E\left[Z(1,g_0,Q_0)(O) - Z(0,g_0,Q_0)(O) \big | d^\eta_0(V) = 1 \right], \end{align}

where $Z(a,g,Q)(O)$ is the doubly robust score for treatment $a$ and

\begin{align} d^\eta_0(v) &= I{B_0(v) > \eta}\ B_0(v) &= E\left[Z(1,g_0,Q_0)(O) - Z(0,g_0,Q_0)(O) \big | V = v \right] \end{align}

Threshold policy learning

In polle the threshold policy $d_\eta$ can be estimated using policy_learn() via the threshold argument, and the average treatment effect in the subgroup can be estimated using policy_eval() setting target = subgroup.

Here we consider an example using simulated data:

par0 <- list(a = 1, b = 0, c = 3)
sim_d <- function(n, par=par0, potential_outcomes = FALSE) {
  W <- runif(n = n, min = -1, max = 1)
  L <- runif(n = n, min = -1, max = 1)
  A <- rbinom(n = n, size = 1, prob = 0.5)
  U1 <- W + L + (par$c*W + par$a*L + par$b) # U^1
  U0 <- W + L # U^0
  U <- A * U1 + (1 - A) * U0 + rnorm(n = n)
  out <- data.table(W = W, L = L, A = A, U = U)
  if (potential_outcomes == TRUE) {
    out$U0 <- U0
    out$U1 <- U1
  }
  return(out)
}

Note that in this simple case $U^{(1)} - U^{(0)} = cW + aL + b$.

set.seed(1)
d <- sim_d(n = 200)
pd <- policy_data(
    d,
    action = "A",
    covariates = list("W", "L"),
    utility = "U"
)

We set a correctly specified policy learner using policy_learn() with type = "blip" and set a threshold of $\eta = 1$:

pl1 <- policy_learn(
  type = "blip",
  control = control_blip(blip_models = q_glm(~ W + L)),
  threshold = 1
)

When then apply the policy learner based on the correctly specified nuisance models. Furthermore, we extract the corresponding policy actions, where $d_N(Z,L) = 1$ identifies the optimal subgroup for $\eta = 1$:

po1 <- pl1(
  policy_data = pd,
  g_models = g_glm(~ 1),
  q_models = q_glm(~ A * (W + L))
)
pf1 <- get_policy(po1)
pa <- pf1(pd)

In the following plot, the black line indicates the boundary for the true optimal subgroup. The dots represent the estimated threshold policy:

library("ggplot2")
plot_data <- data.table(d_N = factor(pa$d), W = d$W, L = d$L)
ggplot(plot_data) +
  geom_point(aes(x = W, y = L, color = d_N)) +
  geom_abline(slope = -3, intercept = 1) +
  theme_bw()

Similarly, we can also use type = "ptl" to fit a policy tree with a given threshold for not choosing the reference action (first action in action set in alphabetical order)

get_action_set(pd)[1] # reference action
pl1_ptl <- policy_learn(
    type = "ptl",
    control = control_ptl(policy_var = c("W", "L")),
    threshold = 1
)
po1_ptl <- pl1_ptl(
  policy_data = pd,
  g_models = g_glm(~ 1),
  q_models = q_glm(~ A * (W + L))
)
po1_ptl$ptl_objects

pf1_ptl <- get_policy(po1_ptl)
pa_ptl <- pf1_ptl(pd)
library("ggplot2")
plot_data <- data.table(d_N = factor(pa_ptl$d), W = d$W, L = d$L)
ggplot(plot_data) +
  geom_point(aes(x = W, y = L, color = d_N)) +
  geom_abline(slope = -3, intercept = 1) +
  theme_bw()

Subgroup average treatment effect

The true subgroup average treatment effect is given by:

\begin{align} E[cW + aL + b | cW + aL + b \geq \eta ], \end{align}

which we can easily approximate:

set.seed(1)
approx <- sim_d(n = 1e7, potential_outcomes = TRUE)
(sate <- with(approx, mean((U1 - U0)[(U1 - U0 >= 1)])))
rm(approx)

The subgroup average treatment effect associated with the learned optimal threshold policy can be directly estimated using policy_eval() via the target argument:

(pe <- policy_eval(
  policy_data = pd,
  policy_learn = pl1,
  target = "subgroup"
 ))

We can also estimate the subgroup average treatment effect for a set of thresholds at once:

pl_set <- policy_learn(
  type = "blip",
  control = control_blip(blip_models = q_glm(~ W + L)),
  threshold = c(0, 1)
)

policy_eval(
  policy_data = pd,
  g_models = g_glm(~ 1),
  q_models = q_glm(~ A * (W + L)),
  policy_learn = pl_set,
  target = "subgroup"
)

Asymptotics

The data adaptive target parameter

\begin{align} E[U^{(1)} - U^{(0)}| d_N(V) = 1] = E[Z_0(1,g,Q)(O) - Z_0(0,g,Q)(O)| d_N(V) = 1] \end{align}

is asymptotically normal with influence function

\begin{align} \frac{1}{P(d'(\cdot) = 1)} I{d'(\cdot) = 1}\left{Z(1,g,Q)(O) - Z(0,g,Q)(O) - E[Z(1,g,Q)(O) - Z(0,g,Q)(O) | d'(\cdot) = 1]\right}, \end{align}

where $d'$ is the limiting policy of $d_N$. The fitted influence curve can be extracted using IC():