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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 (Nordland and Holst 2023) for the syntax and
methodological context.
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
\[\begin{align*} d^\eta_0(v) &= I\{B_0(v) > \eta\}\\ B_0(v) &= E\left[Z(1,g_0,Q_0)(O) - Z(0,g_0,Q_0)(O) \big | V = v \right] \end{align*}\]
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:
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)
## [1] "0"
pl1_ptl <- policy_learn(
type = "ptl",
control = control_ptl(policy_var = c("W", "L")),
threshold = 1
)
## Loading required namespace: policytree
po1_ptl <- pl1_ptl(
policy_data = pd,
g_models = g_glm(~ 1),
q_models = q_glm(~ A * (W + L))
)
po1_ptl$ptl_objects
## $stage_1
## $stage_1$threshold_1
## policy_tree object
## Tree depth: 2
## Actions: 1: 0 2: 1
## Variable splits:
## (1) split_variable: W split_value: -0.0948583
## (2) split_variable: W split_value: -0.107529
## (4) * action: 1
## (5) * action: 2
## (3) split_variable: W split_value: 0.197522
## (6) * action: 1
## (7) * action: 2
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)])))
## [1] 2.082982
The subgroup average treatment effect associated with the learned
optimal threshold policy can be directly estimated using
policy_eval()
via the target
argument:
## Estimate Std.Err 2.5% 97.5% P-value
## E[Z(1)-Z(0)|d=1]: d=blip(eta=1) 1.941 0.2614 1.428 2.453 1.136e-13
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"
)
## Estimate Std.Err 2.5% 97.5% P-value
## E[Z(1)-Z(0)|d=1]: d=blip(eta=0) 1.641 0.2161 1.217 2.064 3.118e-14
## E[Z(1)-Z(0)|d=1]: d=blip(eta=1) 1.935 0.2612 1.423 2.447 1.268e-13
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()
:
## [,1]
## [1,] 0.000000
## [2,] 0.000000
## [3,] 0.000000
## [4,] -1.780405
## [5,] 0.000000
## [6,] 9.520253
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They may not be fully stable and should be used with caution. We make no claims about them.