---
title: "Evaluating Causal Effects of Modified Treatment Policies"
author: "[Nima Hejazi](https://nimahejazi.org) and [David
  Benkeser](https://www.sph.emory.edu/faculty/profile/#!dbenkes)"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{Evaluating Causal Effects of Modified Treatment Policies}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

## Introduction

Stochastic treatment regimes constitute a flexible framework for evaluating the
effects of continuous-valued exposures/treatments. _Modified treatment
policies_, one such technique within this framework, examine the effects
attributable to shifting the observed ("natural") value of a treatment, usually
up or down by some scalar $\delta$. The `txshift` package implements algorithms
for computing one-step or targeted minimum loss-based (TML) estimates of the
counterfactual means induced by additive modified treatment policies (MTPs),
defined by a shifting function $\delta(A,W)$. For a technical presentation, the
interested reader is invited to consult @diaz2018stochastic or the earlier work
of @diaz2012population and @haneuse2013estimation. For background on Targeted
Learning, consider consulting @vdl2011targeted, @vdl2018targeted, and
@vdl2022targeted.

To start, let's load the packages we'll need and set a seed for simulation:

```{r setup}
library(data.table)
library(haldensify)
library(txshift)
set.seed(11249)
```

---

## Data and Notation

We'll consider $n$ observed units $O_1, \ldots, O_n$, where each random variable
$O = (W, A, Y)$ corresponds to the data available on a single unit. Within $O$,
$W$ denotes baseline covariates (e.g., age, biological sex, BMI), $A \in
\mathbb{R}$ a continuous-valued exposure (e.g., dosage of nutritional
supplements taken), and $Y$ an outcome of interest (e.g., disease status).  To
minimize unjustifiable assumptions, we let $O \sim \mathcal{P} \in \mathcal{M}$,
where $\mathcal{P}$ is simply any distribution within the nonparametric
statistical model $\mathcal{M}$. To formalize the definition of stochastic
interventions and their corresponding causal effects, we consider a
nonparametric structural equation model (NPSEM), introduced by
@pearl2000causality, to define how the system changes under interventions of
interest:
\begin{align*}\label{eqn:npsem}
  W &= f_W(U_W) \\ A &= f_A(W, U_A) \\ Y &= f_Y(A, W, U_Y),
\end{align*}
We denote the observed data structure $O = (W, A, Y)$

Assuming that the distribution of $A$ conditional on $W = w$ has support in the
interval $(l(w), u(w))$ -- for convenience, we assume that the minimum natural
value of treatment $A$ for an individual with covariates $W = w$ is $l(w)$,
while, similarly, the maximum is $u(w)$ -- a simple MTP based on a shift
$\delta$, is
\begin{equation}\label{eqn:shift}
  \delta(a, w) =
  \begin{cases}
    a - \delta & \text{if } a > l(w) + \delta \\
    a & \text{if } a \leq l(w) + \delta,
  \end{cases}
\end{equation}
where $0 \leq \delta$ is an arbitrary pre-specified value that defines the
degree to which the observed value $A$ is to be shifted, where possible.

In case-cohort studies, it is common practice to make use of outcome-dependent
two-phase sampling designs, which allow for expensive measurements made on the
exposure (e.g., genomic sequencing of immune markers) to be avoided. As a
complication, such sampling schemes alter the observed data structure from the
simpler $O = (W, A, Y)$ to $O = (W, \Delta A, Y, \Delta)$, where the sampling
indicator $\Delta$ may itself be a function of the variables $\{W, Y\}$. In this
revised data structure, the value of $A$ is only observed for units in the
two-phase sample, for whom $\Delta = 1$. @hejazi2020efficient provide a detailed
investigation of the methodological details of efficient estimation under such
designs in the context of vaccine efficacy trials; their work was used in the
analysis of immune correlates of protection for HIV-1 [@hejazi2020efficient] and
COVID-19 [@gilbert2021covpn]. Of course, one may also account for loss to
follow-up (i.e., censoring), which the `txshift` package supports (through the
`C_cens` argument of the eponymous `txshift` function), though we avoid this
complication in our subsequent examples in the interest of clarity of
exposition. Corrections for both censoring and two-phase sampling make use of
inverse probability of censoring weighting (IPCW), leading to IPCW-augmented
one-step and TML estimators.

### Simulate Data

```{r make_data}
# parameters for simulation example
n_obs <- 200 # number of observations

# baseline covariate -- simple, binary
W <- rbinom(n_obs, 1, 0.5)

# create treatment based on baseline W
A <- rnorm(n_obs, mean = 2 * W, sd = 0.5)

# create outcome as a linear function of A, W + white noise
Y <- A + W + rnorm(n_obs, mean = 0, sd = 1)

# two-phase sampling based on covariates
Delta_samp <- rbinom(n_obs, 1, plogis(W))

# treatment shift parameter
delta <- 0.5
```

## Estimating the Effects of Additive MTPs

The simplest way to compute an efficient estimator for an additive MTP is to fit
each of the nuisance parameters internally. This procedure can be sped up by
using generalized linear models (GLMs) to fit the outcome regression $Q_n$. The
`txshift()` function provides a simple interface.

```{r est_simple}
est_shift <- txshift(
  W = W, A = A, Y = Y, delta = delta,
  g_exp_fit_args = list(
    fit_type = "hal", n_bins = 5,
    grid_type = "equal_mass",
    lambda_seq = exp(seq(-1, -10, length = 100))
  ),
  Q_fit_args = list(
    fit_type = "glm",
    glm_formula = "Y ~ .^2"
  )
)
est_shift
```

### Interlude: Ensemble Machine Learning with `sl3`

To easily incorporate ensemble machine learning into the estimation procedure,
`txshift` integrates with the [`sl3` R
package](https://tlverse.org/sl3/) [@coyle-sl3-rpkg]. For a complete guide on
using the `sl3` R package, consider consulting [the chapter on Super
Learning](https://tlverse.org/tlverse-handbook/sl3.html) in @vdl2022targeted.

```{r setup_sl, eval = FALSE}
library(sl3)

# SL learners to be used for most fits (e.g., IPCW, outcome regression)
mean_learner <- Lrnr_mean$new()
glm_learner <- Lrnr_glm$new()
rf_learner <- Lrnr_ranger$new()
Q_lib <- Stack$new(mean_learner, glm_learner, rf_learner)
sl_learner <- Lrnr_sl$new(learners = Q_lib, metalearner = Lrnr_nnls$new())

# SL learners for fitting the generalized propensity score fit
hose_learner <- make_learner(Lrnr_density_semiparametric,
  mean_learner = glm_learner
)
hese_learner <- make_learner(Lrnr_density_semiparametric,
  mean_learner = rf_learner,
  var_learner = glm_learner
)
g_lib <- Stack$new(hose_learner, hese_learner)
sl_learner_density <- Lrnr_sl$new(
  learners = g_lib,
  metalearner = Lrnr_solnp_density$new()
)
```

### Efficient Effect Estimates with Machine Learning

Using the framework provided by the [`sl3`
package](https://github.com/tlverse/sl3/), the nuisance functions required for
our efficient estimators may be fit with ensemble machine learning. The Super
Learner algorithm [@vdl2007super] implemented in `sl3` uses the asymptotic
optimality of V-fold cross-validation [@dudoit2005asymptotics;
@vdl2004asymptotic; @vdv2006oracle] to select an optimal prediction functions
from a library or to construct an optimal combination of prediction functions.

```{r est_with_sl, eval = FALSE}
est_shift_sl <- txshift(
  W = W, A = A, Y = Y, delta = delta,
  g_exp_fit_args = list(
    fit_type = "sl",
    sl_learners_density = sl_learner_density
  ),
  Q_fit_args = list(
    fit_type = "sl",
    sl_learners = sl_learner
  )
)
est_shift_sl
```

## Estimating the Effects of Additive MTPs Under Two-Phase Sampling

In case-cohort studies in which two-phase sampling is used, the data structure
takes the from $O = (W, \Delta A, Y, \Delta)$ as previously discussed. Under
such sampling, the `txshift()` function may still be used to estimate the causal
effect of an additive MTP -- only a few additional arguments need to be
specified:

```{r ipcw_est_shift}
est_shift_ipcw <- txshift(
  W = W, A = A, Y = Y, delta = delta,
  C_samp = Delta_samp, V = c("W", "Y"),
  samp_fit_args = list(fit_type = "glm"),
  g_exp_fit_args = list(
    fit_type = "hal", n_bins = 5,
    grid_type = "equal_mass",
    lambda_seq = exp(seq(-1, -10, length = 100))
  ),
  Q_fit_args = list(
    fit_type = "glm",
    glm_formula = "Y ~ .^2"
  ),
  eif_reg_type = "glm"
)
est_shift_ipcw
```

Note that we specify a few additional arguments in the call to `txshift()`,
including `C_samp`, the indicator of inclusion in the two-phase sample; `V`, the
set of other variables that may affect the sampling decision (in this case, both
the baseline covariates and the outcome); `samp_fit_args`, which indicates how
the sampling mechanism ought to be estimated; and `eif_reg_type`, which
indicates how a particular reduced-dimension nuisance regression ought to be
estimated (see @rose2011targeted2sd and @hejazi2020efficient for details). This
last argument only has options for using a GLM or the highly adaptive lasso
(HAL), a nonparametric regression estimator, using the [`hal9001`
package](https://github.com/tlverse/hal9001/) [@coyle-hal9001-rpkg;
hejazi2020hal9001-joss]. In practice, we recommend leaving this argument to the
default and fitting this nuisance function with HAL; however, this is
significantly more costly computationally.

## Statistical Inference for One-step and TML Estimators

The efficient estimators implemented in `txshift` are asymptotically linear;
thus, the estimator $\psi_n$ converges to the true parameter value $\psi_0$:
$$\psi_n - \psi_0 = (P_n - P_0) \cdot D(\bar{Q}_n^{\star}, g_n) +
  R(\hat{P}^{\star}, P_0),$$
which yields
$$\psi_n - \psi_0 = (P_n - P_0) \cdot D(P_0) + o_P \left( \frac{1}{\sqrt{n}}
 \right),$$
provided the following conditions,

1. if $D(\bar{Q}_n^{\star}, g_n)$ converges to $D(P_0)$ in $L_2(P_0)$ norm;
2. the size of the class of functions considered for estimation of
   $\bar{Q}_n^{\star}$ and $g_n$ is bounded (technically, $\exists \mathcal{F}$
   such that $D(\bar{Q}_n^{\star}, g_n) \in \mathcal{F}$ with high probability,
   where $\mathcal{F}$ is a Donsker class); and
3. the remainder term $R(\hat{P}^{\star}, P_0)$ decays as $o_P
   \left( \frac{1}{\sqrt{n}} \right)$.

By the central limit theorem, the estimators then have a Gaussian limiting
distribution,
$$\sqrt{n}(\psi_n - \psi) \to N(0, V(D(P_0))),$$
where $V(D(P_0))$ is the variance of the efficient influence function (or
canonical gradient).

The above implies that $\psi_n$ is a $\sqrt{n}$-consistent estimator of $\psi$,
that it is asymptotically normal (as given above), and that it is locally
efficient. This allows us to build Wald-type confidence intervals in a
straightforward manner:

$$\psi_n \pm z_{\alpha} \cdot \frac{\sigma_n}{\sqrt{n}},$$
where $\sigma_n^2$ is an estimator of $V(D(P_0))$. The estimator $\sigma_n^2$
may be obtained using the bootstrap or computed directly via the following

$$\sigma_n^2 = \frac{1}{n} \sum_{i = 1}^{n} D^2(\bar{Q}_n^{\star}, g_n)(O_i).$$

Such confidence intervals may easily be created with the `confint` method:

```{r confint}
(ci_est_shift <- confint(est_shift))
```

## _Advanced Usage:_ User-Specified Nuisance Regressions

In some special cases it may be useful for the experienced user to compute the
treatment mechanism, censoring mechanism, outcome regression, and sampling
mechanism fits separately (i.e., outside of the `txshift` wrapper function),
instead applying the wrapper only to construct an efficient one-step or TML
estimator. In such cases, the optional arguments ending in `_ext`.

```{r fit_external, eval=FALSE}
# compute censoring mechanism and produce IPC weights externally
pi_mech <- plogis(W)
ipcw_out <- pi_mech

# compute treatment mechanism (propensity score) externally
## first, produce the down-shifted treatment data
gn_downshift <- dnorm(A - delta, mean = tx_mult * W, sd = 1)
## next, initialize and produce the up-shifted treatment data
gn_upshift <- dnorm(A + delta, mean = tx_mult * W, sd = 1)
## now, initialize and produce the up-up-shifted (2 * delta) treatment data
gn_upupshift <- dnorm(A + 2 * delta, mean = tx_mult * W, sd = 1)
## then, initialize and produce the un-shifted treatment data
gn_noshift <- dnorm(A, mean = tx_mult * W, sd = 1)
## finally, put it all together into an object like what's produced internally
gn_out <- as.data.table(cbind(
  gn_downshift, gn_noshift, gn_upshift,
  gn_upupshift
))[C == 1, ]
setnames(gn_out, c("downshift", "noshift", "upshift", "upupshift"))

# compute outcome regression externally
# NOTE: transform Y to lie in the unit interval and bound predictions such that
#       no values fall near the bounds of the interval
Qn_noshift <- (W + A - min(Y)) / diff(range(Y))
Qn_upshift <- (W + A + delta - min(Y)) / diff(range(Y))
Qn_noshift[Qn_noshift < 0] <- 0.025
Qn_noshift[Qn_noshift > 1] <- 0.975
Qn_upshift[Qn_upshift < 0] <- 0.025
Qn_upshift[Qn_upshift > 1] <- 0.975
Qn_out <- as.data.table(cbind(Qn_noshift, Qn_upshift))[C == 1, ]
setnames(Qn_out, c("noshift", "upshift"))

# construct efficient estimator by applying wrapper function
est_shift_spec <- txshift(
  W = W, A = A, Y = Y, delta = delta,
  samp_fit_args = NULL,
  samp_fit_ext = ipcw_out,
  g_exp_fit_args = list(fit_type = "external"),
  Q_fit_args = list(fit_type = "external"),
  gn_exp_fit_ext = gn_out,
  Qn_fit_ext = Qn_out
)
```

## References