## Introduction

Stochastic treatment regimes present a relatively simple manner in which to assess the effects of continuous treatments by way of parameters that examine the effects induced by the counterfactual shifting of the observed values of a treatment of interest. Here, we present an implementation of a new algorithm for computing targeted minimum loss-based estimates of treatment shift parameters defined based on a shifting function $$d(A,W)$$. For a technical presentation of the algorithm, the interested reader is invited to consult Dı́az and van der Laan (2018). For additional background on Targeted Learning and previous work on stochastic treatment regimes, please consider consulting van der Laan and Rose (2011), van der Laan and Rose (2018), and Dı́az and van der Laan (2012).

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

library(data.table)
library(sl3)
library(tmle3)
library(tmle3shift)
set.seed(429153)

## Data and Notation

Consider $$n$$ observed units $$O_1, \ldots, O_n$$, where each random variable $$O = (W, A, Y)$$ corresponds to a single observational unit. Let $$W$$ denote baseline covariates (e.g., age, sex, education level), $$A$$ an intervention variable of interest (e.g., nutritional supplements), and $$Y$$ an outcome of interest (e.g., disease status). Though it need not be the case, let $$A$$ be continuous-valued, i.e. $$A \in \mathbb{R}$$. Let $$O_i \sim \mathcal{P} \in \mathcal{M}$$, where $$\mathcal{M}$$ is the nonparametric statistical model defined as the set of continuous densities on $$O$$ with respect to some dominating measure. To formalize the definition of stochastic interventions and their corresponding causal effects, we introduce a nonparametric structural equation model (NPSEM), based on Pearl (2000), to define how the system changes under posited interventions: \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 $$O = (W, A, Y)$$

Letting $$A$$ denote a continuous-valued treatment, we assume that the distribution of $$A$$ conditional on $$W = w$$ has support in the interval $$(l(w), u(w))$$ – for convenience, let this support be a.e. That is, the minimum natural value of treatment $$A$$ for an individual with covariates $$W = w$$ is $$l(w)$$; similarly, the maximum is $$u(w)$$. Then, a simple stochastic intervention, based on a shift $$\delta$$, may be defined $\begin{equation}\label{eqn:shift} d(a, w) = \begin{cases} a - \delta & \text{if } a > l(w) + \delta \\ a & \text{if } a \leq l(w) + \delta, \end{cases} \end{equation}$ where $$0 \leq \delta \leq u(w)$$ is an arbitrary pre-specified value that defines the degree to which the observed value $$A$$ is to be shifted, where possible. For the purpose of using such a shift in practice, the present software provides the functions shift_additive and shift_additive_inv, which define a variation of this shift, assuming that the density of treatment $$A$$, conditional on the covariates $$W$$, has support a.e.

### Simulate Data

# simulate simple data for tmle-shift sketch
n_obs <- 1000 # number of observations
n_w <- 1 # number of baseline covariates
tx_mult <- 2 # multiplier for the effect of W = 1 on the treatment

## baseline covariates -- simple, binary
W <- as.numeric(replicate(n_w, rbinom(n_obs, 1, 0.5)))

## create treatment based on baseline W
A <- as.numeric(rnorm(n_obs, mean = tx_mult * W, sd = 1))

## create outcome as a linear function of A, W + white noise
Y <- A + W + rnorm(n_obs, mean = 0, sd = 0.5)

The above composes our observed data structure $$O = (W, A, Y)$$. To formally express this fact using the tlverse grammar introduced by the tmle3 package, we create a single data object and specify the functional relationships between the nodes in the directed acyclic graph (DAG) via nonparametric structural equation models (NPSEMs), reflected in the node list that we set up:

# organize data and nodes for tmle3
data <- data.table(W, A, Y)
node_list <- list(W = "W", A = "A", Y = "Y")
head(data)
##    W          A          Y
## 1: 1  2.4031607  3.7157578
## 2: 1  4.4973744  5.9651611
## 3: 1  2.0330871  2.2531970
## 4: 0 -0.8089023 -0.8849531
## 5: 1  1.8432067  2.7193091
## 6: 1  1.3555863  2.5705832

We now have an observed data structure (data) and a specification of the role that each variable in the data set plays as the nodes in a DAG.

## Methodology

To start, we will initialize a specification for the TMLE of our parameter of interest (called a tmle3_Spec in the tlverse nomenclature) simply by calling tmle_shift. We specify the argument shift_val = 0.5 when initializing the tmle3_Spec object to communicate that we’re interested in a shift of $$0.5$$ on the scale of the treatment $$A$$ – that is, we specify $$\delta = 0.5$$ (note that this is an arbitrarily chosen value for this example).

# initialize a tmle specification
tmle_spec <- tmle_shift(shift_val = 0.5,
shift_fxn_inv = shift_additive_inv)

As seen above, the tmle_shift specification object (like all tmle3_Spec objects) does not store the data for our specific analysis of interest. Later, we’ll see that passing a data object directly to the tmle3 wrapper function, alongside the instantiated tmle_spec, will serve to construct a tmle3_Task object internally (see the tmle3 documentation for details). Note that, by default, the tmle_spec object is set up to facilitate cross-validated estimation of likelihood components, ensuring certain empirical process conditions may be circumvented by reducing the contribution of an empirical process term to the estimated influence function (Zheng and Laan 2011). In practice, this automatic incorporation of cross-validation (CV-TMLE) means that the user need not be concerned with these theoretical conditions being satisfied; moreover, cross-validated estimation of the efficient influence function is expected to control the estimated variance.

### Interlude: Constructing Optimal Stacked Regressions with sl3

To easily incorporate ensemble machine learning into the estimation procedure, we rely on the facilities provided in the sl3 R package. For a complete guide on using the sl3 R package, consider consulting https://tlverse.org/sl3, or https://tlverse.org for the tlverse ecosystem, of which sl3 is a core component.

Using the framework provided by the sl3 package, the nuisance parameters of the TML estimator may be fit with ensemble learning, using the cross-validation framework of the Super Learner algorithm of van der Laan, Polley, and Hubbard (2007). To estimate the treatment mechanism (denoted g, we must make use of learning algorithms specifically suited to conditional density estimation; a list of such learners may be extracted from sl3 using sl3_list_learners():

sl3_list_learners("density")
##  "Lrnr_density_discretize"     "Lrnr_density_hse"
##  "Lrnr_density_semiparametric" "Lrnr_haldensify"
##  "Lrnr_solnp_density"

To proceed, we’ll select two of the above learners, Lrnr_haldensify for using the highly adaptive lasso for conditional density estimation, based on an algorithm given by Dı́az and van der Laan (2011), and Lrnr_density_semiparametric, an approach for semiparametric conditional density estimation:

# learners used for conditional density regression (i.e., propensity score)
haldensify_lrnr <- Lrnr_haldensify$new( n_bins = 3, grid_type = "equal_mass", lambda_seq = exp(seq(-1, -9, length = 100)) ) hse_lrnr <- Lrnr_density_semiparametric$new(mean_learner = Lrnr_glm$new()) mvd_lrnr <- Lrnr_density_semiparametric$new(mean_learner = Lrnr_glm$new(), var_learner = Lrnr_mean$new())
sl_lrn_dens <- Lrnr_sl$new( learners = list(haldensify_lrnr, hse_lrnr, mvd_lrnr), metalearner = Lrnr_solnp_density$new()
)

We also require an approach for estimating the outcome regression (denoted Q). For this, we build a Super Learner composed of an intercept model, a main terms GLM, and the xgboost algorithm for gradient boosting:

# learners used for conditional expectation regression (e.g., outcome)
mean_lrnr <- Lrnr_mean$new() glm_lrnr <- Lrnr_glm$new()
xgb_lrnr <- Lrnr_xgboost$new() sl_lrn <- Lrnr_sl$new(
learners = list(mean_lrnr, glm_lrnr, xgb_lrnr),
metalearner = Lrnr_nnls\$new()
)

We can make the above explicit with respect to standard notation by bundling the ensemble learners into a list object below.

# specify outcome and treatment regressions and create learner list
Q_learner <- sl_lrn
g_learner <- sl_lrn_dens
learner_list <- list(Y = Q_learner, A = g_learner)

The learner_list object above specifies the role that each of the ensemble learners we’ve generated is to play in computing initial estimators to be used in building a TMLE for the parameter of interest here. In particular, it makes explicit the fact that our Q_learner is used in fitting the outcome regression while our g_learner is used in fitting our treatment mechanism regression.

### Targeted Estimation of Stochastic Interventions Effects

Note that, by default, the

tmle_fit <- tmle3(tmle_spec, data, node_list, learner_list)
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
##
## Iter: 1 fn: 1419.0430     Pars:  0.75107 0.24893
## Iter: 2 fn: 1419.0430     Pars:  0.75107 0.24893
## solnp--> Completed in 2 iterations
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
## Error : Optimal lambda not selected by CV in fitted haldensify model
tmle_fit
## A tmle3_Fit that took 1 step(s)
##    type         param init_est tmle_est         se    lower    upper
## 1:  TSM E[Y_{A=NULL}] 2.051413  2.05467 0.06087795 1.935351 2.173989
##    psi_transformed lower_transformed upper_transformed
## 1:         2.05467          1.935351          2.173989

The print method of the resultant tmle_fit object conveniently displays the results from computing our TML estimator.

### Statistical Inference for Targeted Maximum Likelihood Estimates

Recall that the asymptotic distribution of TML estimators has been studied thoroughly: $\psi_n - \psi_0 = (P_n - P_0) \cdot D(\bar{Q}_n^*, g_n) + R(\hat{P}^*, P_0),$ which, provided the following two conditions:

1. If $$D(\bar{Q}_n^{\star}, g_n)$$ converges to $$D(P_0)$$ in $$L_2(P_0)$$ norm, and
2. the size of the class of functions considered for estimation of $$\bar{Q}_n^{\star}$$ and $$g_n$$ is bounded (technically, $$\exists \mathcal{F}$$ s.t. $$D(\bar{Q}_n^{\star}, g_n) \in \mathcal{F}$$ whp, where $$\mathcal{F}$$ is a Donsker class), readily admits the conclusion that $$\psi_n - \psi_0 = (P_n - P_0) \cdot D(P_0) + R(\hat{P}^{\star}, P_0)$$.

Under the additional condition that the remainder term $$R(\hat{P}^*, P_0)$$ decays as $$o_P \left( \frac{1}{\sqrt{n}} \right),$$ we have that $\psi_n - \psi_0 = (P_n - P_0) \cdot D(P_0) + o_P \left( \frac{1}{\sqrt{n}} \right),$ which, by a central limit theorem, establishes a Gaussian limiting distribution for the estimator:

$\sqrt{n}(\psi_n - \psi) \to N(0, V(D(P_0))),$ where $$V(D(P_0))$$ is the variance of the efficient influence curve (canonical gradient) when $$\psi$$ admits an asymptotically linear representation.

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

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

Having now re-examined these facts, let’s simply examine the results of computing our TML estimator:

tmle_fit
## A tmle3_Fit that took 1 step(s)
##    type         param init_est tmle_est         se    lower    upper
## 1:  TSM E[Y_{A=NULL}] 2.051413  2.05467 0.06087795 1.935351 2.173989
##    psi_transformed lower_transformed upper_transformed
## 1:         2.05467          1.935351          2.173989