In tlverse/tmle3mopttx: Targeted Maximum Likelihood Estimation of the Mean under Optimal Individualized Treatment

options(scipen=999)

Introduction

In this section, we present how to use the tmle3mopttx package when the data is subject to missigness. Currently we support missigness process on the following nodes:

1. Outcome node, $Y$.

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

library(data.table)
library(sl3)
library(tmle3)
library(tmle3mopttx)
library(devtools)
set.seed(111)

---

Data and Notation

Suppose we observe $n$ i.i.d. observations of $O=(W,A,Y) \sim P_0$. We denote $A$ as treatment, where $A \in {0,1}$ and $Y$ is the final outcome. Note that we treat $W$ as all of our collected baseline covariates. We emphasize that we make no assumptions about the distribution of $P_0$, so that $P_0 \in \mathcal{M}$, where $\mathcal{M}$ is the fully nonparametric model. This is in contrast to much of the current literature that relies on parametric assumptions. We can break the data generating distribution $P_0$ into the following parts:

$$P_0(O) = P_0(Y|A,W)P_0(A|W)P_0(W) = Q_0(Y|A,W)g_0(A|W)Q_{W,0}(W)$$ In addition, we also define $\bar{Q}{Y,0}(A,W) \equiv E_0[Y|A,W]$ such that $E_0(Y_a) = E{0,W}(\bar{Q}_{Y,0}(A=a,W))$.

Simulated Data

First, we load the simulated data. Here, our data generating distribution was of the following form:

$$W \sim \mathcal{N}(\bf{0},I_{3 \times 3})$$ $$P(A=1|W) = \frac{1}{1+\exp^{(-0.8*W_1)}}$$

$$P(Y=1|A,W) = 0.5\text{logit}^{-1}[-5I(A=1)(W_1-0.5)+5I(A=0)(W_1-0.5)] + 0.5\text{logit}^{-1}(W_2W_3)$$

data("data_bin")

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_bin

#Add some random missingless:
rr <- sample(nrow(data), 100, replace = FALSE)
data[rr,"Y"]<-NA

node_list <- list(
  W = c("W1", "W2", "W3"),
  A = "A",
  Y = "Y"
)

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. In addition, we introduced some missigness to our outcome node; in particular, we randomly force $10\%$ of our outcomes to be missing.

---

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://sl3.tlverse.org, or https://tlverse.org for the tlverse ecosystem, of which sl3 is a major part.

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 @vdl2007super.

# Define sl3 library and metalearners:
xgboost_50<-Lrnr_xgboost$new(nrounds = 50)
xgboost_100<-Lrnr_xgboost$new(nrounds = 100)
xgboost_500<-Lrnr_xgboost$new(nrounds = 500)
lrn1 <- Lrnr_mean$new()
lrn2<-Lrnr_glm_fast$new()

Q_learner <- Lrnr_sl$new(
  learners = list(xgboost_50,xgboost_100,xgboost_500,
                  lrn1,lrn2),
  metalearner = Lrnr_nnls$new()
)

g_learner <- Lrnr_sl$new(
  learners = list(xgboost_100,lrn2),
  metalearner = Lrnr_nnls$new()
)

b_learner <- Lrnr_sl$new(
  learners = list(xgboost_50,xgboost_100,xgboost_500,
                  lrn1,lrn2),
  metalearner = Lrnr_nnls$new()
)

delta_learner <- Lrnr_sl$new(
  learners = list(xgboost_50,xgboost_100,xgboost_500,
                  lrn1,lrn2),
  metalearner = Lrnr_nnls$new()
)

As seen above, we generate four different ensemble learners that must be fit, corresponding to the learners for the outcome regression, propensity score, blip function, and the missigness process. We 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
learner_list <- list(Y = Q_learner, A = g_learner, B = b_learner, delta_Y=delta_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. In particular, it makes explicit the fact that our Y is used in fitting the outcome regression while our A is used in fitting our treatment mechanism regression, B is used in fitting the blip function, and delta_Y fits the missing outcome process.

Initializing tmle3mopttx through its tmle3_Spec

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 tmle3_mopttx_blip_revere. We specify the argument V = c("W1", "W2", "W3") when initializing the tmle3_Spec object in order to communicate that we're interested in learning a rule dependent on V covariates. We also need to specify the type of pseudo-blip we will use in this estimation problem, and finally the list of learners. Note that for binary treatment, the most natural type of blip to use is the blip1, as specified below.

# initialize a tmle specification
tmle_spec <- tmle3_mopttx_blip_revere(V = c("W1", "W2", "W3"), type = "blip1", 
                                      learners = learner_list, maximize = TRUE, 
                                      complex = TRUE, realistic=FALSE)

As seen above, the tmle3_mopttx_blip_revere 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).

In initializing the specification for the TMLE of our parameter of interest, we have specified the set of covariates the rule depends on ($V$), the type of pseudo-blip to use ("type"), and the learners used for estimating all the relevant parts ($Q$, $g$, blip and missigness). In addition, we need to specify whether we want to maximize the mean outcome under the rule ("maximize=TRUE"), and whether we want to estimate the rule under all the covariates $V$ provided by the user. If FALSE, tmle3mopttx will instead consider all the possible rules under a smaller set of covariates including the static rules, and optimize the mean outcome over all the subsets of $V$. As such, while the user might have provided a full set of collected covariates as input for $V$, it is possible that the true rule only depends on a subset of the set provided by the user. In that case, our returned mean under the optimal individualized rule will be based on the smaller subset.

Targeted Estimation of the Mean under the Optimal ITR with Binary Treatment

One may walk through the step-by-step procedure for fitting the TML estimator of the mean counterfactual outcome under the optimal ITR, using the machinery exposed by the tmle3 R package (see below); however, the step-by-step procedure is often not of interest.

# NOT RUN -- SEE NEXT CODE CHUNK

# Define data:
tmle_task <- tmle_spec$make_tmle_task(data, node_list)

# Define likelihood:
initial_likelihood <- tmle_spec$make_initial_likelihood(tmle_task, learner_list)

# Define updater and targeted likelihood:
updater <- tmle_spec$make_updater()
targeted_likelihood <- tmle_spec$make_targeted_likelihood(initial_likelihood, 
                                                            updater)

tmle_params <- tmle_spec$make_params(tmle_task, likelihood=targeted_likelihood)
updater$tmle_params <- tmle_params

fit <- fit_tmle3(tmle_task, targeted_likelihood, tmle_params, 
                   updater)
fit$summary

Instead, one may invoke the tmle3 convenience function to fit the series of TML estimators in a single function call:

# fit the TML estimator
fit <- tmle3(tmle_spec, data, node_list, learner_list)

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

# fit the TML estimator
print(fit)

We can also look at the distribution of the optimal interventions estimated:

#Return the distribution of the optimal rule:
table(tmle_spec$return_rule)

```

References

tlverse/tmle3mopttx documentation built on Aug. 9, 2022, 3:31 p.m.