# Chapter 4 The TMLE Framework

Based on the `tmle3`

`R`

package.

Updated: 2020-02-20

## 4.1 Introduction

The first step in the estimation procedure is an initial estimate of the data-generating distribution, or the relevant part of this distribution that is needed to evaluate the target parameter. For this initial estimation, we use the super learner (van der Laan, Polley, and Hubbard 2007), as described in the previous section.

With the initial estimate of relevant parts of the data-generating distribution necessary to evaluate the target parameter, we are ready to construct the TMLE!

### 4.1.1 Substitution Estimators

- Beyond a fit of the prediction function, one might also want to estimate more targeted parameters specific to certain scientific questions.
- The approach is to plug into the estimand of interest estimates of the relevant distributions.
- Sometimes, we can use simple empirical distributions, but averaging some function over the observations (e.g., giving weight \(1/n\) for all observations).
- Other parts of the distribution, like conditional means or probabilities, the estimate will require some sort of smoothing due to the curse of dimensionality.

We give one example using an example of the average treatment effect (see above):

- \(\Psi(P_0) = \Psi(Q_0) = \mathbb{E}_0 \big[\mathbb{E}_0[Y \mid A = 1, W] - \mathbb{E}_0[Y \mid A = 0, W]\big]\), where \(Q_0\) represents both the distribution of \(Y \mid A,W\) and distribution of \(W\).
- Let \(\bar{Q}_0(A,W) \equiv \mathbb{E}_0(Y \mid A,W)\) and \(Q_{0,W}(w) = P_0 (W=w)\), then \[ \Psi(Q_0) = \sum_w \{ \bar{Q}_0(1,w)-\bar{Q}_0(0,w)\} Q_{0,W}(w) \]
- The
**Substitution Estimator**plugs in the empirical distribution (weight \(1/n\) for each observation) for \(Q_{0,W}(W_i)\), and some estimate of the regression of \(Y\) on \((A,W)\) (say SL fit): \[ \Psi(Q_n) = \frac{1}{n} \sum_{i=1}^n \{ \bar{Q}_n(1,W_i)-\bar{Q}_n(0,W_i)\} \] - Thus, it becomes the average of the differences in predictions from the fit keeping the observed \(W\), but first replacing \(A=1\) and then the same but all \(A=0\).

### 4.1.2 TMLE

- Though using SL over an arbitrary parametric regression is an improvement, it’s not sufficient to have the properties of an estimator one needs for rigorous inference.
- Because the variance-bias trade-off in the SL is focused on the prediction model, it can, for instance, under-fit portions of the distributions that are critical for estimating the parameter of interest, \(\Psi(P_0)\).
- TMLE keeps the benefits of substitution estimators (it is one), but augments the original estimates to correct for this issue and also results in an asymptotically linear (and thus normally-distributed) estimator with consistent Wald-style confidence intervals.
- Produces a well-defined, unbiased, efficient substitution estimator of target parameters of a data-generating distribution.
- Updates an initial (super learner) estimate of the relevant part of the data-generating distribution possibly using an estimate of a nuisance parameter (like the model of intervention given covariates).
- Removes asymptotic residual bias of initial estimator for the target parameter, if it uses a consistent estimator of \(g_0\).
- If initial estimator was consistent for the target parameter, the additional fitting of the data in the targeting step may remove finite sample bias, and preserves consistency property of the initial estimator.
- If the initial estimator and the estimator of \(g_0\) are both consistent, then it is also asymptotically efficient according to semi-parametric statistical model efficiency theory.
- Thus, every effort is made to achieve minimal bias and the asymptotic semi-parametric efficiency bound for the variance.

### 4.1.3 Inference

- The estimators we discuss are
**asymptotically linear**, meaning that the difference in the estimate \(\Psi(P_n)\) and the true parameter (\(\Psi(P_0)\)) can be represented in first order by a i.i.d. sum: \[\begin{equation}\label{eqn:IC} \Psi(P_n) - \Psi(P_0) = \frac{1}{n} \sum_{i=1}^n IC(O_i; \nu) + o_p(1/\sqrt{n}) \end{equation}\]

where \(IC(O_i; \nu)\) (the influence curve or function) is a function of the data and possibly other nuisance parameters \(\nu\). Importantly, such estimators have mean-zero Gaussian limiting distributions; thus, in the univariate case, one has that \[\begin{equation}\label{eqn:limit_dist} \sqrt{n}(\Psi(P_n) - \Psi(P_0)) \xrightarrow[]{D}N(0,\mathbb{V}IC(O_i;\nu)), \end{equation}\] so that inference for the estimator of interest may be obtained in terms of the influence function. For this simple case, a 95% confidence interval may be derived as: \[\begin{equation}\label{eqn:CI} \Psi(P^{\star}_n) \pm z_{1 - \frac{\alpha}{2}} \sqrt{\frac{\hat{\sigma}^2}{n}}, \end{equation}\] where \(SE=\sqrt{\frac{\hat{\sigma}^2}{n}}\) and \(\hat{\sigma}^2\) is the sample variance of the estimated IC’s: \(IC(O; \hat{\nu})\). One can use the functional delta method to derive the influence curve if a parameter of interest may be written as a function of other asymptotically linear estimators.

- Thus, we can derive robust inference for parameters that are estimated by fitting complex, machine learning algorithms and these methods are computationally quick (do not rely on re-sampling based methods like the bootstrap).

## 4.2 Learning Objectives

- Use
`tmle3`

to estimate an Average Treatment Effect (ATE) - Understand
`tmle3`

“Specs” - Fit
`tmle3`

for a custom set of parameters - Use the delta method to estimate transformations of parameters

## 4.3 Easy-Bake Example: `tmle3`

for ATE

We’ll illustrate the most basic use of TMLE using the WASH benefits example data introduced earlier and estimating an Average Treatment Effect (ATE).

As a reminder, the ATE is identified with the following statistical parameter (under assumptions): \(ATE = \mathbb{E}_0(Y(1)-Y(0)) = \mathbb{E}_0\left(\mathbb{E}_0[Y \mid A=1,W]-\mathbb{E}_0[Y \mid A=0,W] \right)\)

This Easy-Bake implementation consists of the following steps:

- Load the necessary libraries and data
- Define the variable roles
- Create a “Spec” object
- Define the super learners
- Fit the TMLE
- Evaluate the TMLE estimates

### 0. Load the Data

We’ll use the same WASH Benefits data as the earlier chapters:

### 1. Define the variable roles

We’ll use the common \(W\) (covariates), \(A\) (treatment/intervention), \(Y\)
(outcome) data structure. `tmle3`

needs to know what variables in the dataset
correspond to each of these roles. We use a list of character vectors to tell
it. We call this a **“Node List”** as it corresponds to the nodes in a Directed
Acyclic Graph (DAG), a way of displaying causal relationships between variables.

```
node_list <- list(
W = c(
"month", "aged", "sex", "momage", "momedu",
"momheight", "hfiacat", "Nlt18", "Ncomp", "watmin",
"elec", "floor", "walls", "roof", "asset_wardrobe",
"asset_table", "asset_chair", "asset_khat",
"asset_chouki", "asset_tv", "asset_refrig",
"asset_bike", "asset_moto", "asset_sewmach",
"asset_mobile"
),
A = "tr",
Y = "whz"
)
```

#### Handling Missingness

Currently, missingness in `tlverse`

is handled in a fairly simple way:

- Missing covariates are median (for continuous) or mode (for discrete) imputed, and additional covariates indicating imputation are generated
- Observations missing treatment variables are excluded.
- We implement an IPCW-TMLE to more efficiently handle missingness in the outcome variables.

These steps are implemented in the `process_missing`

function in `tmle3`

:

### 2. Create a “Spec” Object

`tmle3`

is general, and allows most components of the TMLE procedure to be
specified in a modular way. However, most end-users will not be interested in
manually specifying all of these components. Therefore, `tmle3`

implements a
`tmle3_Spec`

object that bundles a set of components into a **specification**
that, with minimal additional detail, can be run by an end-user.

We’ll start with using one of the specs, and then work our way down into the
internals of `tmle3`

.

### 3. Define the Relevant Super Learners

Currently, the only other thing a user must define are the `sl3`

learners used
to estimate the relevant factors of the likelihood: Q and g.

This takes the form of a list of `sl3`

learners, one for each likelihood factor
to be estimated with `sl3`

:

```
# choose base learners
lrnr_mean <- make_learner(Lrnr_mean)
lrnr_xgboost <- make_learner(Lrnr_xgboost)
# define metalearners appropriate to data types
ls_metalearner <- make_learner(Lrnr_nnls)
mn_metalearner <- make_learner(
Lrnr_solnp, metalearner_linear_multinomial,
loss_loglik_multinomial
)
sl_Y <- Lrnr_sl$new(
learners = list(lrnr_mean, lrnr_xgboost),
metalearner = ls_metalearner
)
sl_A <- Lrnr_sl$new(
learners = list(lrnr_mean, lrnr_xgboost),
metalearner = mn_metalearner
)
learner_list <- list(A = sl_A, Y = sl_Y)
```

Here, we use a Super Learner as defined in the previous `sl3`

section. In the
future, we plan to include reasonable default learners.

### 4. Fit the TMLE

We now have everything we need to fit the tmle using `tmle3`

:

### 5. Evaluate the Estimates

We can see the summary results by printing the fit object. Alternatively, we can extra results from the summary by indexing into it:

```
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est
1: ATE ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}] 0.002493012 0.005841083
se lower upper psi_transformed lower_transformed
1: 0.05035828 -0.09285934 0.1045415 0.005841083 -0.09285934
upper_transformed
1: 0.1045415
```

`[1] 0.005841083`

## 4.4 `tmle3`

Components

Now that we’ve successfully used a spec to obtain a TML estimate, let’s look under the hood at the components. The spec has a number of functions that generate the objects necessary to define and fit a TMLE.

### 4.4.1 `tmle3_task`

First is, a `tmle3_Task`

, analogous to an `sl3_Task`

, containing the data we’re
fitting the TMLE to, as well as an NPSEM generated from the `node_list`

defined
above, describing the variables and their relationships.

```
$W
tmle3_Node: W
Variables: month, aged, sex, momedu, hfiacat, Nlt18, Ncomp, watmin, elec, floor, walls, roof, asset_wardrobe, asset_table, asset_chair, asset_khat, asset_chouki, asset_tv, asset_refrig, asset_bike, asset_moto, asset_sewmach, asset_mobile, momage, momheight, delta_momage, delta_momheight
Parents:
$A
tmle3_Node: A
Variables: tr
Parents: W
$Y
tmle3_Node: Y
Variables: whz
Parents: A, W
```

### 4.4.2 Initial Likelihood

Next, is an object representing the likelihood, factorized according to the NPSEM described above:

```
initial_likelihood <- ate_spec$make_initial_likelihood(
tmle_task,
learner_list
)
print(initial_likelihood)
```

```
W: Lf_emp
A: LF_fit
Y: LF_fit
```

These components of the likelihood indicate how the factors were estimated: the
marginal distribution of \(W\) was estimated using NP-MLE, and the conditional
distributions of \(A\) and \(Y\) were estimated using `sl3`

fits (as defined with
the `learner_list`

) above.

We can use this in tandem with the `tmle_task`

object to obtain likelihood
estimates for each observation:

```
W A Y
1: 0.0002129925 0.2488774 -0.6607409
2: 0.0002129925 0.2533803 -0.6334247
3: 0.0002129925 0.2563182 -0.6210798
4: 0.0002129925 0.2701605 -0.6006003
5: 0.0002129925 0.2526940 -0.5434273
---
4691: 0.0002129925 0.1323222 -0.4626731
4692: 0.0002129925 0.1265871 -0.4816926
4693: 0.0002129925 0.1267474 -0.5671211
4694: 0.0002129925 0.1588095 -0.8185901
4695: 0.0002129925 0.1290488 -0.5404857
```

### 4.4.3 Targeted Likelihood (updater)

We also need to define a “Targeted Likelihood” object. This is a special type
of likelihood that is able to be updated using an `tmle3_Update`

object. This
object defines the update strategy (e.g. submodel, loss function, CV-TMLE or
not, etc).

When constructing the targeted likelihood, you can specify different update
options. See the documentation for `tmle3_Update`

for details of the different
options. For example, you can disable CV-TMLE (the default in `tmle3`

) as
follows:

### 4.4.4 Parameter Mapping

Finally, we need to define the parameters of interest. Here, the spec defines a single parameter, the ATE. In the next section, we’ll see how to add additional parameters.

```
[[1]]
Param_ATE: ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}]
```

### 4.4.5 Putting it all together

Having used the spec to manually generate all these components, we can now
manually fit a `tmle3`

:

```
tmle_fit_manual <- fit_tmle3(
tmle_task, targeted_likelihood, tmle_params,
targeted_likelihood$updater
)
print(tmle_fit_manual)
```

```
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est
1: ATE ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}] 0.002443709 0.0004502056
se lower upper psi_transformed lower_transformed
1: 0.05056788 -0.09866102 0.09956143 0.0004502056 -0.09866102
upper_transformed
1: 0.09956143
```

The result is equivalent to fitting using the `tmle3`

function as above.

## 4.5 Fitting `tmle3`

with multiple parameters

Above, we fit a `tmle3`

with just one parameter. `tmle3`

also supports fitting
multiple parameters simultaneously. To illustrate this, we’ll use the
`tmle_TSM_all`

spec:

```
tsm_spec <- tmle_TSM_all()
targeted_likelihood <- Targeted_Likelihood$new(initial_likelihood)
all_tsm_params <- tsm_spec$make_params(tmle_task, targeted_likelihood)
print(all_tsm_params)
```

```
[[1]]
Param_TSM: E[Y_{A=Control}]
[[2]]
Param_TSM: E[Y_{A=Handwashing}]
[[3]]
Param_TSM: E[Y_{A=Nutrition}]
[[4]]
Param_TSM: E[Y_{A=Nutrition + WSH}]
[[5]]
Param_TSM: E[Y_{A=Sanitation}]
[[6]]
Param_TSM: E[Y_{A=WSH}]
[[7]]
Param_TSM: E[Y_{A=Water}]
```

This spec generates a Treatment Specific Mean (TSM) for each level of the exposure variable. Note that we must first generate a new targeted likelihood, as the old one was targeted to the ATE. However, we can recycle the initial likelihood we fit above, saving us a super learner step.

### 4.5.1 Delta Method

We can also define parameters based on Delta Method Transformations of other parameters. For instance, we can estimate a ATE using the delta method and two of the above TSM parameters:

```
ate_param <- define_param(
Param_delta, targeted_likelihood,
delta_param_ATE,
list(all_tsm_params[[1]], all_tsm_params[[4]])
)
print(ate_param)
```

`Param_delta: E[Y_{A=Nutrition + WSH}] - E[Y_{A=Control}]`

This can similarly be used to estimate other derived parameters like Relative Risks, and Population Attributable Risks

### 4.5.2 Fit

We can now fit a TMLE simultaneously for all TSM parameters, as well as the above defined ATE parameter

```
all_params <- c(all_tsm_params, ate_param)
tmle_fit_multiparam <- fit_tmle3(
tmle_task, targeted_likelihood, all_params,
targeted_likelihood$updater
)
print(tmle_fit_multiparam)
```

```
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est
1: TSM E[Y_{A=Control}] -0.594554632 -0.6241981460
2: TSM E[Y_{A=Handwashing}] -0.607121184 -0.6388781558
3: TSM E[Y_{A=Nutrition}] -0.602632984 -0.6185483623
4: TSM E[Y_{A=Nutrition + WSH}] -0.592110924 -0.6236397328
5: TSM E[Y_{A=Sanitation}] -0.587928739 -0.5869769593
6: TSM E[Y_{A=WSH}] -0.529022211 -0.4467983685
7: TSM E[Y_{A=Water}] -0.576234083 -0.5293518574
8: ATE E[Y_{A=Nutrition + WSH}] - E[Y_{A=Control}] 0.002443709 0.0005584132
se lower upper psi_transformed lower_transformed
1: 0.02989455 -0.68279039 -0.56560590 -0.6241981460 -0.68279039
2: 0.04190379 -0.72100808 -0.55674824 -0.6388781558 -0.72100808
3: 0.04275451 -0.70234566 -0.53475106 -0.6185483623 -0.70234566
4: 0.04091421 -0.70383012 -0.54344935 -0.6236397328 -0.70383012
5: 0.04240558 -0.67009036 -0.50386356 -0.5869769593 -0.67009036
6: 0.04547852 -0.53593464 -0.35766210 -0.4467983685 -0.53593464
7: 0.03887313 -0.60554178 -0.45316193 -0.5293518574 -0.60554178
8: 0.05055319 -0.09852402 0.09964085 0.0005584132 -0.09852402
upper_transformed
1: -0.56560590
2: -0.55674824
3: -0.53475106
4: -0.54344935
5: -0.50386356
6: -0.35766210
7: -0.45316193
8: 0.09964085
```

## 4.6 Stratified Effect Estimates

TMLE can also be applied to estimate effects in in strata of a baseline covariate. The tmle_stratified spec makes it easy to extend an existing spec with stratification.

For instance, we can estimate strata specific ATEs as follows: \(ATE = \mathbb{E}_0(Y(1)-Y(0) \mid V=v ) = \mathbb{E}_0\left(\mathbb{E}_0[Y \mid A=1,W]-\mathbb{E}_0[Y \mid A=0,W] \mid V=v \right)\)

For example, we can stratify the above ATE spec to estimate the ATE in strata of sex:

```
stratified_ate_spec <- tmle_stratified(ate_spec, "sex")
stratified_fit <- tmle3(stratified_ate_spec, washb_data, node_list, learner_list)
print(stratified_fit)
```

```
A tmle3_Fit that took 1 step(s)
type param
1: ATE ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}]
2: stratified ATE ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}] | V=male
3: stratified ATE ATE[Y_{A=Nutrition + WSH}-Y_{A=Control}] | V=female
init_est tmle_est se lower upper psi_transformed
1: 0.002379980 8.674050e-06 0.05100856 -0.09996627 0.09998361 8.674050e-06
2: 0.002071090 2.674460e-02 0.07644424 -0.12308336 0.17657257 2.674460e-02
3: 0.002687687 -2.662495e-02 0.06758710 -0.15909323 0.10584333 -2.662495e-02
lower_transformed upper_transformed
1: -0.09996627 0.09998361
2: -0.12308336 0.17657257
3: -0.15909323 0.10584333
```

This TMLE is consistent for both the marginal ATE as well as the ATEs in strata of V. For continuous V, this could be extended using a working Marginal Structural Model (MSM), although that has not yet been implemented in `tmle3`

.

## 4.7 Exercises

### 4.7.1 Estimation of the ATE with `tmle3`

Follow the steps below to estimate an average treatment effect using data from
the Collaborative Perinatal Project (CPP), available in the `sl3`

package. To
simplify this example, we define a binary intervention variable, `parity01`

–
an indicator of having one or more children before the current child and a
binary outcome, `haz01`

– an indicator of having an above average height for
age.

```
# load the data set
data(cpp)
cpp <- cpp[!is.na(cpp[, "haz"]), ]
cpp$parity01 <- as.numeric(cpp$parity > 0)
cpp[is.na(cpp)] <- 0
cpp$haz01 <- as.numeric(cpp$haz > 0)
```

- Define the variable roles \((W,A,Y)\) by creating a list of these nodes.
Include the following baseline covariates in \(W\):
`apgar1`

,`apgar5`

,`gagebrth`

,`mage`

,`meducyrs`

,`sexn`

. Both \(A\) and \(Y\) are specified above. - Define a
`tmle3_Spec`

object for the ATE,`tmle_ATE()`

. - Using the same base learning libraries defined above, specify
`sl3`

base learners for estimation of \(Q = E(Y|A,Y)\) and \(g=P(A|W)\). - Define the metalearner like below.

```
metalearner <- make_learner(Lrnr_solnp,
loss_function = loss_loglik_binomial,
learner_function = metalearner_logistic_binomial
)
```

- Define one super learner for estimating \(Q\) and another for estimating \(g\). Use the metalearner above for both \(Q\) and \(g\) super learners.
- Create a list of the two super learners defined in Step 5 and call this
object
`learner_list`

. The list names should be`A`

(defining the super learner for estimating \(g\)) and`Y`

(defining the super learner for estimating \(Q\)). - Fit the tmle with the
`tmle3`

function by specifying (1) the`tmle3_Spec`

, which we defined in Step 2; (2) the data; (3) the list of nodes, which we specified in Step 1; and (4) the list of super learners for estimating \(g\) and \(Q\), which we defined in Step 6.*Note*: Like before, you will need to make a data copy to deal with`data.table`

weirdness (`cpp2 <- data.table::copy(cpp)`

) and use`cpp2`

as the data.

### 4.7.2 Estimation of Strata-Specific ATEs with `tmle3`

For this exercise, we will work with a random sample of 5,000 patients who
participated in the International Stroke Trial (IST). This data is described in
the Chapter 3.2 of the `tlverse`

handbook. We included the
data below and a summarized description that is relevant for this exercise.

The outcome, \(Y\), indicates recurrent ischemic stroke within 14 days after
randomization (`DRSISC`

); the treatment of interest, \(A\), is the randomized
aspirin vs. no aspirin treatment allocation (`RXASP`

in `ist`

); and the
adjustment set, \(W\), consists simply of other variables measured at baseline. In
this data, the outcome is occasionally missing, but there is no need to create a
variable indicating this missingness (such as \(\Delta\)) for analyses in the
`tlverse`

, since the missingness is automatically detected when `NA`

are present
in the outcome. Covariates with missing values (`RATRIAL`

, `RASP3`

and `RHEP24`

)
have already been imputed. Additional covariates were created
(`MISSING_RATRIAL_RASP3`

and `MISSING_RHEP24`

), which indicate whether or not
the covariate was imputed. The missingness was identical for `RATRIAL`

and
`RASP3`

, which is why only one covariate indicating imputation for these two
covariates was created.

- Estimate the average effect of randomized asprin treatment (
`RXASP`

= 1) on recurrent ischemic stroke. Even though the missingness mechanism on \(Y\), \(\Delta\), does not need to be specified in the node list, it does still need to be accounted for in the TMLE. In other words, for this estimation problem, \(\Delta\) is a relevant factor of the likelihood in addition to \(Q\), \(g\). Thus, when defining the list of`sl3`

learners for each likelihood factor, be sure to include a list of learners for estimation of \(\Delta\), say`sl_Delta`

, and specify something like`learner_list <- list(A = sl_A, delta_Y = sl_Delta, Y = sl_Y)`

. - Recall that this RCT was conducted internationally. Suposse there is concern
that the dose of asprin may have varied across geographical regions, and an
average across all geographical regions may not be warranted. Calculate the
strata specific ATEs according to geographical region (
`REGION`

).

## 4.8 Summary

`tmle3`

is a general purpose framework for generating TML estimates. The
easiest way to use it is to use a predefined spec, allowing you to just fill in
the blanks for the data, variable roles, and `sl3`

learners. However, digging
under the hood allows users to specify a wide range of TMLEs. In the next
sections, we’ll see how this framework can be used to estimate advanced
parameters such as optimal treatments and shift interventions.

### References

van der Laan, Mark J, Eric C Polley, and Alan E Hubbard. 2007. “Super Learner.” *Statistical Applications in Genetics and Molecular Biology* 6 (1).