The tmle3
package differs from previous tmle
software efforts in that it attempts to directly model the key objects defined in the mathematical TMLE framework. That is, rather than focus on implementing a specific TML estimator, or a small set of related estimators, the focus is on model the TMLE framework itself.
Therefore, we explicitly define objects to model the NPSEM, the factorized likelihood, counterfactual interventions, parameters, and TMLE update procedures. The hope is that, in so doing, it will be possible to support a substantial subset of the vast array of TML estimators currently present in the literature (cite TL books), as well as those that have yet to be developed. In this vignette, we describe these mathematical objets, their software analogs in tmle3
, and illustrate with a motivating example, described below. At the end, we describe how these objects can be bundled up into a specification of a TML estimation procedure that can be easily applied by an end user.
We use 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.
library(tmle3)
library(sl3)
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)
TMLE requires the specification of a Nonparametric Structual Equation Model (NPSEM), which specifies our knowledge of relationships between the variables.
We start with a set of endogenous variables, \(X=(X_1,\ldots,X_J)\), that we want to model the relationship between. Each \(X_j\) is at least partially observed in the dataset. The NPSEM defines each variable (\(X_j\)) by a deterministic function (\(f_{X_j}\)) of its parent nodes (\(Pa(X_j)\)) and an exogenous random variable (\(U_{X_j}\)):
\[X_j=f_{X_j}(Pa(X_j), U_{X_j}),\;\; j\in \{1, \ldots, J\}\]
The exact functional form of the functions \(f_{X_j}\) is left unspecified at this step. If there is a priori knowledge for some of these functions, that can be specified during the likelihood step below.
The collection of exogenous random variables defined by the NPSEM is \(U=(U_{X_1}, \ldots,U_{X_J})\). Typically, non-testable assumptions about the joint distribution of \(U\) are necessary for identifiability of causal parameters with statistical parameters of the observed data. These assumptions are not managed in the tmle3
framework, which instead focus on the statistical estimation problem. Therefore, those developing tools for end users need to be clear about the additional causal assumptions necessary for causal intepretation of estimates.
In the case of our CPP example, we use the classic point treatment NPSEM which defines three nodes: \(X=(W,A,Y)\), where \(W\) is a set of baseline covariates, \(A\) is our exposure of interest (parity01), and \(Y\) is our outcome of interest (haz01). We define the following SCM:
\[W=f_W(U_W)\] \[A=f_A(W,U_A)\] \[Y=f_Y(W,U_Y)\]
In tmle3
, this is done using the define_node
function for each node. define_node
allows a user to specify the node_name, which columns in the data comprise the node, and a list of parent nodes.
npsem <- list(
define_node("W", c(
"apgar1", "apgar5", "gagebrth", "mage",
"meducyrs", "sexn"
)),
define_node("A", c("parity01"), c("W")),
define_node("Y", c("haz01"), c("A", "W"))
)
Nodes also track information about the data types of the variables (continuous, categorical, binomial, etc). Here, that information is being estimated automatically from the data. In the future, each node will also contain information about censoring indicators, where applicable, but this is not yet implemented.
tmle3_Task
A tmle3_Task
is an object comprised of data, and the NPSEM defined above:
task <- tmle3_Task$new(cpp, npsem = npsem)
This task object contains methods to help subset the data as needed for various tmle steps:
#get the outcome node data
head(task$get_tmle_node("Y"))
## [1] 1 1 1 0 0 1
#get the sl3 task corresponding to an outcome regression
task$get_regression_task("Y")
## A sl3 Task with 1441 obs and these nodes:
## $covariates
## A W1 W2 W3 W4 W5
## "parity01" "apgar1" "apgar5" "gagebrth" "mage" "meducyrs"
## W6
## "sexn"
##
## $outcome
## [1] "haz01"
##
## $id
## NULL
##
## $weights
## NULL
##
## $offset
## NULL
Having defined the NPSEM, we can now define a joint likelihood (probability density function) over the observed variables \(X\): \[P(X_1, \ldots,X_J\in D)=\int_D f_{X_1, \ldots,X_J}(x_1,\ldots,x_J) dx_1,\ldots,dx_J\] This can then be factorized into a series of conditional densities according to the NPSEM: \[f_{X_1, \ldots,X_J}=\prod_j^J f_{X_j|Pa(X_j)}(x|Pa(x_j))\]
Where each \(f_{X_j|Pa(X_j)\) is a conditional pdf (or probability mass function for discrete \(X_j\)), where the conditioning set is all parent nodes as defined in the NPSEM. We refer to these objects as likelihood factors.
TMLE depends on estimates (or a priori knowledge) of the functional form of these likelihood factors. However, not all factors of the likelihood are always necessary for estimation, and only those necessary will be estimated.
tmle3
models this likelihood as a list of likelihood factor objects, where each likeliood factor object describes either a priori knowledge or an estimation strategy for the corresponding likelihood factor. These objects all inherit from the LF_base
base class, and there are different types depending on which of a range of estimation strategies or a priori knowledge is appropriate.
In some cases, a full conditional density for a particular factor is not necessary. Instead, a conditional mean (a much easier quantity to estimate), is all that’s required. Although conditional means are not truely likelihood factors, conditional means are also modelled using using likelihood factor objects.
LF_emp
LF_emp
represents a likelihood factor to be estimated using NP-MLE. That is, probability mass \(1/n\) is placed on each observation in the observed dataset. Going forward, weights will be used if specified, although this is not yet supported. \(LF_emp\) only supports marginal densities. That is the conditioning set \(Pa(X_j)=\{\emptyset\}\) must be empty. Therefore, it is only appropriate for estimation of the marginal density of baseline covariates.
LF_fit
LF_fit
represents a likelihood factor to be estimated using the sl3
framework. Based on the learner type used, this can fit a pmf (for binomial or categorical data, see sl3_list_learners("binomial")
and sl3_list_learners("categorical")
for lists), a conditional mean (most learners), or a conditional density (using condensier
via Lrnr_condensier
). LF_fit
takes a sl3
learner object as an argument, which is fit to the data in the tmle3_Task
automatically.
In tmle3
, interventions are modeled as likelihoods where one or more likelihood factors is replaced with a counterfactual version representing some intervention.
tmle3
defines the CF_Likelihood
class, which inherits from Likelihood
, and takes an observed_likelihood
and an intervention_list
.
Below, we describe some examples of additional likelihood factors intended to be used to describe interventions. We expect this list to grow as tmle3
is extended to additional use cases.
Psi(Pn)
Definition of a target parameter requires specification of a mapping Ψ ap- plied to P0. Ψ maps any P ∈ M into a vector of numbers Ψ(P). We write the mapping as Ψ : M → Rd for a d-dimensional parameter. • ψ0 is the evaluation of Ψ(P0), i.e., the true value of our parameter. The statistical estimation problem is to map the observed data O1, . . . , On into an estimator of Ψ(P0) that incorporates the knowledge that P0 ∈ M, accompanied by an assessment of the uncertainty in the estimator.