7 Stochastic Treatment Regimes (optional)
Nima Hejazi
Based on the tmle3shift R package.
Updated: 2024-11-14
7.1 Learning Objectives
- Differentiate stochastic treatment regimes from static, dynamic, and optimal treatment regimes.
- Describe how estimating causal effects of stochastic interventions informs a real-world data analysis.
- Contrast a population level stochastic intervention policy from a modified treatment policy.
- Estimate causal effects under stochastic treatment regimes with the
tmle3shiftRpackage. - Specify a grid of counterfactual shift interventions to be used for defining a set of stochastic intervention policies.
- Interpret a set of effect estimates from a grid of counterfactual shift interventions.
- Construct marginal structural models to measure variable importance in terms of stochastic interventions, using a grid of shift interventions.
- Implement a shift intervention at the individual level, to facilitate shifting each individual to a value that’s supported by the data.
- Define novel shift intervention functions to extend the
tmle3shiftRpackage.
7.2 Introduction
In this section, we examine a simple example of stochastic treatment regimes in
the context of a continuous treatment variable of interest, defining an
intuitive causal effect through which to examine stochastic interventions more
generally. As a first step to using stochastic
treatment regimes in practice, we present the tmle3shift R
package, which features an
implementation of a recently developed algorithm for computing targeted minimum
loss-based estimates of a causal effect based on a stochastic treatment regime
that shifts the natural value of the treatment based on a shifting function
\(d(A,W)\). We will also use tmle3shift to construct marginal structural models
for variable importance measures, implement shift interventions at the
individual level, and define novel shift intervention functions.
7.3 Stochastic Interventions
Present a relatively simple yet extremely flexible manner by which realistic causal effects (and contrasts thereof) may be defined.
May be applied to nearly any manner of treatment variable – continuous, ordinal, categorical, binary – allowing for a rich set of causal effects to be defined through this formalism.
Arguably the most general of the classes of interventions through which causal effects may be defined, and are conceptually simple.
-
We may consider stochastic interventions in two ways:
The equation \(f_A\), which produces \(A\), is replaced by a probabilistic mechanism \(g_{A,\delta}(A \mid W)\) that differs from the original \(g_A(A \mid W)\). The stochastically modified value of the treatment \(A_{\delta}\) is drawn from a user-specified distribution \(g_{A,\delta}(A \mid W)\), which may depend on the original distribution \(g_A(A \mid W)\) and is indexed by a user-specified parameter \(\delta\). In this case, the stochastically modified value of the treatment \(A_{\delta} \sim g_{A,\delta}(\cdot \mid W)\).
The observed value \(A\) is replaced by a new value \(A_{d(A,W)}\) based on applying a user-defined function \(d(A,W)\) to \(A\). In this case, the stochastic treatment regime may be viewed as an intervention in which \(A\) is set equal to a value based on a hypothetical regime \(d(A, W)\), where regime \(d\) depends on the treatment level \(A\) that would be assigned in the absence of the regime as well as the covariates \(W\). Stochastic interventions of this variety may be referred to as depending on the natural value of treatment or as modified treatment policies (Haneuse and Rotnitzky, 2013; Young et al., 2014).
7.3.1 Identifying the Causal Effect of a Stochastic Intervention
The stochastic intervention generates a counterfactual random variable \(Y_{d(A,W)} := f_Y(d(A,W), W, U_Y) \equiv Y_{g_{A,\delta}} := f_Y(A_{\delta}, W, U_Y)\), where \(Y_{d(A,W)} \sim P_0^{\delta}\), for \(P_0^{\delta}\) being the counterfactual distribution implied by the intervention on \(A\).
The target causal estimand of our analysis is \(\psi_{0, \delta} := \mathbb{E}_{P_0^{\delta}}\{Y_{d(A,W)}\}\), the mean of the counterfactual outcome variable \(Y_{d(A, W)}\). The statistical target parameter may also be denoted \(\Psi(P_0) = \mathbb{E}_{P_0}{\overline{Q}_{Y,0}(d(A, W), W)}\), where \(\overline{Q}_{Y,0}(d(A, W), W)\) is the counterfactual outcome value of a given individual under the stochastic intervention distribution (Dı́az and van der Laan, 2018).
In prior work, Dı́az and van der Laan (2012) showed that the causal quantity of interest \(\mathbb{E}_{P_0} \{Y_{d(A, W)}\}\) is identified by a functional of the distribution of the observed data \(O\): \[\begin{align*}\label{eqn:identification2012} \psi_{0,\delta} = \int_{\mathcal{W}} \int_{\mathcal{A}} & \mathbb{E}_{P_0} \{Y \mid A = d(a, w), W = w\} \\ &g_{A,0}(a \mid W = w) q_{W,0}(w) d\mu(a)d\nu(w). \end{align*}\]
-
Two standard assumptions are necessary in order to establish identifiability of the causal parameter from the observed data via the statistical functional above. These were
Definition 7.1 (Strong Ignorability) \(A_i \perp \!\!\! \perp Y^{d(a_i, w_i)}_i \mid W_i\), for \(i = 1, \ldots, n\).
Definition 7.2 (Treatment Positivity (or Overlap)) \(a_i \in \mathcal{A} \implies d(a_i, w_i) \in \mathcal{A}\) for all \(w \in \mathcal{W}\), where \(\mathcal{A}\) denotes the support of \(A \mid W = w_i \quad \forall i = 1, \ldots n\).
With the identification assumptions satisfied, Dı́az and van der Laan (2018) provide an efficient influence function with respect to the nonparametric model \(\mathcal{M}\) as \[\begin{equation*}\label{eqn:eif} D(P_0)(o) = H(a, w)({y - \overline{Q}_{Y,0}(a, w)}) + \overline{Q}_{Y,0}(d(a, w), w) - \Psi(P_0), \end{equation*}\] where the auxiliary covariate \(H(a,w)\) may be expressed \[\begin{equation*}\label{eqn:aux_covar_full} H(a,w) = \mathbb{I}(a + \delta < u(w)) \frac{g_{A,0}(a - \delta \mid w)} {g_{A,0}(a \mid w)} + \mathbb{I}(a + \delta \geq u(w)), \end{equation*}\] which may be reduced to \[\begin{equation*}\label{eqn:aux_covar_simple} H(a,w) = \frac{g_{A,0}(a - \delta \mid w)}{g_{A,0}(a \mid w)} + 1 \end{equation*}\] in the case that the treatment is in the limits that arise from conditioning on \(W\), i.e., for \(A_i \in (u(w) - \delta, u(w))\).
7.4 Estimating the Causal Effect of a Stochastic Intervention with tmle3shift
We use tmle3shift to construct a targeted maximum likelihood (TML) estimator of
of a causal effect of a stochastic treatment regime that shifts the natural
value of the treatment based on a shifting function \(d(A,W)\). We will follow
the recipe provided by Dı́az and van der Laan (2018), tailored to the tmle3 framework:
- Construct initial estimators \(g_{A,n}\) of \(g_{A,0}(A, W)\) and \(\overline{Q}_{Y,n}\) of \(\overline{Q}_{Y,0}(A, W)\), perhaps using data-adaptive regression techniques.
- For each observation \(i\), compute an estimate \(H_n(a_i, w_i)\) of the auxiliary covariate \(H(a_i,w_i)\).
- Estimate the parameter \(\epsilon\) in the logistic regression model \[ \text{logit}\overline{Q}_{Y, n, \epsilon}(a, w) = \text{logit}\overline{Q}_{Y,n}(a, w) + \epsilon H_n(a, w),\] or an alternative regression model incorporating weights.
- Compute TML estimator \(\Psi_n\) of the target parameter, defining update \(\overline{Q}_{Y,n}^{\star}\) of the initial estimate \(\overline{Q}_{Y, n, \epsilon_n}\): \[\begin{equation*}\label{eqn:tmle} \psi_n^{\star} = \Psi(P_n^{\star}) = \frac{1}{n} \sum_{i = 1}^n \overline{Q}_{Y,n}^{\star}(d(A_i, W_i), W_i). \end{equation*}\]
To start, let’s load the packages we’ll use and set a seed for simulation:
library(tidyverse)
library(data.table)
library(sl3)
library(tmle3)
library(tmle3shift)
set.seed(429153)1. Construct initial estimators \(g_{A,n}\) of \(g_{A,0}(A, W)\) and \(\overline{Q}_{Y,n}\) of \(\overline{Q}_{Y,0}(A, W)\).
We need to estimate two components of the likelihood in order to construct a TML estimator.
- The outcome regression, \(\overline{Q}_{Y,n}\), which is a simple regression of the form \(\mathbb{E}[Y \mid A,W]\).
# learners used for conditional expectation regression
mean_learner <- Lrnr_mean$new()
fglm_learner <- Lrnr_glm_fast$new()
xgb_learner <- Lrnr_xgboost$new(nrounds = 200)
sl_regression_learner <- Lrnr_sl$new(
learners = list(mean_learner, fglm_learner, xgb_learner)
)- The second of these is an estimate of the treatment mechanism, \(g_{A,n}\),
i.e., the generalized propensity score. In the case of a continuous
intervention node \(A\), such a quantity takes the form \(p(A \mid W)\), which is
a conditional density of treatment, given covariates. Generally speaking,
conditional density estimation is a challenging problem that has received
much attention in the literature. To estimate the treatment mechanism, we
must make use of learning algorithms specifically suited to conditional
density estimation; a list of such learners may be extracted from
sl3by usingsl3_list_learners():
sl3_list_learners("density")
[1] "Lrnr_density_discretize" "Lrnr_density_hse"
[3] "Lrnr_density_semiparametric" "Lrnr_haldensify"
[5] "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 implemented in Hejazi et al. (2022), and
semiparametric location-scale conditional density estimators implemented in the
sl3 package. A Super Learner may be
constructed by pooling estimates from each of these modified conditional
density estimation techniques.
# learners used for conditional densities (i.e., generalized propensity score)
haldensify_learner <- Lrnr_haldensify$new(
n_bins = c(3, 5),
lambda_seq = exp(seq(-1, -10, length = 200))
)
# semiparametric density estimator based on homoscedastic errors (HOSE)
hose_learner_xgb <- make_learner(Lrnr_density_semiparametric,
mean_learner = xgb_learner
)
# semiparametric density estimator based on heteroscedastic errors (HESE)
hese_learner_xgb_fglm <- make_learner(Lrnr_density_semiparametric,
mean_learner = xgb_learner,
var_learner = fglm_learner
)
# SL for the conditional treatment density
sl_density_learner <- Lrnr_sl$new(
learners = list(haldensify_learner, hose_learner_xgb,
hese_learner_xgb_fglm),
metalearner = Lrnr_solnp_density$new()
)Finally, we construct a learner_list object for use in constructing a TML
estimator of our target parameter of interest:
Q_learner <- sl_regression_learner
g_learner <- sl_density_learner
learner_list <- list(Y = Q_learner, A = g_learner)7.4.1 Simulate Data
# simulate simple data for tmle-shift sketch
n_obs <- 1000 # number of observations
tx_mult <- 2 # multiplier for the effect of W = 1 on the treatment
## baseline covariates -- simple, binary
W <- replicate(2, rbinom(n_obs, 1, 0.5))
## create treatment based on baseline W
A <- rnorm(n_obs, mean = tx_mult * W, sd = 1)
## create outcome as a linear function of A, W + white noise
Y <- rbinom(n_obs, 1, prob = plogis(A + W))
# organize data and nodes for tmle3
data <- data.table(W, A, Y)
setnames(data, c("W1", "W2", "A", "Y"))
node_list <- list(W = c("W1", "W2"), A = "A", Y = "Y")
head(data)
W1 W2 A Y
<int> <int> <num> <int>
1: 1 1 3.58065 1
2: 1 0 3.20718 1
3: 1 1 1.03584 1
4: 0 0 -0.65785 1
5: 1 1 3.01990 1
6: 1 1 2.78031 1We 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 directed acyclic
graph (DAG) via nonparametric structural equation models (NPSEMs).
To start, we will initialize a specification for the TMLE of our parameter of
interest (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 \(\delta = 0.5\).
# initialize a tmle specification
tmle_spec <- tmle_shift(
shift_val = 0.5,
shift_fxn = shift_additive,
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).
7.4.2 Targeted Estimation of Stochastic Interventions Effects
tmle_fit <- tmle3(tmle_spec, data, node_list, learner_list)
Iter: 1 fn: 1341.6709 Pars: 0.579923107 0.000003629 0.420073262
Iter: 2 fn: 1341.6709 Pars: 0.579923129 0.000002394 0.420074477
solnp--> Completed in 2 iterations
tmle_fit
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est se lower upper
<char> <char> <num> <num> <num> <num> <num>
1: TSM E[Y_{A=NULL}] 0.79815 0.79748 0.012927 0.77215 0.82282
psi_transformed lower_transformed upper_transformed
<num> <num> <num>
1: 0.79748 0.77215 0.82282The print method of the resultant tmle_fit object conveniently displays the
results from computing our TML estimator.
7.5 Stochastic Interventions over a Grid of Counterfactual Shifts
Consider an arbitrary scalar \(\delta\) that defines a counterfactual outcome \(\psi_n = \mathbb{E}_P \overline{Q}_{Y,n}(d(A, W), W)\), where, for simplicity, let \(d(A, W) = A + \delta\). A simplified expression of the auxiliary covariate for the TML estimator of \(\psi\) is \(H_n := \frac{g^{\star}_{A,0}(a \mid w)}{g_{A,0}(a \mid w)}\), where \(g^{\star}_{A,0}(a \mid w)\) defines the treatment mechanism with the stochastic intervention implemented. In this manner, we can specify a grid of shifts \(\delta\) to define a set of stochastic intervention policies in an a priori manner.
To ascertain whether a given choice of the shift \(\delta\) is acceptable, let there be a bound \(C(\delta) = \frac{g^{\star}_{A,0}(a \mid w)}{g_{A,0}(a \mid w)} \leq M\), where \(g^{\star}_{A,0}(a \mid w)\) is a function of \(\delta\) in part, and \(M\) is a user-specified upper bound of \(C(\delta)\). Then, \(C(\delta)\) is a measure of the influence of a given observation (under a bound of the ratio of the conditional densities), which provides a way to limit the maximum influence of a given observation through a choice of the shift \(\delta\).
For the purpose of using such a shift in practice, the present software provides the functions
shift_additive_boundedandshift_additive_bounded_inv, which define a variation of this shift: \[\begin{equation} \delta(a, w) = \begin{cases} \delta, & C(\delta) \leq M \\ 0, \text{otherwise} \\ \end{cases}, \end{equation}\] which corresponds to an intervention in which the natural value of treatment of a given observational unit is shifted by a value \(\delta\) in the case that the ratio of the intervened density \(g^{\star}_{A,0}(a \mid w)\) to the natural density \(g_{A,0}(a \mid w)\) (that is, \(C(\delta)\)) does not exceed a bound \(M\). In the case that the ratio \(C(\delta)\) exceeds the bound \(M\), the stochastic intervention policy does not apply to the given unit and they remain at their natural value of treatment \(a\).
7.5.1 Initializing vimshift 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
tmle_shift. We specify the argument shift_grid = seq(-1, 1, by = 1)
when initializing the tmle3_Spec object to communicate that we’re interested
in assessing the mean counterfactual outcome over a grid of shifts -1, 0, 1 on the scale of the treatment \(A\).
# what's the grid of shifts we wish to consider?
delta_grid <- seq(from = -1, to = 1, by = 1)
# initialize a tmle specification
tmle_spec <- tmle_vimshift_delta(
shift_grid = delta_grid,
max_shifted_ratio = 2
)7.5.2 Targeted Estimation of Stochastic Intervention Effects
One may walk through the step-by-step procedure for fitting the TML estimator
of the mean counterfactual outcome under each shift in the grid, using the
machinery exposed by the tmle3 R package, or
simply invoke the tmle3 wrapper function to fit the series of TML estimators
(one for each parameter defined by the grid delta) in a single function call.
For convenience, we choose the latter:
tmle_fit <- tmle3(tmle_spec, data, node_list, learner_list)
Iter: 1 fn: 1338.5738 Pars: 0.6164612 0.0006504 0.3828885
Iter: 2 fn: 1338.5734 Pars: 0.61647167 0.00000852 0.38351981
Iter: 3 fn: 1338.5734 Pars: 0.616471717 0.000005384 0.383522899
solnp--> Completed in 3 iterations
tmle_fit
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est se lower upper
<char> <char> <num> <num> <num> <num> <num>
1: TSM E[Y_{A=NULL}] 0.60402 0.60198 0.0161588 0.57031 0.63366
2: TSM E[Y_{A=NULL}] 0.73900 0.73900 0.0138950 0.71177 0.76623
3: TSM E[Y_{A=NULL}] 0.84934 0.84818 0.0116570 0.82533 0.87102
4: MSM_linear MSM(intercept) 0.73078 0.72972 0.0131926 0.70386 0.75558
5: MSM_linear MSM(slope) 0.12266 0.12310 0.0054675 0.11238 0.13381
psi_transformed lower_transformed upper_transformed
<num> <num> <num>
1: 0.60198 0.57031 0.63366
2: 0.73900 0.71177 0.76623
3: 0.84818 0.82533 0.87102
4: 0.72972 0.70386 0.75558
5: 0.12310 0.11238 0.13381Remark: The print method of the resultant tmle_fit object conveniently
displays the results from computing our TML estimator.
7.5.3 Inference with Marginal Structural Models
Since we consider estimating the mean counterfactual outcome \(\psi_n\) under several values of the intervention \(\delta\), taken from the aforementioned \(\delta\)-grid, one approach for obtaining inference on a single summary measure of these estimated quantities involves leveraging working marginal structural models (MSMs). Summarizing the estimates \(\psi_n\) through a working MSM allows for inference on the trend imposed by a \(\delta\)-grid to be evaluated via a simple hypothesis test on a parameter of this working MSM. Letting \(\psi_{\delta}(P_0)\) be the mean outcome under a shift \(\delta\) of the treatment, we have \(\vec{\psi}_{\delta} = (\psi_{\delta}: \delta)\) with corresponding estimators \(\vec{\psi}_{n, \delta} = (\psi_{n, \delta}: \delta)\). Further, let \(\beta(\vec{\psi}_{\delta}) = \phi((\psi_{\delta}: \delta))\). By a straightforward application of the delta method (discussed previously), we may write the efficient influence function of the MSM parameter \(\beta\) in terms of the EIFs of each of the corresponding point estimates. Based on this, inference from a working MSM is rather straightforward. To wit, the limiting distribution for \(m_{\beta}(\delta)\) may be expressed \[\sqrt{n}(\beta_n - \beta_0) \to N(0, \Sigma),\] where \(\Sigma\) is the empirical covariance matrix of \(\text{EIF}_{\beta}(O)\).
tmle_fit$summary[4:5, ]
type param init_est tmle_est se lower upper
<char> <char> <num> <num> <num> <num> <num>
1: MSM_linear MSM(intercept) 0.73078 0.72972 0.0131926 0.70386 0.75558
2: MSM_linear MSM(slope) 0.12266 0.12310 0.0054675 0.11238 0.13381
psi_transformed lower_transformed upper_transformed
<num> <num> <num>
1: 0.72972 0.70386 0.75558
2: 0.12310 0.11238 0.133817.5.4 Directly Targeting the MSM Parameter \(\beta\)
Note that in the above, a working MSM is fit to the individual TML estimates of the mean counterfactual outcome under a given value of the shift \(\delta\) in the supplied grid. The parameter of interest \(\beta\) of the MSM is asymptotically linear (and, in fact, a TML estimator) as a consequence of its construction from individual TML estimators. In smaller samples, it may be prudent to perform a TML estimation procedure that targets the parameter \(\beta\) directly, as opposed to constructing it from several independently targeted TML estimates. An approach for constructing such an estimator is proposed in the sequel.
Suppose a simple working MSM \(\mathbb{E}Y_{g^0_{\delta}} = \beta_0 + \beta_1 \delta\), then a TML estimator targeting \(\beta_0\) and \(\beta_1\) may be constructed as \[\overline{Q}_{n, \epsilon}(A,W) = \overline{Q}_n(A,W) + \epsilon (H_1(g), H_2(g),\] for all \(\delta\), where \(H_1(g)\) is the auxiliary covariate for \(\beta_0\) and \(H_2(g)\) is the auxiliary covariate for \(\beta_1\).
To construct a targeted maximum likelihood estimator that directly targets the
parameters of the working marginal structural model, we may use the
tmle_vimshift_msm Spec (instead of the tmle_vimshift_delta Spec that
appears above):
# initialize a tmle specification
tmle_msm_spec <- tmle_vimshift_msm(
shift_grid = delta_grid,
max_shifted_ratio = 2
)
# fit the TML estimator and examine the results
tmle_msm_fit <- tmle3(tmle_msm_spec, data, node_list, learner_list)
Iter: 1 fn: 1338.9482 Pars: 0.62810695 0.00002675 0.37186630
Iter: 2 fn: 1338.9482 Pars: 0.62810718 0.00001349 0.37187934
solnp--> Completed in 2 iterations
tmle_msm_fit
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est se lower upper
<char> <char> <num> <num> <num> <num> <num>
1: MSM_linear MSM(intercept) 0.73087 0.73079 0.0133021 0.70472 0.75686
2: MSM_linear MSM(slope) 0.12250 0.12242 0.0056199 0.11141 0.13344
psi_transformed lower_transformed upper_transformed
<num> <num> <num>
1: 0.73079 0.70472 0.75686
2: 0.12242 0.11141 0.133447.5.5 Example with the WASH Benefits Data
To complete our walk through, let’s turn to using stochastic interventions to
investigate the data from the WASH Benefits trial. To start, let’s load the
data, convert all columns to be of class numeric, and take a quick look at it
washb_data <- fread("https://raw.githubusercontent.com/tlverse/tlverse-data/master/wash-benefits/washb_data_subset.csv", stringsAsFactors = TRUE)
washb_data <- washb_data[!is.na(momage) & !is.na(momheight), ]
head(washb_data, 3)
whz tr fracode month aged sex momage momedu momheight
<num> <fctr> <fctr> <int> <int> <fctr> <int> <fctr> <num>
1: -0.94 Handwashing N06505 7 237 male 21 Primary (1-5y) 146.00
2: -1.13 Control N06505 8 310 female 26 No education 148.90
3: -1.61 Control N06524 3 162 male 25 Primary (1-5y) 153.75
hfiacat Nlt18 Ncomp watmin elec floor walls roof asset_wardrobe
<fctr> <int> <int> <int> <int> <int> <int> <int> <int>
1: Food Secure 1 25 2 1 0 1 1 0
2: Food Secure 1 7 4 1 0 0 1 0
3: Food Secure 0 15 2 0 0 1 1 0
asset_table asset_chair asset_khat asset_chouki asset_tv asset_refrig
<int> <int> <int> <int> <int> <int>
1: 1 0 0 1 0 0
2: 1 1 0 1 0 0
3: 1 0 1 1 0 0
asset_bike asset_moto asset_sewmach asset_mobile
<int> <int> <int> <int>
1: 0 0 0 0
2: 0 0 0 1
3: 0 0 0 0Next, we specify our NPSEM via the node_list object. For our example analysis,
we’ll consider the outcome to be the weight-for-height Z-score (as in previous
sections), the intervention of interest to be the mother’s age at time of
child’s birth, and take all other covariates to be potential confounders.
node_list <- list(
W = names(washb_data)[!(names(washb_data) %in%
c("whz", "momage"))],
A = "momage", Y = "whz"
)Were we to consider the counterfactual weight-for-height Z-score under shifts in the age of the mother at child’s birth, how would we interpret estimates of our parameter?
To simplify our interpretation, consider a shift (up or down) of two years in the mother’s age (i.e., \(\delta = \{-2, 0, 2\}\)); in this setting, a stochastic intervention would correspond to a policy advocating that potential mothers defer or accelerate plans of having a child for two calendar years, possibly implemented through the deployment of an encouragement design.
First, let’s try a simple upward shift of just two years:
# initialize a tmle specification for just a single delta shift
washb_shift_spec <- tmle_shift(
shift_val = 2,
shift_fxn = shift_additive,
shift_fxn_inv = shift_additive_inv
)To examine the effect modification approach we looked at in previous chapters,
we’ll estimate the effect of this shift \(\delta = 2\) while stratifying on the
mother’s education level (momedu, a categorical variable with three levels).
For this, we augment our initialized tmle3_Spec object like so
# initialize effect modification specification around previous specification
washb_shift_strat_spec <- tmle_stratified(washb_shift_spec)Prior to running our analysis, we’ll modify the learner_list object we had
created to include just one of the semiparametric location-scale conditional
density estimators, as fitting of these estimators is much faster than the more
computationally intensive approach implemented in the
haldensify package
(Hejazi et al., 2022).
# learners used for conditional density regression (i.e., propensity score),
# but we need to turn on cross-validation for this conditional density learner
hose_learner_xgb_cv <- Lrnr_cv$new(
learner = hose_learner_xgb,
full_fit = TRUE
)
# modify learner list, using existing SL for Q fit
learner_list <- list(Y = Q_learner, A = hose_learner_xgb_cv)Now we’re ready to construct a TML estimate of the shift parameter at \(\delta = 2\), stratified across levels of our variable of interest:
# fit stratified TMLE
strat_node_list <- copy(node_list)
strat_node_list$W <- setdiff(strat_node_list$W,"momedu")
strat_node_list$V <- "momedu"
washb_shift_strat_fit <- tmle3(washb_shift_strat_spec, washb_data, strat_node_list,
learner_list)
washb_shift_strat_fit
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est se
<char> <char> <num> <num> <num>
1: TSM E[Y_{A=NULL}] -0.56851 -0.56380 0.047206
2: stratified TSM E[Y_{A=NULL}] | V=Primary (1-5y) -0.60154 -0.69115 0.075572
3: stratified TSM E[Y_{A=NULL}] | V=No education -0.64661 -0.84248 0.126252
4: stratified TSM E[Y_{A=NULL}] | V=Secondary (>5y) -0.52532 -0.40382 0.066083
lower upper psi_transformed lower_transformed upper_transformed
<num> <num> <num> <num> <num>
1: -0.65633 -0.47128 -0.56380 -0.65633 -0.47128
2: -0.83927 -0.54303 -0.69115 -0.83927 -0.54303
3: -1.08992 -0.59503 -0.84248 -1.08992 -0.59503
4: -0.53334 -0.27430 -0.40382 -0.53334 -0.27430For the next example, we’ll use the variable importance strategy of considering
a grid of stochastic interventions to evaluate the weight-for-height Z-score
under a shift in the mother’s age down by two years (\(\delta = -2\)) through up
by two years (\(\delta = 2\)), incrementing by a single year between the two. To
do this, we simply initialize a Spec tmle_vimshift_delta similar to how we
did in a previous example:
# initialize a tmle specification for the variable importance parameter
washb_vim_spec <- tmle_vimshift_delta(
shift_grid = seq(from = -2, to = 2, by = 1),
max_shifted_ratio = 2
)Having made the above preparations, we’re now ready to estimate the
counterfactual mean of the weight-for-height Z-score under a small grid of
shifts in the mother’s age at child’s birth. Just as before, we do this through
a simple call to our tmle3 wrapper function:
washb_tmle_fit <- tmle3(washb_vim_spec, washb_data, node_list, learner_list)
washb_tmle_fit
A tmle3_Fit that took 1 step(s)
type param init_est tmle_est se lower
<char> <char> <num> <num> <num> <num>
1: TSM E[Y_{A=NULL}] -0.5606885 -0.5569651 0.047829 -0.6507078
2: TSM E[Y_{A=NULL}] -0.5629913 -0.5679171 0.046885 -0.6598101
3: TSM E[Y_{A=NULL}] -0.5652941 -0.5652941 0.046631 -0.6566901
4: TSM E[Y_{A=NULL}] -0.5675969 -0.5766783 0.047440 -0.6696587
5: TSM E[Y_{A=NULL}] -0.5698998 -0.5637526 0.048028 -0.6578856
6: MSM_linear MSM(intercept) -0.5652941 -0.5661215 0.046638 -0.6575301
7: MSM_linear MSM(slope) -0.0023028 -0.0022336 0.002743 -0.0076097
upper psi_transformed lower_transformed upper_transformed
<num> <num> <num> <num>
1: -0.4632225 -0.5569651 -0.6507078 -0.4632225
2: -0.4760241 -0.5679171 -0.6598101 -0.4760241
3: -0.4738982 -0.5652941 -0.6566901 -0.4738982
4: -0.4836980 -0.5766783 -0.6696587 -0.4836980
5: -0.4696196 -0.5637526 -0.6578856 -0.4696196
6: -0.4747128 -0.5661215 -0.6575301 -0.4747128
7: 0.0031425 -0.0022336 -0.0076097 0.00314257.6 Exercises
Exercise 7.1 Set the sl3 library of algorithms for the Super Learner to a simple,
interpretable library and use this new library to estimate the counterfactual
mean of mother’s age at child’s birth (momage) under a shift \(\delta = 0\).
What does this counterfactual mean equate to in terms of the observed data?
Exercise 7.2 Describe two (equivalent) ways in which the causal effects of stochastic interventions may be interpreted.
Exercise 7.3 Using a grid of values of the shift parameter \(\delta\) (e.g., \(\{-1, 0, +1\}\)),
repeat the analysis on the variable of interest (momage), summarizing the
trend for this sequence of shifts using a marginal structural model.
Exercise 7.4 For either the grid of shifts in the example preceding the exercises or that estimated in (3) above, plot the resultant estimates against their respective counterfactual shifts. Graphically add to the scatterplot a line with slope and intercept equivalent to the MSM fit through the individual TML estimates.
Exercise 7.5 How does the marginal structural model we used to summarize the trend along the sequence of shifts previously help to contextualize the estimated effect for a single shift? That is, how does access to estimates across several shifts and the marginal structural model parameters allow us to more richly interpret our findings?