ATE_internal: Estimating the Average Treatment Effect (ATE) in an internal target population using multi-source data

Description

Doubly-robust and efficient estimator for the ATE in each internal target population using multi-source data.

Usage

ATE_internal(
  X,
  Y,
  S,
  A,
  cross_fitting = FALSE,
  replications = 10L,
  source_model = "MN.glmnet",
  source_model_args = list(),
  treatment_model_type = "separate",
  treatment_model_args = list(),
  outcome_model_args = list(),
  show_progress = TRUE
)

Value

An object of class "ATE_internal". This object is a list with the following elements:

df_dif: A data frame containing the treatment effect (mean difference) estimates for the internal populations.
df_A0: A data frame containing the potential outcome mean estimates under A = 0 for the internal populations.
df_A1: A data frame containing the potential outcome mean estimates under A = 1 for the internal populations.
fit_outcome: Fitted outcome model.
fit_source: Fitted source model.
fit_treatment: Fitted treatment model(s).

Arguments

X: Data frame (or matrix) containing the covariate data in the multi-source data. It should have $n$ rows and $p$ columns. Character variables will be converted to factors.
Y: Vector of length $n$ containing the outcome.
S: Vector of length $n$ containing the source indicator. If S is a factor, it will maintain its level order; otherwise it will be converted to a factor with the default level order. The order will be carried over to the outputs and plots.
A: Vector of length $n$ containing the binary treatment (1 for treated and 0 for untreated).
cross_fitting: Logical specifying whether sample splitting and cross fitting should be used.
replications: Integer specifying the number of sample splitting and cross fitting replications to perform, if cross_fitting = TRUE. The default is 10L.
source_model: Character string specifying the (penalized) multinomial logistic regression for estimating the source model. It has two options: "MN.glmnet" (default) and "MN.nnet", which use glmnet and nnet respectively.
source_model_args: List specifying the arguments for the source model (in glmnet or nnet).
treatment_model_type: Character string specifying how the treatment model is estimated. Options include "separate" (default) and "joint". If "separate", the treatment model (i.e., $P(A=1|X, S=s)$) is estimated by regressing $A$ on $X$ within each specific internal population $S=s$. If "joint", the treatment model is estimated by regressing $A$ on $X$ and $S$ using the multi-source population.
treatment_model_args: List specifying the arguments for the treatment model (in SuperLearner).
outcome_model_args: List specifying the arguments for the outcome model (in SuperLearner).
show_progress: Logical specifying whether to print a progress bar for the cross-fit replicates completed, if cross_fitting = TRUE.

Details

Data structure:

The multi-source dataset consists the outcome Y, source S, treatment A, and covariates X ($n \times p$) in the internal populations. The data sources can be trials, observational studies, or a combination of both.

Estimation of nuissance parameters:

The following models are fit:

Propensity score model: $\eta_a(X)=P(A=a|X)$. We perform the decomposition $P(A=a|X)=\sum_{s} P(A=a|X, S=s)P(S=s|X)$ and estimate $P(A=1|X, S=s)$ (i.e., the treatment model) and $q_s(X)=P(S=s|X)$ (i.e., the source model).
Outcome model: $\mu_a(X)=E(Y|X, A=a)$

The models are estimated by SuperLearner with the exception of the source model which is estimated by glmnet or nnet.

ATE estimation:

The ATE estimator is $$ \dfrac{\widehat \kappa}{n}\sum\limits_{i=1}^{n} \Bigg[ I(S_i = s) \widehat \mu_a(X_i) +I(A_i = a) \dfrac{\widehat q_{s}(X_i)}{\widehat \eta_a(X_i)} \Big\{ Y_i - \widehat \mu_a(X_i) \Big\} \Bigg], $$ where $\widehat \kappa=\{n^{-1} \sum_{i=1}^n I(S_i=s)\}^{-1}$. The estimator is doubly robust and non-parametrically efficient.

To achieve non-parametric efficiency and asymptotic normality, it requires that $||\widehat \mu_a(X) -\mu_a(X)||\big\{||\widehat \eta_a(X) -\eta_a(X)||+||\widehat q_s(X) -q_s(X)||\big\}=o_p(n^{-1/2})$. In addition, sample splitting and cross-fitting can be performed to avoid the Donsker class assumption.

When a data source is a randomized trial, it is still recommended to estimate the propensity score for optimal efficiency.

References

Robertson, S.E., Steingrimsson, J.A., Joyce, N.R., Stuart, E.A. and Dahabreh, I.J. (2021). Center-specific causal inference with multicenter trials: Reinterpreting trial evidence in the context of each participating center. arXiv preprint arXiv:2104.05905.

Wang, G., McGrath, S., Lian, Y. and Dahabreh, I. (2024) CausalMetaR: An R package for performing causally interpretable meta-analyses, arXiv preprint arXiv:2402.04341.

Examples

Run this code

# \donttest{
ai <- ATE_internal(
  X = dat_multisource[, 1:10],
  Y = dat_multisource$Y,
  S = dat_multisource$S,
  A = dat_multisource$A,
  source_model = "MN.glmnet",
  source_model_args = list(),
  treatment_model_type = "separate",
  treatment_model_args = list(
    family = binomial(),
    SL.library = c("SL.glmnet", "SL.nnet", "SL.glm"),
    cvControl = list(V = 5L)
  ),
  outcome_model_args = list(
    family = gaussian(),
    SL.library = c("SL.glmnet", "SL.nnet", "SL.glm"),
    cvControl = list(V = 5L)
  )
)
# }

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