ATE_external {CausalMetaR}

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

Description

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

Usage

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

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.

X_external

Data frame (or matrix) containing the covariate data in the external target population. It should have n_0 rows and p columns. This is the external data counterpart to the X argument.

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).

external_model_args

List specifying the arguments for the external 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.

The external dataset contains only covariates X_external (n_0 \times p).

Estimation of nuissance parameters:

The following models are fit:

• External model: q(X)=P(R=1|X), where R takes value 1 if the subject belongs to any of the internal dataset and 0 if the subject belongs to the external dataset

• 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 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(R_i = 0) \widehat \mu_a(X_i) +I(A_i = a, R_i=1) \dfrac{1-\widehat q(X_i)}{\widehat \eta_a(X_i)\widehat q(X_i)} \Big\{ Y_i - \widehat \mu_a(X_i) \Big\} \Bigg],

where N=n+n_0, and \widehat \kappa=\{N^{-1} \sum_{i=1}^N I(R_i=0)\}^{-1}.

This 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(X) -q(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.

Value

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

df_dif

A data frame containing the treatment effect (mean difference) estimates for the extenal data.

df_A0

A data frame containing the potential outcome mean estimates under A = 0 for the extenal data.

df_A1

A data frame containing the potential outcome mean estimates under A = 1 for the extenal data.

fit_outcome

Fitted outcome model.

fit_source

Fitted source model.

fit_treatment

Fitted treatment model(s).

fit_external

Fitted external model.

References

Dahabreh, I.J., Robertson, S.E., Petito, L.C., Hernán, M.A. and Steingrimsson, J.A. (2019) Efficient and robust methods for causally interpretable meta‐analysis: Transporting inferences from multiple randomized trials to a target population, Biometrics.

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

ae <- ATE_external(
  X = dat_multisource[, 1:10],
  Y = dat_multisource$Y,
  S = dat_multisource$S,
  A = dat_multisource$A,
  X_external = dat_external[, 1:10],
  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)
  ),
  external_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)
  )
)