npRR: model-robust inference for the conditional relative risk using targeted learning

Nonparametric inference for the conditional relative risk function using targeted machine learning

This package contains the R function npRRWorkingModel(), which implements a targeted maximum likelihood estimator (TMLE) for the coefficients of the projection of the conditional log relative risk function onto a user-specified linear working model. This TMLE can be viewed as a model-robust version of the semiparametric TMLE studied in Tuglus, Porter, van der Laan (2011).

The function requires initial estimates of the outcome regression and propensity score which can be passed directly to npRRWorkingModel() or can be estimated internally using the sl3 ensemble machine learning R package.

Data-structure

This package is based on the data structure (W, A, Y) where W is a covariate, A is a binary treatment assignment, and Y is a binary or nonnegative outcome. The estimand of interest is the conditional relative risk function: RR(w) := E[Y | A = 1, W = w] / E[Y | A = 0, W = w]. This package supports observation weights, which can be used to adjust for outcome missingness/censoring.

The function npRRWorkingModel() provides estimate and inference for a working Poisson likelihood-based projection of the conditional relative risk onto a user-speciifed log-linear parametric model. Coefficient estimates (b) are returned based on the working model RR(w) ~ exp{b^T f(w)} where f is a known, user-specified transformation of the covariate vector W.

Inputs:

The npRRWorkingModel() function requires the following arguments:

1. formula_LRR: A formula object specifying the linear working model for the log relative risk function.
2. W: numeric matrix containing covariate information (e.g. possible confounders).
3. A: binary vector of treatment assignments.
4. Y: A vector of binary or nonnegative outcome values.
5. weights: observation weights. No weights corresponds with the argument weights = rep(1, length(Y)).
6. EY1W : vector of estimates for E[Y|A=1,W] (see also sl3_Learner_EYAW argument).
7. EY0W : vector of estimates for E[Y|A=0,W] (see also sl3_Learner_EYAW argument).
8. pA1W : vector of estimates for P(A=1|W) (see also sl3_Learner_pA1W argument).

Output of npRRWorkingModel()

The output of npRRWorkingModel() is a list object that includes:

1. Coefficient estimates
2. Z-scores and p-values for coefficients
3. 95% confidence intervals for coefficients

Example code

library(npRR)
library(sl3)
# relative risk
n <- 250
W <- runif(n, min = -1,  max = 1)
A <- rbinom(n, size = 1, prob = plogis(W))
Y <- rpois(n, lambda = exp( A * (1 + W + 2*W^2)  + sin(5 * W)))
formula = ~ 1 + W
# Estimate nuisance function internally using sl3 package and generalized additive models
fit <- npRRWorkingModel(formula_LRR = formula,
                        W = W, A = A, Y = Y,
                        weights = rep(1,n), 
                        sl3_Learner_EYAW = Lrnr_gam$new(),
                        sl3_Learner_pA1W = Lrnr_gam$new()
                        )
coef(fit)

References

1. Tuglus, Cathy, Kristin E. Porter, and Mark J. van der Laan. "Targeted Maximum Likelihood Estimation of Conditional Relative Risk in a Semi-parametric Regression Model." (2011).
2. Van der Laan, Mark J., and Sherri Rose. Targeted learning: causal inference for observational and experimental data. Vol. 4. New York: Springer, 2011.
3. Van Der Laan, Mark J., and Daniel Rubin. "Targeted maximum likelihood learning." The international journal of biostatistics 2.1 (2006).
4. Coyle, Jeremy R and Hejazi, Nima S and Malenica, Ivana and Phillips, Rachael V and Sofrygin, Oleg , sl3: Modern Pipelines for Machine Learning and {Super Learning, 2021, R package version 1.4.2