Utilities for Improving Estimation Efficiency via Augmentation and for Propensity Score Estimation

In personalized: Estimation and Validation Methods for Subgroup Identification and Personalized Medicine

Efficiency augmentation

To demonstrate how to use efficiency augmentation and the propensity score utilities available in the personalized package, we simulate data with two treatments. The treatment assignments are based on covariates and hence mimic an observational setting with no unmeasured confounders.

library(personalized)

In this simulation, the treatment assignment depends on covariates and hence we must model the propensity score $\pi(x) = Pr(T = 1 | X = x)$. In this simulation we will assume that larger values of the outcome are better.

library(personalized)

set.seed(1)
n.obs  <- 500
n.vars <- 10
x <- matrix(rnorm(n.obs * n.vars, sd = 3), n.obs, n.vars)

# simulate non-randomized treatment
xbetat   <- 0.5 + 0.25 * x[,9] - 0.25 * x[,1]
trt.prob <- exp(xbetat) / (1 + exp(xbetat))
trt      <- rbinom(n.obs, 1, prob = trt.prob)

# simulate delta
delta <- (0.5 + x[,2] - 0.5 * x[,3] - 1 * x[,1] + 1 * x[,1] * x[,4] )

# simulate main effects g(X)
xbeta <- 2 * x[,1] + 3 * x[,4] - 0.25 * x[,2]^2 + 2 * x[,3] + 0.25 * x[,5] ^ 2
xbeta <- xbeta + delta * (2 * trt - 1)

# simulate continuous outcomes
y <- drop(xbeta) + rnorm(n.obs, sd = 3)

Propensity score utilities

Estimation of the propensity score is a fundamental aspect of the estimation of individualized treatment rules (ITRs). The personalized package offers support tools for construction of the propensity score function used by the fit.subgroup() function. The support is via the create.propensity.function() function. This tool allows for estimation of the propensity score in high dimensions via glmnet. In high dimensions it can be important to account for regularization bias via cross-fitting (https://arxiv.org/abs/1608.00060); the create.propensity.function() offers a cross-fitting approach for high-dimensional propensity score estimation. A basic usage of this function with cross-fitting (with 4 folds; normally we would set this larger, but have reduced it to make computation time shorter) is as follows:

# arguments can be passed to cv.glmnet via `cv.glmnet.args`
prop.func <- create.propensity.function(crossfit = TRUE,
                                        nfolds.crossfit = 4,
                                        cv.glmnet.args = list(type.measure = "auc", nfolds = 3))

prop.func can then be passed to fit.subgroup() as follows:

We have set nfolds to 3 for computational reasons; it should generally be higher, such as 10.

subgrp.model <- fit.subgroup(x = x, y = y,
                             trt = trt,
                             propensity.func = prop.func,
                             loss   = "sq_loss_lasso",
                             nfolds = 3)    # option for cv.glmnet (for ITR estimation)

summary(subgrp.model)

Augmentation utilities

Efficiency in estimating ITRs can be improved by including an augmentation term. The optimal augmentation term generally is a function of the main effects of the full outcome regression model marginalized over the treatment. Especially in high dimensions, regularization bias can potentially have a negative impact on performance. Cross-fitting is again another reasonable approach to circumventing this issue. Augmentation functions can be constructed (with cross-fitting as an option) via the create.augmentation.function() function, which works similarly as the create.propensity.function() function. The basic usage is as follows:

aug.func <- create.augmentation.function(family = "gaussian",
                                         crossfit = TRUE,
                                         nfolds.crossfit = 4,
                                         cv.glmnet.args = list(type.measure = "mae", nfolds = 3))

We have set nfolds to 3 for computational reasons; it should generally be higher, such as 10.

aug.func can be used for augmentation by passing it to fit.subgroup() like:

subgrp.model.aug <- fit.subgroup(x = x, y = y,
                             trt = trt,
                             propensity.func = prop.func,
                             augment.func = aug.func,
                             loss   = "sq_loss_lasso",
                             nfolds = 3)    # option for cv.glmnet (for ITR estimation)

summary(subgrp.model.aug)

Comparing performance with augmentation

We first run the training/testing procedure to assess the performance of the non-augmented estimator:

valmod <- validate.subgroup(subgrp.model, B = 3,
                            method = "training_test",
                            train.fraction = 0.75)
valmod

Then we compare with the augmented estimator. Although this is based on just 3 replications, we can see that the augmented estimator is better at discriminating between benefitting and non-benefitting patients, as evidenced by the large treatment effect among those predicted to benefit (and smaller standard error) by the augmented estimator versus the smaller conditional treatment effect above.

valmod.aug <- validate.subgroup(subgrp.model.aug, B = 3,
                                method = "training_test",
                                train.fraction = 0.75)
valmod.aug

personalized documentation built on June 28, 2022, 1:06 a.m.