Collaborative Targeted Maximum Likelihood Estimation (C-TMLE) is an extention of Targeted Maximum Likelihood Estimation (TMLE). It applies variable/model selection for nuisance parameter (e.g. the propensity score) estimation in a 'collaborative' way, by directly optimizing the empirical metric on the causal estimator.
In this package, we implemented the general template of C-TMLE, for the estimation of the average treatment effect (ATE).
The package also offers convenient functions for discrete C-TMLE for variable selection, and LASSO-C-TMLE for model selection of LASSO, in estimation of the propensity score (PS).
To install the CRAN release version of ctmle:
install.packages('ctmle')To install the development version (requires the devtools package):
devtools::install_github('jucheng1992/ctmle')In this section, we start with examples of discrete C-TMLE for variable selection, using greedy forward searching, and scalable discrete C-TMLE with pre-ordering option.
library(ctmle)
#> Loading required package: SuperLearner
#> Loading required package: nnls
#> Super Learner
#> Version: 2.0-22
#> Package created on 2017-07-18
#> Loading required package: tmle
#> Welcome to the tmle package, version 1.2.0-5
#>
#> Use tmleNews() to see details on changes and bug fixes
#> Loading required package: glmnet
#> Loading required package: Matrix
#> Loading required package: foreach
#> Loaded glmnet 2.0-10
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
set.seed(123)
N <- 1000
p = 5
Wmat <- matrix(rnorm(N * p), ncol = p)
beta1 <- 4+2*Wmat[,1]+2*Wmat[,2]+2*Wmat[,5]
beta0 <- 2+2*Wmat[,1]+2*Wmat[,2]+2*Wmat[,5]
tau <- 2
gcoef <- matrix(c(-1,-1,rep(-(3/((p)-2)),(p)-2)),ncol=1)
W <- as.matrix(Wmat)
g <- 1/(1+exp(W%*%gcoef /3))
A <- rbinom(N, 1, prob = g)
epsilon <-rnorm(N, 0, 1)
Y <- beta0 + tau * A + epsilon
# With initial estimate of Q
Q <- cbind(rep(mean(Y[A == 0]), N), rep(mean(Y[A == 1]), N))
time_greedy <- system.time(
ctmle_discrete_fit1 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat), Q = Q,
preOrder = FALSE, detailed = TRUE)
)
ctmle_discrete_fit2 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat),
preOrder = FALSE, detailed = TRUE)
time_preorder <- system.time(
ctmle_discrete_fit3 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat), Q = Q,
preOrder = TRUE,
order = rev(1:p), detailed = TRUE)
)Scalable (discrete) C-TMLE takes much less computation time:
time_greedy
#> user system elapsed
#> 1.589 0.045 1.646
time_preorder
#> user system elapsed
#> 0.994 0.012 1.008Show the brief results from greedy CTMLE:
ctmle_discrete_fit1
#> C-TMLE result:
#> parameter estimate: 1.99472
#> estimated variance: 0.00838
#> p-value: <2e-16
#> 95% conf interval: (1.81533, 2.1741)Summary function offers detial information of which variable is selected.
summary(ctmle_discrete_fit1)
#>
#> Number of candidate TMLE estimators created: 6
#> A candidate TMLE estimator was created at each move, as each new term
#> was incorporated into the model for g.
#> ----------------------------------------------------------------------
#> term added cleverCovar estimate cv-RSS cv-varIC cv-penRSS
#> cand 1 (intercept) 1 4.22 19.9 0.0788 14045
#> cand 2 X2 1 3.22 19.6 0.0851 13818
#> cand 3 X5 1 2.61 19.1 0.0870 13485
#> cand 4 X1 1 2.00 18.3 0.0955 12945
#> cand 5 X4 2 1.99 18.3 0.0950 12937
#> cand 6 X3 3 2.01 18.3 0.1008 12941
#> ----------------------------------------------------------------------
#> Selected TMLE estimator is candidate 5
#>
#> Each TMLE candidate was created by fluctuating the initial fit, Q0(A,W)=E[Y|A,W], obtained in stage 1.
#>
#> cand 1: Q1(A,W) = Q0(A,W) + epsilon1a * h1a
#> h1a is based on an intercept-only model for treatment mechanism g(A,W)
#>
#> cand 2: Q2(A,W) = Q0(A,W) + epsilon1b * h1b
#> h1b is based on a treatment mechanism model containing covariates X2
#>
#> cand 3: Q3(A,W) = Q0(A,W) + epsilon1c * h1c
#> h1c is based on a treatment mechanism model containing covariates X2, X5
#>
#> cand 4: Q4(A,W) = Q0(A,W) + epsilon1d * h1d
#> h1d is based on a treatment mechanism model containing covariates X2, X5, X1
#>
#> cand 5: Q5(A,W) = Q0(A,W) + epsilon1d * h1d + epsilon2 * h2 = Q4(A,W) + epsilon2 * h2,
#> h2 is based on a treatment mechanism model containing covariates X2, X5, X1, X4
#>
#> cand 6: Q6(A,W) = Q0(A,W) + epsilon1d * h1d + epsilon2 * h2 + epsilon3 * h3 = Q5(A,W) + epsilon3 * h3,
#> h3 is based on a treatment mechanism model containing covariates X2, X5, X1, X4, X3
#>
#> ----------
#> C-TMLE result:
#> parameter estimate: 1.99472
#> estimated variance: 0.00838
#> p-value: <2e-16
#> 95% conf interval: (1.81533, 2.1741)In this section, we introduce the LASSO-C-TMLE algorithm for model selection of LASSO in the estimation of the propensity score. We implemented three variations of the LASSO-C-TMLE algorithm. For simplicity, we call them C-TMLE1-3. See technical details in the corresponding references.
# Generate high-dimensional data
set.seed(123)
N <- 1000
p = 100
Wmat <- matrix(rnorm(N * p), ncol = p)
beta1 <- 4 + 2 * Wmat[,1] + 2 * Wmat[,2] + 2 * Wmat[,5] + 2 * Wmat[,6] + 2 * Wmat[,8]
beta0 <- 2 + 2 * Wmat[,1] + 2 * Wmat[,2] + 2 * Wmat[,5] + 2 * Wmat[,6] + 2 * Wmat[,8]
tau <- 2
gcoef <- matrix(c(-1,-1,rep(-(3/((p)-2)),(p)-2)),ncol=1)
W <- as.matrix(Wmat)
g <- 1/(1+exp(W%*%gcoef /3))
A <- rbinom(N, 1, prob = g)
epsilon <-rnorm(N, 0, 1)
Y <- beta0 + tau * A + epsilon
# With initial estimate of Q
Q <- cbind(rep(mean(Y[A == 0]), N), rep(mean(Y[A == 1]), N))
glmnet_fit <- cv.glmnet(y = A, x = W, family = 'binomial', nlambda = 20)We start build a sequence of lambdas from the lambda selected by cross-validation, as the model selected by cv.glmnet would over-smooth w.r.t. the target parameter.
lambdas <- glmnet_fit$lambda[(which(glmnet_fit$lambda==glmnet_fit$lambda.min)):length(glmnet_fit$lambda)]We fit C-TMLE1 algorithm by feed the algorithm with a vector of lambda, in decreasing order:
time_ctmlelasso1 <- system.time(
ctmle_fit1 <- ctmleGlmnet(Y = Y, A = A,
W = data.frame(W = W),
Q = Q, lambdas = lambdas, ctmletype=1,
family="gaussian",gbound=0.025, V=5)
)We fit C-TMLE2 algorithm:
time_ctmlelasso2 <- system.time(
ctmle_fit2 <- ctmleGlmnet(Y = Y, A = A,
W = data.frame(W = W),
Q = Q, lambdas = lambdas, ctmletype=2,
family="gaussian",gbound=0.025, V=5)
)For C-TMLE3, we need two gn estimators, one with lambda selected by cross-validation, and the other with lambda slightly different from the selected lambda:
gcv <- predict.cv.glmnet(glmnet_fit, newx=W, s="lambda.min",type="response")
gcv <- bound(gcv,c(0.025,0.975))
s_prev <- glmnet_fit$lambda[(which(glmnet_fit$lambda == glmnet_fit$lambda.min))] * (1+5e-2)
gcvPrev <- predict.cv.glmnet(glmnet_fit,newx = W,s = s_prev,type="response")
gcvPrev <- bound(gcvPrev,c(0.025,0.975))
time_ctmlelasso3 <- system.time(
ctmle_fit3 <- ctmleGlmnet(Y = Y, A = A, W = W, Q = Q,
ctmletype=3, g1W = gcv, g1WPrev = gcvPrev,
family="gaussian",
gbound=0.025, V = 5)
)Les't compare the running time for each LASSO-C-TMLE
time_ctmlelasso1
#> user system elapsed
#> 15.005 0.104 15.266
time_ctmlelasso2
#> user system elapsed
#> 18.351 0.083 18.528
time_ctmlelasso3
#> user system elapsed
#> 0.005 0.000 0.006Finally, we compare three C-TMLE estimates:
ctmle_fit1
#> C-TMLE result:
#> parameter estimate: 2.20368
#> estimated variance: 0.09796
#> p-value: 1.9124e-12
#> 95% conf interval: (1.59022, 2.81714)
ctmle_fit2
#> C-TMLE result:
#> parameter estimate: 2.16669
#> estimated variance: 0.05327
#> p-value: <2e-16
#> 95% conf interval: (1.71429, 2.61908)
ctmle_fit3
#> C-TMLE result:
#> parameter estimate: 2.02388
#> estimated variance: 0.04972
#> p-value: <2e-16
#> 95% conf interval: (1.58684, 2.46093)Show which regularization parameter (lambda) is selected by C-TMLE1:
lambdas[ctmle_fit1$best_k]
#> [1] 0.004409285In comparison, we show which regularization parameter (lambda) is selected by cv.glmnet:
glmnet_fit$lambda.min
#> [1] 0.03065303In this section, we briefly introduce the general template of C-TMLE. In this function, the gn candidates could be a user-specified matrix, each column stand for the estimated PS for each unit. The estimators should be ordered by their empirical fit.
As C-TMLE requires cross-validation, it needs two gn estimate: one from cross-validated prediction, one from a vanilla prediction. For example, consider 5-folds cross-validation, where argument folds is the list of indices for each folds, then the (i,j)-th element in input gn_candidates_cv should be the predicted value of i-th unit, predicted by j-th unit, trained by other 4 folds where all of them do not contain i-th unit. gn_candidates should be just the predicted PS for each estimator trained on the whole data.
We could easily use SuperLearner package and build_gn_seq function to easily achieve this:
lasso_fit <- cv.glmnet(x = as.matrix(W), y = A, alpha = 1, nlambda = 100, nfolds = 10)
lasso_lambdas <- lasso_fit$lambda[lasso_fit$lambda <= lasso_fit$lambda.min][1:5]
# Build SL template for glmnet
SL.glmnet_new <- function(Y, X, newX, family, obsWeights, id, alpha = 1,
nlambda = 100, lambda = 0,...){
# browser()
if (!is.matrix(X)) {
X <- model.matrix(~-1 + ., X)
newX <- model.matrix(~-1 + ., newX)
}
fit <- glmnet::glmnet(x = X, y = Y,
lambda = lambda,
family = family$family, alpha = alpha)
pred <- predict(fit, newx = newX, type = "response")
fit <- list(object = fit)
class(fit) <- "SL.glmnet"
out <- list(pred = pred, fit = fit)
return(out)
}
# Use a sequence of estimator to build gn sequence:
SL.cv1lasso <- function (... , alpha = 1, lambda = lasso_lambdas[1]){
SL.glmnet_new(... , alpha = alpha, lambda = lambda)
}
SL.cv2lasso <- function (... , alpha = 1, lambda = lasso_lambdas[2]){
SL.glmnet_new(... , alpha = alpha, lambda = lambda)
}
SL.cv3lasso <- function (... , alpha = 1, lambda = lasso_lambdas[3]){
SL.glmnet_new(... , alpha = alpha, lambda = lambda)
}
SL.cv4lasso <- function (... , alpha = 1, lambda = lasso_lambdas[4]){
SL.glmnet_new(... , alpha = alpha, lambda = lambda)
}
SL.library = c('SL.cv1lasso', 'SL.cv2lasso', 'SL.cv3lasso', 'SL.cv4lasso', 'SL.glm')Construct the object folds, which is a list of indices for each fold
V = 5
folds <-by(sample(1:N,N), rep(1:V, length=N), list)Use folds and SuperLearner template to compute gn_candidates and gn_candidates_cv
gn_seq <- build_gn_seq(A = A, W = W, SL.library = SL.library, folds = folds)
#> Number of covariates in All is: 100
#> CV SL.cv1lasso_All
#> CV SL.cv2lasso_All
#> CV SL.cv3lasso_All
#> CV SL.cv4lasso_All
#> CV SL.glm_All
#> Number of covariates in All is: 100
#> CV SL.cv1lasso_All
#> CV SL.cv2lasso_All
#> CV SL.cv3lasso_All
#> CV SL.cv4lasso_All
#> CV SL.glm_All
#> Number of covariates in All is: 100
#> CV SL.cv1lasso_All
#> CV SL.cv2lasso_All
#> CV SL.cv3lasso_All
#> CV SL.cv4lasso_All
#> CV SL.glm_All
#> Number of covariates in All is: 100
#> CV SL.cv1lasso_All
#> CV SL.cv2lasso_All
#> CV SL.cv3lasso_All
#> CV SL.cv4lasso_All
#> CV SL.glm_All
#> Number of covariates in All is: 100
#> CV SL.cv1lasso_All
#> CV SL.cv2lasso_All
#> CV SL.cv3lasso_All
#> CV SL.cv4lasso_All
#> CV SL.glm_All
#> Non-Negative least squares convergence: TRUE
#> full SL.cv1lasso_All
#> full SL.cv2lasso_All
#> full SL.cv3lasso_All
#> full SL.cv4lasso_All
#> full SL.glm_AllLets look at the output of build_gn_seq
gn_seq$gn_candidates %>% dim
#> [1] 1000 5
gn_seq$gn_candidates_cv %>% dim
#> [1] 1000 5
gn_seq$folds %>% length
#> [1] 5Then we could use ctmleGeneral algorithm. As input estimator is already trained, it is much faster than previous C-TMLE algorithms.
Note: we recommand use the same folds as build_gn_seq for ctmleGeneral, to make cross-validation objective.
ctmle_general_fit1 <- ctmleGeneral(Y = Y, A = A, W = W, Q = Q,
ctmletype = 1,
gn_candidates = gn_seq$gn_candidates,
gn_candidates_cv = gn_seq$gn_candidates_cv,
folds = folds, V = 5)
ctmle_general_fit1
#> C-TMLE result:
#> parameter estimate: 2.19494
#> estimated variance: 0.08348
#> p-value: 3.0302e-14
#> 95% conf interval: (1.62865, 2.76122)If you used ctmle package in your research, please cite:
Ju, Cheng; Susan, Gruber; van der Laan, Mark J.; ctmle: Collaborative Targeted Maximum Likelihood Estimation. R package version 0.1.1, https://CRAN.R-project.org/package=ctmle.
@Manual{,
title = {ctmle: Collaborative Targeted Maximum Likelihood Estimation},
author = {Cheng Ju and Susan Gruber and Mark van der Laan},
year = {2017},
note = {R package version 0.1.1},
url = {https://CRAN.R-project.org/package=ctmle},
}Ju, Cheng, Schwab, Joshua, & van der Laan, Mark J. (2017). On Adaptive Propensity Score Truncation in Causal Inference. arXiv preprint arXiv:1707.05861 (2017).
Ju, Cheng; Wyss, Richard; Franklin, Jessica M.; Schneeweiss, Sebastian; Häggström, Jenny; van der Laan, Mark J.. "Collaborative-controlled LASSO for Constructing Propensity Score-based Estimators in High-Dimensional Data", arXiv preprint arXiv: 1706.10029 (2017). (To appear in Statistical Methods in Medical Research)
Ju, Cheng; Gruber, Susan; Lendle, Samuel D.; Chambaz, Antoine; Franklin, Jessica M.; Wyss, Richard; Schneeweiss, Sebastian; and van der Laan, Mark J.. "Scalable Collaborative Targeted Learning for High-dimensional Data", Statistical Methods in Medical Research (2017), https://doi.org/10.1177/0962280217729845.
Susan, Gruber, and van rder Laan, Mark J.. "An Application of Collaborative Targeted Maximum Likelihood Estimation in Causal Inference and Genomics." The International Journal of Biostatistics 6.1 (2010): 1-31.
van der Laan, Mark J., and Susan Gruber. "Collaborative double robust targeted maximum likelihood estimation." The international journal of biostatistics 6.1 (2010): 1-71.
In preperation