Ctmle
Collaborative Targeted Maximum Likelihood Estimation
Install / Use
/learn @jucheng1992/CtmleREADME
Collaborative Targeted Maximum Likelihood Estimation
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).
Installation
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')
C-TMLE for variable selection
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 1.008
Show 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)
LASSO-C-TMLE for model selection of LASSO
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$lambda.min))] * (1+5e-2)
gcvPrev <- predict.cv.glmnet(glmnet_fit,newx = W,s = s_prev,type="response")
gcvPrev <- bound(gcvPrev,c(0.025,0.975))
time_ctmlelasso3 <- system.time(
ctmle_fit3 <- ctmleGlmnet(Y = Y, A = A, W = W, Q = Q,
ctmletype=3, g1W = gcv, g1WPrev = gcvPrev,
family="gaussian",
gbound=0.025, V = 5)
)
Les't compare the running time for each LASSO-C-TMLE
time_ctmlelasso1
#> user system elapsed
#> 15.005 0.104 15.266
time_ctmlelasso2
#> user system elapsed
#> 18.351 0.083 18.528
time_ctmlelasso3
#> user system elapsed
#> 0.005 0.000 0.006
Finally, we compare three C-TMLE estimates:
ctmle_fit1
#> C-TMLE result:
#> parameter estimate: 2.20368
#> estimated variance: 0.09796
#> p-value: 1.9124e-12
#> 95% conf interval: (1.59022, 2.81714)
ctmle_fit2
#> C-TMLE result:
#> parameter estimate: 2.16669
#> estimated variance: 0.05327
#> p-value: <2e-16
#> 95% conf interval: (1.71429, 2.61908)
ctmle_fit3
#> C-TMLE result:
#> parameter estimate: 2.02388
#> estimated variance: 0.04972
#> p-value: <2e-16
#> 95% conf interval: (1.58684, 2.46093)
Show which regularization parameter (lambda) is selected by C-TMLE1:
lambdas[ctmle_fit1$best_k]
#> [1] 0.004409285
In comparison, we show which regularization parameter (lambda) is selected by cv.glmnet:
glmnet_fit$lambda.min
#> [1] 0.03065303
