CTMLE Vignette

In ctmle: Collaborative Targeted Maximum Likelihood Estimation

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')

Collaborative Targeted Maximum Likelihood Estimation

In this package, we implemented the general template of C-TMLE, for estimation of average additive treatment effect (ATE). The package also offers the functions for discrete C-TMLE, which could be used for variable selection, and C-TMLE for model selection of LASSO.

C-TMLE for variable selection

In this section, we start with examples of discrete C-TMLE for variable selection, using greedy forward searhcing, and scalable discrete C-TMLE with pre-ordering option.

library(ctmle)
library(dplyr)
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
time_preorder

Show the brief results from greedy CTMLE:

ctmle_discrete_fit1

Summary function offers detial information of which variable is selected.

summary(ctmle_discrete_fit1)

C-TMLE LASSO for model selection of LASSO

In this section, we introduce the C-TMLE algorithms for model selection of LASSO in the estimation of propensity core, and for simplicity we call them LASSO C-TMLE algorithm. We have three variacions of C-TMLE LASSO algorithms, 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 suggest 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 <- stats::predict(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 <- stats::predict(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
time_ctmlelasso2
time_ctmlelasso3

Finally, we compared three C-TMLE estimates:

ctmle_fit1
ctmle_fit2
ctmle_fit3

Show which regularization parameter (lambda) is selected by C-TMLE1:

lambdas[ctmle_fit1$best_k]

In comparison, show which regularization parameter (lambda) is selected by cv.glmnet:

glmnet_fit$lambda.min

Advanced topic: the general template of C-TMLE

In 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)

Lets look at the output of build_gn_seq

gn_seq$gn_candidates %>% dim
gn_seq$gn_candidates_cv %>% dim
gn_seq$folds %>% length

Then 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

Citation

If you used ctmle package in your research, please cite:

Ju, Cheng; Susan, Gruber; van der Laan, Mark J.; ctmle: Variable and Model Selection for Causal Inference with Collaborative Targeted Maximum Likelihood Estimation

References

C-TMLE LASSO and C-TMLE for Model Selection

Ju, Cheng; Benkeser, David; van der Laan, Mark; "Robust inference on the average treatment effect using the outcome highly adaptive lasso", Biometrics, https://doi.org/10.1111/biom.13121

Scalable Discrete C-TMLE with Pre-ordering

Ju, Cheng; Gruber, Susan; Lendle. S. D.; et al. Scalable collaborative targeted learning for high-dimensional data. Statistical methods in medical research, 2019, 28(2): 532-554.

Discrete C-TMLE with Greedy Search

Susan, Gruber, and van der 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.

General Template of C-TMLE

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.

ctmle documentation built on Dec. 16, 2019, 1:19 a.m.