# 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)
## Super Learner
## Version: 2.0-25
## Package created on 2019-08-05
## Welcome to the tmle package, version 1.4.0.1
##
## Use tmleNews() to see details on changes and bug fixes
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
##   1.149   0.067   1.220
time_preorder
##    user  system elapsed
##   0.754   0.017   0.772

Show the brief results from greedy CTMLE:

ctmle_discrete_fit1
## C-TMLE result:
##  parameter estimate:  1.99642
##  estimated variance:  0.00905
##             p-value:  <2e-16
##   95% conf interval: (1.80998, 2.18287)

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.
## ----------------------------------------------------------------------
## cand 1 (intercept)           1     4.22   19.8   0.0794     13991
## cand 2          X2           1     3.22   19.5   0.0855     13751
## cand 3          X5           1     2.61   19.0   0.0897     13414
## cand 4          X1           1     2.00   18.2   0.0956     12877
## cand 5          X4           2     1.99   18.3   0.1112     12930
## cand 6          X3           3     2.01   18.3   0.1018     12930
## ----------------------------------------------------------------------
## Selected TMLE estimator is candidate 4
##
## 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:  1.99642
##  estimated variance:  0.00905
##             p-value:  <2e-16
##   95% conf interval: (1.80998, 2.18287)

## 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)
)
## Warning in c(-1.96, 1.96) * sqrt(var.psi): Recycling array of length 1 in vector-array arithmetic is deprecated.
##   Use c() or as.vector() instead.

Les't compare the running time for each LASSO-C-TMLE

time_ctmlelasso1
##    user  system elapsed
##  14.745   0.296  15.084
time_ctmlelasso2
##    user  system elapsed
##  19.292   0.453  19.829
time_ctmlelasso3
##    user  system elapsed
##   0.005   0.000   0.005

Finally, we compared three C-TMLE estimates:

ctmle_fit1
## C-TMLE result:
##  parameter estimate:  2.19644
##  estimated variance:  0.10065
##             p-value:  4.4125e-12
##   95% conf interval: (1.57462, 2.81826)
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] 0.00271545

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

glmnet_fit\$lambda.min
## [1] 0.03065303

## 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)
## 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_All

Lets 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] 5

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
## C-TMLE result:
##  parameter estimate:  2.19494
##  estimated variance:  0.08348
##             p-value:  3.0302e-14
##   95% conf interval: (1.62865, 2.76122)

## Citation

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.