README.md
In ck37/varimpact: Variable Importance Estimation Using Targeted Causal Inference (TMLE)
varimpact - variable importance through causal inference
Summary
varimpact uses causal inference statistics to generate variable
importance estimates for a given dataset and outcome. It answers the
question: which of my Xs are most related to my Y? Each variable’s
influence on the outcome is estimated semiparametrically, without
assuming a linear relationship or other functional form, and the
covariate list is ranked by order of importance. This can be used for
exploratory data analysis, for dimensionality reduction, for
experimental design (e.g. to determine blocking and re-randomization),
to reduce variance in an estimation procedure, etc. See Hubbard,
Kennedy, & van der Laan (2018) for more details, or Hubbard & van der
Laan (2016) for an earlier description.
Details
Each covariate is analyzed using targeted minimum loss-based estimation
(TMLE) as though it were a
treatment, with all other variables serving as adjustment variables via
SuperLearner. Then the
statistical significance of the estimated treatment effect for each
covariate determines the variable importance ranking. This formulation
allows the asymptotics of TMLE to provide valid standard errors and
p-values, unlike other variable importance algorithms.
The results provide raw p-values as well as p-values adjusted for false
discovery rate using the Benjamini-Hochberg (1995) procedure. Adjustment
variables are automatically clustered hierarchically using HOPACH (van
der Laan & Pollard 2003) in order to reduce dimensionality. The package
supports multi-core and multi-node parallelization, which are detected
and used automatically when a parallel backend is registered. Missing
values are automatically imputed using K-nearest neighbors (Troyanskaya
et al. 2001, Jerez et al. 2010) and missingness indicator variables are
incorporated into the analysis.
varimpact is under active development so please submit any bug reports
or feature requests to the issue
queue, or email Alan and/or
Chris directly.
Installation
GitHub
# Install remotes if necessary:
# install.packages("remotes")
remotes::install_github("ck37/varimpact")
CRAN
Forthcoming fall 2022
Examples
Example: basic functionality
library(varimpact)
#> Loading required package: SuperLearner
#> Loading required package: nnls
#> Super Learner
#> Version: 2.0-27-9000
#> Package created on 2021-03-28
####################################
# Create test dataset.
set.seed(1, "L'Ecuyer-CMRG")
N <- 300
num_normal <- 5
X <- as.data.frame(matrix(rnorm(N * num_normal), N, num_normal))
Y <- rbinom(N, 1, plogis(.2*X[, 1] + .1*X[, 2] - .2*X[, 3] + .1*X[, 3]*X[, 4] - .2*abs(X[, 4])))
# Add some missing data to X so we can test imputation.
for (i in 1:10) X[sample(nrow(X), 1), sample(ncol(X), 1)] <- NA
####################################
# Basic example
vim <- varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
# Review consistent and significant results.
vim
#> No significant and consistent results.
#> All results:
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered 0.17058432 (-0.0518 - 0.393) 0.06639069 0.3319535 1.3174241
#> V1 Ordered 0.03831094 (-0.158 - 0.234) 0.35081119 0.8770280 1.0724707
#> V3 Ordered -0.05171291 (-0.339 - 0.235) 0.63807247 0.8835731 0.9673808
#> V2 Ordered -0.06678388 (-0.307 - 0.174) 0.70685848 0.8835731 0.9305320
#> V5 Ordered -0.12419619 (-0.304 - 0.0561) 0.91152962 0.9115296 0.8468485
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.953 - 1.82) 0.0474590 0.2372950 TRUE
#> V1 (0.446 - 2.58) 0.4379413 0.8101835 TRUE
#> V3 (0.654 - 1.43) 0.5658283 0.8101835 FALSE
#> V2 (0.642 - 1.35) 0.6481468 0.8101835 FALSE
#> V5 (0.634 - 1.13) 0.8696742 0.8696742 FALSE
# Look at all results.
vim$results_all
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered 0.17058432 (-0.0518 - 0.393) 0.06639069 0.3319535 1.3174241
#> V1 Ordered 0.03831094 (-0.158 - 0.234) 0.35081119 0.8770280 1.0724707
#> V3 Ordered -0.05171291 (-0.339 - 0.235) 0.63807247 0.8835731 0.9673808
#> V2 Ordered -0.06678388 (-0.307 - 0.174) 0.70685848 0.8835731 0.9305320
#> V5 Ordered -0.12419619 (-0.304 - 0.0561) 0.91152962 0.9115296 0.8468485
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.953 - 1.82) 0.0474590 0.2372950 TRUE
#> V1 (0.446 - 2.58) 0.4379413 0.8101835 TRUE
#> V3 (0.654 - 1.43) 0.5658283 0.8101835 FALSE
#> V2 (0.642 - 1.35) 0.6481468 0.8101835 FALSE
#> V5 (0.634 - 1.13) 0.8696742 0.8696742 FALSE
# Plot the V2 impact.
plot_var("V2", vim)
# Generate latex tables with results.
exportLatex(vim)
#> NULL
# Clean up - will get a warning if there were no consistent results.
suppressWarnings({
file.remove(c("varimpByFold.tex", "varImpAll.tex", "varimpConsistent.tex"))
})
#> [1] TRUE TRUE FALSE
Example: customize outcome and propensity score estimation
Q_lib = c("SL.mean", "SL.glmnet", "SL.ranger", "SL.rpartPrune")
g_lib = c("SL.mean", "SL.glmnet")
set.seed(1, "L'Ecuyer-CMRG")
(vim = varimpact(Y = Y, data = X, Q.library = Q_lib, g.library = g_lib))
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
#> No significant and consistent results.
#> All results:
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered -0.02595001 (-0.25 - 0.198) 0.5897958 0.9863267 0.9926791
#> V3 Ordered -0.12688688 (-0.391 - 0.137) 0.8267887 0.9863267 0.8304033
#> V2 Ordered -0.11547591 (-0.355 - 0.124) 0.8277832 0.9863267 0.8529795
#> V1 Ordered -0.17014276 (-0.397 - 0.0571) 0.9288760 0.9863267 0.6823365
#> V5 Ordered -0.19094845 (-0.361 - -0.0213) 0.9863267 0.9863267 0.6210945
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.719 - 1.37) 0.5177707 0.9838749 FALSE
#> V3 (0.55 - 1.25) 0.7944201 0.9838749 FALSE
#> V2 (0.584 - 1.25) 0.8117625 0.9838749 FALSE
#> V1 (0.44 - 1.06) 0.9560802 0.9838749 TRUE
#> V5 (0.402 - 0.96) 0.9838749 0.9838749 FALSE
Example: parallel via multicore
library(future)
plan("multiprocess")
vim = varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
Example: parallel via SNOW
library(RhpcBLASctl)
# Detect the number of physical cores on this computer using RhpcBLASctl.
cl = parallel::makeCluster(get_num_cores())
plan("cluster", workers = cl)
vim = varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
parallel::stopCluster(cl)
Example: mlbench breast cancer
data(BreastCancer, package = "mlbench")
data = BreastCancer
# Create a numeric outcome variable.
data$Y = as.integer(data$Class == "malignant")
# Use multicore parallelization to speed up processing.
plan("multiprocess")
(vim = varimpact(Y = data$Y, data = subset(data, select = -c(Y, Class, Id))))
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 9
#> - Numeric variables: 0
#>
#> Estimating variable importance for 9 factors.
#> Significant and consistent results:
#> Type Estimate CI95 P-value Adj. p-value
#> Bare.nuclei Factor 0.6284939 (0.503 - 0.754) 0.000000e+00 0.000000e+00
#> Mitoses Factor 0.4097166 (0.336 - 0.483) 0.000000e+00 0.000000e+00
#> Cl.thickness Factor 0.5344847 (0.378 - 0.691) 1.040124e-11 2.340278e-11
#> Cell.size Factor 0.5577438 (0.386 - 0.729) 8.920165e-11 1.605630e-10
#> Est. RR CI95 RR P-value RR Adj. p-value RR
#> Bare.nuclei 3.697125 (2.15 - 6.35) 0.000000e+00 0.000000e+00
#> Mitoses 2.095869 (1.85 - 2.37) 7.227108e-12 3.252199e-11
#> Cl.thickness 3.103819 (2.23 - 4.31) 4.128421e-07 9.288948e-07
#> Cell.size 3.326385 (1.93 - 5.73) 1.062140e-06 1.911853e-06
plot_var("Mitoses", vim)
Authors
Alan E. Hubbard and Chris J. Kennedy, University of California, Berkeley
References
Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery
rate: a practical and powerful approach to multiple testing. Journal of
the royal statistical society. Series B (Methodological), 289-300.
Gruber, S., & van der Laan, M. J. (2012). tmle: An R Package for
Targeted Maximum Likelihood Estimation. Journal of Statistical Software,
51(i13).
Hubbard, A. E., Kennedy, C. J., van der Laan, M. J. (2018).
Data-adaptive target parameters. In M. van der Laan & S. Rose (2018)
Targeted Learning in Data Science. Springer.
Hubbard, A. E., Kherad-Pajouh, S., & van der Laan, M. J. (2016).
Statistical Inference for Data Adaptive Target Parameters. The
international journal of biostatistics, 12(1), 3-19.
Hubbard, A., Munoz, I. D., Decker, A., Holcomb, J. B., Schreiber, M. A.,
Bulger, E. M., … & Rahbar, M. H. (2013). Time-Dependent Prediction and
Evaluation of Variable Importance Using SuperLearning in High
Dimensional Clinical Data. The journal of trauma and acute care surgery,
75(1 0 1), S53.
Hubbard, A. E., & van der Laan, M. J. (2016). Mining with inference:
data-adaptive target parameters (pp. 439-452). In P. Bühlmann et
al. (Ed.), Handbook of Big Data. CRC Press, Taylor & Francis Group, LLC:
Boca Raton, FL.
Jerez, J. M., Molina, I., García-Laencina, P. J., Alba, E., Ribelles,
N., Martín, M., & Franco, L. (2010). Missing data imputation using
statistical and machine learning methods in a real breast cancer
problem. Artificial intelligence in medicine, 50(2), 105-115.
Rozenholc, Y., Mildenberger, T., & Gather, U. (2010). Combining regular
and irregular histograms by penalized likelihood. Computational
Statistics & Data Analysis, 54(12), 3313-3323.
Troyanskaya, O., Cantor, M., Sherlock, G., Brown, P., Hastie, T.,
Tibshirani, R., Botstein, D., & Altman, R. B. (2001). Missing value
estimation methods for DNA microarrays. Bioinformatics, 17(6), 520-525.
van der Laan, M. J. (2006). Statistical inference for variable
importance. The International Journal of Biostatistics, 2(1).
van der Laan, M. J., & Pollard, K. S. (2003). A new algorithm for hybrid
hierarchical clustering with visualization and the bootstrap. Journal of
Statistical Planning and Inference, 117(2), 275-303.
van der Laan, M. J., Polley, E. C., & Hubbard, A. E. (2007). Super
learner. Statistical applications in genetics and molecular biology,
6(1).
van der Laan, M. J., & Rose, S. (2011). Targeted learning: causal
inference for observational and experimental data. Springer Science &
Business Media.
ck37/varimpact documentation built on June 23, 2022, 4:41 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
varimpact - variable importance through causal inference
Summary
varimpact uses causal inference statistics to generate variable importance estimates for a given dataset and outcome. It answers the question: which of my Xs are most related to my Y? Each variable’s influence on the outcome is estimated semiparametrically, without assuming a linear relationship or other functional form, and the covariate list is ranked by order of importance. This can be used for exploratory data analysis, for dimensionality reduction, for experimental design (e.g. to determine blocking and re-randomization), to reduce variance in an estimation procedure, etc. See Hubbard, Kennedy, & van der Laan (2018) for more details, or Hubbard & van der Laan (2016) for an earlier description.
Details
Each covariate is analyzed using targeted minimum loss-based estimation (TMLE) as though it were a treatment, with all other variables serving as adjustment variables via SuperLearner. Then the statistical significance of the estimated treatment effect for each covariate determines the variable importance ranking. This formulation allows the asymptotics of TMLE to provide valid standard errors and p-values, unlike other variable importance algorithms.
The results provide raw p-values as well as p-values adjusted for false discovery rate using the Benjamini-Hochberg (1995) procedure. Adjustment variables are automatically clustered hierarchically using HOPACH (van der Laan & Pollard 2003) in order to reduce dimensionality. The package supports multi-core and multi-node parallelization, which are detected and used automatically when a parallel backend is registered. Missing values are automatically imputed using K-nearest neighbors (Troyanskaya et al. 2001, Jerez et al. 2010) and missingness indicator variables are incorporated into the analysis.
varimpact is under active development so please submit any bug reports or feature requests to the issue queue, or email Alan and/or Chris directly.
Installation
GitHub
# Install remotes if necessary:
# install.packages("remotes")
remotes::install_github("ck37/varimpact")
CRAN
Forthcoming fall 2022
Examples
Example: basic functionality
library(varimpact)
#> Loading required package: SuperLearner
#> Loading required package: nnls
#> Super Learner
#> Version: 2.0-27-9000
#> Package created on 2021-03-28
####################################
# Create test dataset.
set.seed(1, "L'Ecuyer-CMRG")
N <- 300
num_normal <- 5
X <- as.data.frame(matrix(rnorm(N * num_normal), N, num_normal))
Y <- rbinom(N, 1, plogis(.2*X[, 1] + .1*X[, 2] - .2*X[, 3] + .1*X[, 3]*X[, 4] - .2*abs(X[, 4])))
# Add some missing data to X so we can test imputation.
for (i in 1:10) X[sample(nrow(X), 1), sample(ncol(X), 1)] <- NA
####################################
# Basic example
vim <- varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
# Review consistent and significant results.
vim
#> No significant and consistent results.
#> All results:
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered 0.17058432 (-0.0518 - 0.393) 0.06639069 0.3319535 1.3174241
#> V1 Ordered 0.03831094 (-0.158 - 0.234) 0.35081119 0.8770280 1.0724707
#> V3 Ordered -0.05171291 (-0.339 - 0.235) 0.63807247 0.8835731 0.9673808
#> V2 Ordered -0.06678388 (-0.307 - 0.174) 0.70685848 0.8835731 0.9305320
#> V5 Ordered -0.12419619 (-0.304 - 0.0561) 0.91152962 0.9115296 0.8468485
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.953 - 1.82) 0.0474590 0.2372950 TRUE
#> V1 (0.446 - 2.58) 0.4379413 0.8101835 TRUE
#> V3 (0.654 - 1.43) 0.5658283 0.8101835 FALSE
#> V2 (0.642 - 1.35) 0.6481468 0.8101835 FALSE
#> V5 (0.634 - 1.13) 0.8696742 0.8696742 FALSE
# Look at all results.
vim$results_all
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered 0.17058432 (-0.0518 - 0.393) 0.06639069 0.3319535 1.3174241
#> V1 Ordered 0.03831094 (-0.158 - 0.234) 0.35081119 0.8770280 1.0724707
#> V3 Ordered -0.05171291 (-0.339 - 0.235) 0.63807247 0.8835731 0.9673808
#> V2 Ordered -0.06678388 (-0.307 - 0.174) 0.70685848 0.8835731 0.9305320
#> V5 Ordered -0.12419619 (-0.304 - 0.0561) 0.91152962 0.9115296 0.8468485
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.953 - 1.82) 0.0474590 0.2372950 TRUE
#> V1 (0.446 - 2.58) 0.4379413 0.8101835 TRUE
#> V3 (0.654 - 1.43) 0.5658283 0.8101835 FALSE
#> V2 (0.642 - 1.35) 0.6481468 0.8101835 FALSE
#> V5 (0.634 - 1.13) 0.8696742 0.8696742 FALSE
# Plot the V2 impact.
plot_var("V2", vim)
# Generate latex tables with results.
exportLatex(vim)
#> NULL
# Clean up - will get a warning if there were no consistent results.
suppressWarnings({
file.remove(c("varimpByFold.tex", "varImpAll.tex", "varimpConsistent.tex"))
})
#> [1] TRUE TRUE FALSE
Example: customize outcome and propensity score estimation
Q_lib = c("SL.mean", "SL.glmnet", "SL.ranger", "SL.rpartPrune")
g_lib = c("SL.mean", "SL.glmnet")
set.seed(1, "L'Ecuyer-CMRG")
(vim = varimpact(Y = Y, data = X, Q.library = Q_lib, g.library = g_lib))
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
#> No significant and consistent results.
#> All results:
#> Type Estimate CI95 P-value Adj. p-value Est. RR
#> V4 Ordered -0.02595001 (-0.25 - 0.198) 0.5897958 0.9863267 0.9926791
#> V3 Ordered -0.12688688 (-0.391 - 0.137) 0.8267887 0.9863267 0.8304033
#> V2 Ordered -0.11547591 (-0.355 - 0.124) 0.8277832 0.9863267 0.8529795
#> V1 Ordered -0.17014276 (-0.397 - 0.0571) 0.9288760 0.9863267 0.6823365
#> V5 Ordered -0.19094845 (-0.361 - -0.0213) 0.9863267 0.9863267 0.6210945
#> CI95 RR P-value RR Adj. p-value RR Consistent
#> V4 (0.719 - 1.37) 0.5177707 0.9838749 FALSE
#> V3 (0.55 - 1.25) 0.7944201 0.9838749 FALSE
#> V2 (0.584 - 1.25) 0.8117625 0.9838749 FALSE
#> V1 (0.44 - 1.06) 0.9560802 0.9838749 TRUE
#> V5 (0.402 - 0.96) 0.9838749 0.9838749 FALSE
Example: parallel via multicore
library(future)
plan("multiprocess")
vim = varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
Example: parallel via SNOW
library(RhpcBLASctl)
# Detect the number of physical cores on this computer using RhpcBLASctl.
cl = parallel::makeCluster(get_num_cores())
plan("cluster", workers = cl)
vim = varimpact(Y = Y, data = X)
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 0
#> - Numeric variables: 5
#>
#> No factor variables - skip VIM estimation.
#>
#> Estimating variable importance for 5 numerics.
parallel::stopCluster(cl)
Example: mlbench breast cancer
data(BreastCancer, package = "mlbench")
data = BreastCancer
# Create a numeric outcome variable.
data$Y = as.integer(data$Class == "malignant")
# Use multicore parallelization to speed up processing.
plan("multiprocess")
(vim = varimpact(Y = data$Y, data = subset(data, select = -c(Y, Class, Id))))
#> Finished pre-processing variables.
#>
#> Processing results:
#> - Factor variables: 9
#> - Numeric variables: 0
#>
#> Estimating variable importance for 9 factors.
#> Significant and consistent results:
#> Type Estimate CI95 P-value Adj. p-value
#> Bare.nuclei Factor 0.6284939 (0.503 - 0.754) 0.000000e+00 0.000000e+00
#> Mitoses Factor 0.4097166 (0.336 - 0.483) 0.000000e+00 0.000000e+00
#> Cl.thickness Factor 0.5344847 (0.378 - 0.691) 1.040124e-11 2.340278e-11
#> Cell.size Factor 0.5577438 (0.386 - 0.729) 8.920165e-11 1.605630e-10
#> Est. RR CI95 RR P-value RR Adj. p-value RR
#> Bare.nuclei 3.697125 (2.15 - 6.35) 0.000000e+00 0.000000e+00
#> Mitoses 2.095869 (1.85 - 2.37) 7.227108e-12 3.252199e-11
#> Cl.thickness 3.103819 (2.23 - 4.31) 4.128421e-07 9.288948e-07
#> Cell.size 3.326385 (1.93 - 5.73) 1.062140e-06 1.911853e-06
plot_var("Mitoses", vim)
Authors
Alan E. Hubbard and Chris J. Kennedy, University of California, Berkeley
References
Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the royal statistical society. Series B (Methodological), 289-300.
Gruber, S., & van der Laan, M. J. (2012). tmle: An R Package for Targeted Maximum Likelihood Estimation. Journal of Statistical Software, 51(i13).
Hubbard, A. E., Kennedy, C. J., van der Laan, M. J. (2018). Data-adaptive target parameters. In M. van der Laan & S. Rose (2018) Targeted Learning in Data Science. Springer.
Hubbard, A. E., Kherad-Pajouh, S., & van der Laan, M. J. (2016). Statistical Inference for Data Adaptive Target Parameters. The international journal of biostatistics, 12(1), 3-19.
Hubbard, A., Munoz, I. D., Decker, A., Holcomb, J. B., Schreiber, M. A., Bulger, E. M., … & Rahbar, M. H. (2013). Time-Dependent Prediction and Evaluation of Variable Importance Using SuperLearning in High Dimensional Clinical Data. The journal of trauma and acute care surgery, 75(1 0 1), S53.
Hubbard, A. E., & van der Laan, M. J. (2016). Mining with inference: data-adaptive target parameters (pp. 439-452). In P. Bühlmann et al. (Ed.), Handbook of Big Data. CRC Press, Taylor & Francis Group, LLC: Boca Raton, FL.
Jerez, J. M., Molina, I., García-Laencina, P. J., Alba, E., Ribelles, N., Martín, M., & Franco, L. (2010). Missing data imputation using statistical and machine learning methods in a real breast cancer problem. Artificial intelligence in medicine, 50(2), 105-115.
Rozenholc, Y., Mildenberger, T., & Gather, U. (2010). Combining regular and irregular histograms by penalized likelihood. Computational Statistics & Data Analysis, 54(12), 3313-3323.
Troyanskaya, O., Cantor, M., Sherlock, G., Brown, P., Hastie, T., Tibshirani, R., Botstein, D., & Altman, R. B. (2001). Missing value estimation methods for DNA microarrays. Bioinformatics, 17(6), 520-525.
van der Laan, M. J. (2006). Statistical inference for variable importance. The International Journal of Biostatistics, 2(1).
van der Laan, M. J., & Pollard, K. S. (2003). A new algorithm for hybrid hierarchical clustering with visualization and the bootstrap. Journal of Statistical Planning and Inference, 117(2), 275-303.
van der Laan, M. J., Polley, E. C., & Hubbard, A. E. (2007). Super learner. Statistical applications in genetics and molecular biology, 6(1).
van der Laan, M. J., & Rose, S. (2011). Targeted learning: causal inference for observational and experimental data. Springer Science & Business Media.
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