by Sherri Rose
Assistant Professor of Health Care Policy
Harvard Medical School
Targeted learning methods build machine-learning-based estimators of parameters defined as features of the probability distribution of the data, while also providing influence-curve or bootstrap-based confidence internals. The theory offers a general template for creating targeted maximum likelihood estimators for a data structure, nonparametric or semiparametric statistical model, and parameter mapping. These estimators of causal inference parameters are double robust and have a variety of other desirable statistical properties.
Targeted maximum likelihood estimation built on the loss-based “super learning” system such that lower-dimensional parameters could be targeted (e.g., a marginal causal effect); the remaining bias for the (low-dimensional) target feature of the probability distribution was removed. Targeted learning for effect estimation and causal inference allows for the complete integration of machine learning advances in prediction while providing statistical inference for the target parameter(s) of interest. Further details about these methods can be found in the many targeted learning papers as well as the 2011 targeted learning book.
Practical tools for the implementation of targeted learning methods for effect estimation and causal inference have developed alongside the theoretical and methodological advances. While some work has been done to develop computational tools for targeted learning in proprietary programming languages, such as SAS, the majority of the code has been built in R.
Of key importance are the two R packages SuperLearner and tmle. Ensembling with SuperLearner allows us to use many algorithms to generate an ideal prediction function that is a weighted average of all the algorithms considered. The SuperLearner package, authored by Eric Polley (NCI), is flexible, allowing for the integration of dozens of prespecified potential algorithms found in other packages as well as a system of wrappers that provide the user with the ability to design their own algorithms, or include newer algorithms not yet added to the package. The package returns multiple useful objects, including the cross-validated predicted values, final predicted values, vector of weights, and fitted objects for each of the included algorithms, among others.
Below is sample code with the ensembling prediction package SuperLearner using a small simulated data set.
library(SuperLearner) ##Generate simulated data## set.seed(27) n<-500 data <- data.frame(W1=runif(n, min = .5, max = 1), W2=runif(n, min = 0, max = 1), W3=runif(n, min = .25, max = .75), W4=runif(n, min = 0, max = 1)) data <- transform(data, #add W5 dependent on W2, W3 W5=rbinom(n, 1, 1/(1+exp(1.5*W2-W3)))) data <- transform(data, #add Y dependent on W1, W2, W4, W5 Y=rbinom(n, 1,1/(1+exp(-(-.2*W5-2*W1+4*W5*W1-1.5*W2+sin(W4)))))) summary(data) ##Specify a library of algorithms## SL.library <- c("SL.nnet", "SL.glm", "SL.randomForest") ##Run the super learner to obtain predicted values for the super learner as well as CV risk for algorithms in the library## fit.data.SL<-SuperLearner(Y=data[,6],X=data[,1:5],SL.library=SL.library, family=binomial(),method="method.NNLS", verbose=TRUE) ##Run the cross-validated super learner to obtain its CV risk## fitSL.data.CV <- CV.SuperLearner(Y=data[,6],X=data[,1:5], V=10, SL.library=SL.library,verbose = TRUE, method = "method.NNLS", family = binomial()) ##Cross validated risks## mean((data[,6]-fitSL.data.CV$SL.predict)^2) #CV risk for super learner fit.data.SL #CV risks for algorithms in the library
The final lines of code return the cross-validated risks for the super learner as well as each algorithm considered within the super learner. While a trivial example with a small data set and few covariates, these results demonstrate that the super learner, which takes a weighted average of the algorithms in the library, has the smallest cross-validated risk and outperforms each individual algorithm.
The tmle package, authored by Susan Gruber (Reagan-Udall Foundation), allows for the estimation of both average treatment effects and parameters defined by a marginal structural model in cross-sectional data with a binary intervention. This package also includes the ability to incorporate missingness in the outcome and the intervention, use SuperLearner to estimate the relevant components of the likelihood, and use data with a mediating variable. Additionally, TMLE and collaborative TMLE R code specifically tailored to answer quantitative trait loci mapping questions, such as those discussed in Wang et al 2011, is available in the supplementary material of that paper.
The multiPIM package, authored by Stephan Ritter (Omicia, Inc.), is designed specifically for variable importance analysis, and estimates an attributable-risk-type parameter using TMLE. This package also allows the use of SuperLearner to estimate nuisance parameters and produces additional estimates using estimating-equation-based estimators and g-computation. The package includes its own internal bootstrapping function to calculate standard errors if this is preferred over the use of influence curves, or influence curves are not valid for the chosen estimator.
Four additional prediction-focused packages are casecontrolSL, cvAUC, subsemble, and h2oEnsemble, all primarily authored by Erin LeDell (Berkeley). The casecontrolSL package relies on SuperLearner and performs subsampling in a case-control design with inverse-probability-of-censoring-weighting, which may be particularly useful in settings with rare outcomes. The cvAUC package is a tool kit to evaluate area under the ROC curve estimators when using cross-validation. The subsemble package was developed based on a new approach to ensembling that fits each algorithm on a subset of the data and combines these fits using cross-validation. This technique can be used in data sets of all size, but has been demonstrated to be particularly useful in smaller data sets. A new implementation of super learner can be found in the Java-based h2oEnsemble package, which was designed for big data. The package uses the H2O R interface to run super learning in R with a selection of prespecified algorithms.
Another TMLE package is ltmle, primarily authored by Joshua Schwab (Berkeley). This package mainly focuses on parameters in longitudinal data structures, including the treatment-specific mean outcome and parameters defined by a marginal structural model. The package returns estimates for TMLE, g-computation, and estimating-equation-based estimators.
The text above is a modified excerpt from the chapter "Targeted Learning for Variable Importance" by Sherri Rose in the forthcoming Handbook of Big Data (2015) edited by Peter Buhlmann, Petros Drineas, Michael John Kane, and Mark Van Der Laan to be published by CRC Press.