5 Super (Machine) Learning

Rachael Phillips

Based on the sl3 R package by Jeremy Coyle, Nima Hejazi, Ivana Malenica, Rachael Phillips, and Oleg Sofrygin.

Updated: 2021-01-08

5.1 Learning Objectives

By the end of this chapter you will be able to:

  1. Select a loss function that is appropriate for the functional parameter to be estimated.
  2. Assemble an ensemble of learners based on the properties that identify what features they support.
  3. Customize learner hyperparameters to incorporate a diversity of different settings.
  4. Select a subset of available covariates and pass only those variables to the modeling algorithm.
  5. Fit an ensemble with nested cross-validation to obtain an estimate of the performance of the ensemble itself.
  6. Obtain sl3 variable importance metrics.
  7. Interpret the discrete and continuous Super Learner fits.
  8. Rationalize the need to remove bias from the Super Learner to make an optimal bias–variance tradeoff for the parameter of interest.

5.2 Motivation

  • A common task in statistical data analysis is estimator selection (e.g., for prediction).
  • There is no universally optimal machine learning algorithm for density estimation or prediction.
  • For some data, one needs learners that can model a complex function.
  • For others, possibly as a result of noise or insufficient sample size, a simple, parametric model might fit best.
  • The Super Learner, an ensemble learner, solves this issue, by allowing a combination of learners from the simplest (intercept-only) to most complex (neural nets, random forests, SVM, etc).
  • It works by using cross-validation in a manner which guarantees that the resulting fit will be as good as possible, given the learners provided.

5.3 Introduction

In Chapter 1, we introduced the Roadmap for Targeted Learning as a general template to translate real-world data applications into formal statistical estimation problems. The first steps of this roadmap define the statistical estimation problem, which establish

  1. Data as a realization of a random variable, or equivalently, an outcome of a particular experiment.
  2. A statistical model, representing the true knowledge about the data-generating experiment.
  3. A translation of the scientific question, which is often causal, into a target parameter.

Note that if the target parameter is causal, step 3 also requires establishing identifiability of the target quantity from the observed data distribution, under possible non-testable assumptions that may not necessarily be reasonable. Still, the target quantity does have a valid statistical interpretation. See causal target parameters for more detail on causal models and identifiability.

Now that we have defined the statistical estimation problem, we are ready to construct the TMLE; an asymptotically linear and efficient substitution estimator of this target quantity. The first step in this estimation procedure is an initial estimate of the data-generating distribution, or the relevant part of this distribution that is needed to evaluate the target parameter. For this initial estimation, we use the Super Learner (van der Laan, Polley, and Hubbard 2007).

The Super Learner provides an important step in creating a robust estimator. It is a loss-function-based tool that uses cross-validation to obtain the best prediction of our target parameter, based on a weighted average of a library of machine learning algorithms.

The library of machine learning algorithms consists of functions (“learners” in the sl3 nomenclature) that we think might be consistent with the true data-generating distribution (i.e. algorithms selected based on contextual knowledge of the experiment that generated the data). Also, the library should contain a large set of “default” algorithms that may range from a simple linear regression model to multi-step algorithms involving screening covariates, penalizations, optimizing tuning parameters, etc.

The ensembling of the collection of algorithms with weights (“metalearning” in the sl3 nomenclature) has been shown to be adaptive and robust, even in small samples (Polley and van der Laan 2010). The Super Learner is proven to be asymptotically as accurate as the best possible prediction algorithm in the library (van der Laan and Dudoit 2003; van der Vaart, Dudoit, and van der Laan 2006).

5.3.1 Background

A loss function \(L\) is defined as a function of the observed data and a candidate parameter value \(\psi\), which has unknown true value \(\psi_0\), \(L(\psi)(O)\). We can estimate the loss by substituting the empirical distribution \(P_n\) for the true (but unknown) distribution of the observed data \(P_0\). A valid loss function will have expectation (risk) that is minimized at the true value of the parameter \(\psi_0\). For example, the conditional mean minimizes the risk of the squared error loss. Thus, it is a valid loss function when estimating the conditional mean.

The discrete Super Learner, or cross-validation selector, is the algorithm in the library that minimizes the cross-validated empirical risk.

The cross-validated empirical risk of an algorithm is defined as the empirical mean over a validation sample of the loss of the algorithm fitted on the training sample, averaged across the splits of the data.

The continuous/ensemble Super Learner, often referred to as Super Learner is a weighted average of the library of algorithms, where the weights are chosen to minimize the cross-validated empirical risk of the library. Restricting the weights to be positive and sum to one (i.e., a convex combination) has been shown to improve upon the discrete Super Learner (Polley and van der Laan 2010; van der Laan, Polley, and Hubbard 2007). This notion of weighted combinations was introduced in Wolpert (1992) for neural networks and adapted for regressions in Breiman (1996).

Cross-validation is proven to be optimal for selection among estimators. This result was established through the oracle inequality for the cross-validation selector among a collection of candidate estimators (van der Laan and Dudoit 2003; van der Vaart, Dudoit, and van der Laan 2006). The only condition is that loss function is uniformly bounded, which is guaranteed in sl3.

5.3.1.1 General Overview of the Algorithm

What is cross-validation and how does it work?

There are many different cross-validation schemes, designed to accommodate different study designs and data structures. The figure below shows an example of 10-fold cross-validation.

General step-by-step overview of the Super Learner algorithm:

  • Break up the sample evenly into V-folds (say V=10).
  • For each of these 10 folds, remove that portion of the sample (kept out as validation sample) and the remaining will be used to fit learners (training sample).
  • Fit each learner on the training sample (note, some learners will have their own internal cross-validation procedure or other methods to select tuning parameters).
  • For each observation in the corresponding validation sample, predict the outcome using each of the learners, so if there are \(p\) learners, then there would be \(p\) predictions.
  • Take out another validation sample and repeat until each of the V-sets of data are removed.
  • Compare the cross-validated fit of the learners across all observations based on specified loss function (e.g., squared error, negative log-likelihood, …) by calculating the corresponding average loss (risk).
  • Either:

    • choose the learner with smallest risk and apply that learner to entire data set (resulting SL fit),
    • do a weighted average of the learners to minimize the cross-validated risk (construct an ensemble of learners), by

      • re-fitting the learners on the original data set, and
      • use the weights above to get the SL fit.

This entire procedure can be itself cross-validated to get a consistent estimate of the future performance of the Super Learner, and we implement this procedure later in this chapter.

5.3.2 Super Learner for Prediction

Say we observe a learning data set \(X_i=(Y_i,W_i)\), for \(i=1, ..., n\), where \(Y_i\) is the outcome of interest, \(W_i\) is a \(p\)-dimensional set of covariates, and our objective is to estimate the function \(\psi_0(W) = E(Y|W)\). This function can be expressed as the minimizer of the expected loss: \(\psi_0(W) = \text{argmin}_{\psi} E[L(X,\psi(W))]\). Here, the loss function is represented as \(L\) (e.g., squared error loss, \(L: (Y-\psi(W))^2)\)).

For prediction, one can use the cross-validated risk to empirically determine the relative performance of the Super Learner. When we have tested different algorithms on actual data and looked at the performance (e.g., MSE of prediction), never does one algorithm always win. Below, we show the results of such a study, comparing the fits of several different learners, including the SL algorithms.

For more detail on Super Learner we refer the reader to van der Laan, Polley, and Hubbard (2007) and Polley and van der Laan (2010). The optimality results for the cross-validation selector among a family of algorithms were established in van der Laan and Dudoit (2003) and extended in van der Vaart, Dudoit, and van der Laan (2006).

5.4 sl3 “Microwave Dinner” Implementation

We begin by illustrating the core functionality of the super learner algorithm as implemented in sl3. For those who are interested in the internals of sl3, see this sl3 introductory tutorial.

The sl3 implementation consists of the following steps:

  1. Load the necessary libraries and data
  2. Define the machine learning task
  3. Make a super learner by creating library of base learners and a metalearner
  4. Train the super learner on the machine learning task
  5. Obtain predicted values

5.4.1 WASH Benefits Study Example

Using the WASH data, we are interested in predicting weight-for-height z-score whz using the available covariate data. More information on this dataset, and all other data that we will work with in this handbook, is contained in Chapter 3. Let’s begin!

0. Load the necessary libraries and data

First, we will load the relevant R packages, set a seed, and load the data.

If you would like to use newer sl3 functionality that is available in the devel branch of the sl3 GitHub repository, you need to install that version of the package (e.g. devtools::install_github(tlverse/sl3@devel)), re-start your R session, and then re-load the sl3 package.

library(data.table)
library(tidyverse)
library(origami)
library(SuperLearner)
library(sl3)
library(ggplot2)
library(knitr)
library(kableExtra)

# load data set and take a peek
washb_data <- fread("https://raw.githubusercontent.com/tlverse/tlverse-data/master/wash-benefits/washb_data.csv",
                    stringsAsFactors = TRUE)
head(washb_data) %>%
  kable(digits = 4) %>%
  kableExtra:::kable_styling(fixed_thead = T) %>%
  scroll_box(width = "100%", height = "300px")
whz tr fracode month aged sex momage momedu momheight hfiacat Nlt18 Ncomp watmin elec floor walls roof asset_wardrobe asset_table asset_chair asset_khat asset_chouki asset_tv asset_refrig asset_bike asset_moto asset_sewmach asset_mobile
0.00 Control N05265 9 268 male 30 Primary (1-5y) 146.40 Food Secure 3 11 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1
-1.16 Control N05265 9 286 male 25 Primary (1-5y) 148.75 Moderately Food Insecure 2 4 0 1 0 1 1 0 1 0 1 1 0 0 0 0 0 1
-1.05 Control N08002 9 264 male 25 Primary (1-5y) 152.15 Food Secure 1 10 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1
-1.26 Control N08002 9 252 female 28 Primary (1-5y) 140.25 Food Secure 3 5 0 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1
-0.59 Control N06531 9 336 female 19 Secondary (>5y) 150.95 Food Secure 2 7 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1
-0.51 Control N06531 9 304 male 20 Secondary (>5y) 154.20 Severely Food Insecure 0 3 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1

1. Define the machine learning task

To define the machine learning “task” (predict weight-for-height z-score whz using the available covariate data), we need to create an sl3_Task object.

The sl3_Task keeps track of the roles the variables play in the machine learning problem, the data, and any metadata (e.g., observational-level weights, IDs, offset).

Also, if we had missing outcomes, we would need to set drop_missing_outcome = TRUE when we create the task. In the next analysis, with the IST stroke trial data, we do have a missing outcome. In the following chapter, we estimate the missingness mechanism and account for it in the TMLE.

# specify the outcome and covariates
outcome <- "whz"
covars <- colnames(washb_data)[-which(names(washb_data) == outcome)]

# create the sl3 task
washb_task <- make_sl3_Task(
  data = washb_data,
  covariates = covars,
  outcome = outcome
)
#> Warning in process_data(data, nodes, column_names = column_names, flag = flag, :
#> Missing covariate data detected: imputing covariates.

This warning is important. The task just imputed missing covariates for us. Specifically, for each covariate column with missing values, sl3 uses the median to impute missing continuous covariates, and the mode to impute binary and categorical covariates.

Also, for each covariate column with missing values, sl3 adds an additional column indicating whether or not the value was imputed, which is particularly handy when the missingness in the data might be informative.

Also, notice that we did not specify the number of folds, or the loss function in the task. The default cross-validation scheme is V-fold, with the number of folds \(V=10\).

Let’s visualize our washb_task.

washb_task
#> A sl3 Task with 4695 obs and these nodes:
#> $covariates
#>  [1] "tr"              "fracode"         "month"           "aged"           
#>  [5] "sex"             "momage"          "momedu"          "momheight"      
#>  [9] "hfiacat"         "Nlt18"           "Ncomp"           "watmin"         
#> [13] "elec"            "floor"           "walls"           "roof"           
#> [17] "asset_wardrobe"  "asset_table"     "asset_chair"     "asset_khat"     
#> [21] "asset_chouki"    "asset_tv"        "asset_refrig"    "asset_bike"     
#> [25] "asset_moto"      "asset_sewmach"   "asset_mobile"    "delta_momage"   
#> [29] "delta_momheight"
#> 
#> $outcome
#> [1] "whz"
#> 
#> $id
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $offset
#> NULL
#> 
#> $time
#> NULL

2. Make a Super Learner

Now that we have defined our machine learning problem with the task, we are ready to “make” the Super Learner. This requires specification of

  • A library of base learning algorithms that we think might be consistent with the true data-generating distribution.
  • A metalearner, to ensemble the base learners.

We might also incorporate

  • Feature selection, to pass only a subset of the predictors to the algorithm.
  • Hyperparameter specification, to tune base learners.

Learners have properties that indicate what features they support. We may use sl3_list_properties() to get a list of all properties supported by at least one learner.

sl3_list_properties()
#>  [1] "binomial"             "categorical"          "continuous"          
#>  [4] "cv"                   "density"              "ids"                 
#>  [7] "multivariate_outcome" "offset"               "preprocessing"       
#> [10] "sampling"             "timeseries"           "weights"             
#> [13] "wrapper"

Since we have a continuous outcome, we may identify the learners that support this outcome type with sl3_list_learners().

sl3_list_learners("continuous")
#>  [1] "Lrnr_arima"                     "Lrnr_bartMachine"              
#>  [3] "Lrnr_bilstm"                    "Lrnr_bound"                    
#>  [5] "Lrnr_caret"                     "Lrnr_condensier"               
#>  [7] "Lrnr_cv_selector"               "Lrnr_dbarts"                   
#>  [9] "Lrnr_earth"                     "Lrnr_expSmooth"                
#> [11] "Lrnr_gam"                       "Lrnr_gbm"                      
#> [13] "Lrnr_glm"                       "Lrnr_glm_fast"                 
#> [15] "Lrnr_glmnet"                    "Lrnr_grf"                      
#> [17] "Lrnr_h2o_glm"                   "Lrnr_h2o_grid"                 
#> [19] "Lrnr_hal9001"                   "Lrnr_HarmonicReg"              
#> [21] "Lrnr_lstm"                      "Lrnr_mean"                     
#> [23] "Lrnr_nnls"                      "Lrnr_optim"                    
#> [25] "Lrnr_pkg_SuperLearner"          "Lrnr_pkg_SuperLearner_method"  
#> [27] "Lrnr_pkg_SuperLearner_screener" "Lrnr_polspline"                
#> [29] "Lrnr_randomForest"              "Lrnr_ranger"                   
#> [31] "Lrnr_rpart"                     "Lrnr_rugarch"                  
#> [33] "Lrnr_screener_corP"             "Lrnr_screener_corRank"         
#> [35] "Lrnr_screener_randomForest"     "Lrnr_solnp"                    
#> [37] "Lrnr_stratified"                "Lrnr_svm"                      
#> [39] "Lrnr_tsDyn"                     "Lrnr_xgboost"

Now that we have an idea of some learners, we can construct them using the make_learner function.

# choose base learners
lrnr_glm <- make_learner(Lrnr_glm)
lrnr_mean <- make_learner(Lrnr_mean)

We can customize learner hyperparameters to incorporate a diversity of different settings. Documentation for the learners and their hyperparameters can be found in the sl3 Learners Reference.

lrnr_ranger50 <- make_learner(Lrnr_ranger, num.trees = 50)
lrnr_hal_simple <- make_learner(Lrnr_hal9001, max_degree = 2, n_folds = 2)
lrnr_lasso <- make_learner(Lrnr_glmnet) # alpha default is 1
lrnr_ridge <- make_learner(Lrnr_glmnet, alpha = 0)
lrnr_elasticnet <- make_learner(Lrnr_glmnet, alpha = .5)

We can also include learners from the SuperLearner R package.

lrnr_bayesglm <- Lrnr_pkg_SuperLearner$new("SL.bayesglm")

Here is a fun trick to create customized learners over a grid of parameters.

# I like to crock pot my super learners
grid_params <- list(cost = c(0.01, 0.1, 1, 10, 100, 1000),
                    gamma = c(0.001, 0.01, 0.1, 1),
                    kernel = c("polynomial", "radial", "sigmoid"),
                    degree = c(1, 2, 3))
grid <- expand.grid(grid_params, KEEP.OUT.ATTRS = FALSE)
params_default <- list(nthread = getOption("sl.cores.learners", 1))
svm_learners <- apply(grid, MARGIN = 1, function(params_tune) {
  do.call(Lrnr_svm$new, c(params_default, as.list(params_tune)))})
grid_params <- list(max_depth = c(2, 4, 6, 8),
                    eta = c(0.001, 0.01, 0.1, 0.2, 0.3),
                    nrounds = c(20, 50))
grid <- expand.grid(grid_params, KEEP.OUT.ATTRS = FALSE)
params_default <- list(nthread = getOption("sl.cores.learners", 1))
xgb_learners <- apply(grid, MARGIN = 1, function(params_tune) {
  do.call(Lrnr_xgboost$new, c(params_default, as.list(params_tune)))})

Did you see Lrnr_caret when we called sl3_list_learners(c("binomial"))? All we need to specify is the algorithm to use, which is passed as method to caret::train(). The default method for parameter selection criterion with is set to “CV” instead of the caret::train() default boot. The summary metric to used to select the optimal model is RMSE for continuous outcomes and Accuracy for categorical and binomial outcomes.

# I have no idea how to tune a neural net (or BART machine..) 
lrnr_caret_nnet <- make_learner(Lrnr_caret, algorithm = "nnet")
lrnr_caret_bartMachine <- make_learner(Lrnr_caret, algorithm = "bartMachine",
                                       method = "boot", metric = "Accuracy",
                                       tuneLength = 10)

In order to assemble the library of learners, we need to “stack” them together.

A Stack is a special learner and it has the same interface as all other learners. What makes a stack special is that it combines multiple learners by training them simultaneously, so that their predictions can be either combined or compared.

stack <- make_learner(
  Stack,
  lrnr_glm, lrnr_mean, lrnr_ridge, lrnr_lasso, xgb_learners[[10]]
)

We can optionally select a subset of available covariates and pass only those variables to the modeling algorithm.

Let’s consider screening covariates based on their randomForest variable importance ranking (ordered by mean decrease in accuracy). We select the top 5 most important covariates according to this ranking, and we decreased the ntree to 20.

Before you think it – I will confess. Bob Ross and I both know that 20 trees makes for a lonely forest, and I shouldn’t even consider it, but these are the sacrifices I have to make for this chapter to build in under 50 minutes!

screen_rf <- make_learner(Lrnr_screener_randomForest, nVar = 5, ntree = 20)
# which covariates are selected on the full data?
screen_rf$train(washb_task)
#> [1] "Lrnr_screener_randomForest_5_20"
#> $selected
#> [1] "month"     "aged"      "momage"    "momheight" "Ncomp"

To “pipe” only the selected covariates to the modeling algorithm, we need to make a Pipeline, which is a just set of learners to be fit sequentially, where the fit from one learner is used to define the task for the next learner.

screen_rf_pipeline <- make_learner(Pipeline, screen_rf, stack)

Now our learners will be preceded by a screening step.

We also consider the original stack, to compare how the feature selection methods perform in comparison to the methods without feature selection, and because

Analogous to what we have seen before, we have to stack the pipeline and original stack together, so we may use them as base learners in our super learner.

fancy_stack <- make_learner(Stack, screen_rf_pipeline, stack)
# we can visualize the stack
dt_stack <- delayed_learner_train(fancy_stack, washb_task)
plot(dt_stack, color = FALSE, height = "400px", width = "90%")

We will use the default metalearner, which uses Lrnr_solnp() to provide fitting procedures for a pairing of loss function and metalearner function. This default metalearner selects a loss and metalearner pairing based on the outcome type. Note that any learner can be used as a metalearner.

We have made a library/stack of base learners, so we are ready to make the super learner. The super learner algorithm fits a metalearner on the validation-set predictions.

sl <- make_learner(Lrnr_sl,
  learners = fancy_stack
)

We can also use Lrnr_cv to build a super learner, cross-validate a stack of learners to compare performance of the learners in the stack, or cross-validate any single learner (see “Cross-validation” section of this sl3 introductory tutorial).

Furthermore, we can Define New sl3 Learners which can be used in all the places you could otherwise use any other sl3 learners, including Pipelines, Stacks, and the Super Learner.

In the plot below, we visualize the steps for executing the Super Learner in the tlverse/delayed framework. For those like myself who are not particularly keen on understanding the intricacies of delayed, let’s focus on the main point of this figure: we can see there are 10 realizations of the stack which represent the 10 cross-validation folds and there is a separate hold-out (top branch of the figure) that will not be used to fit the Super Learner.

dt_sl <- delayed_learner_train(sl, washb_task)
plot(dt_sl, color = FALSE, height = "400px", width = "90%")

3. Train the Super Learner on the machine learning task

The Super Learner algorithm fits a metalearner on the validation-set predictions in a cross-validated manner, thereby avoiding overfitting.

Now we are ready to “train” our Super Learner on our sl3_task object, washb_task.

sl_fit <- sl$train(washb_task)

4. Obtain predicted values

Now that we have fit the super learner, we are ready to calculate the predicted outcome for each subject.

# we did it! now we have super learner predictions
sl_preds <- sl_fit$predict()
head(sl_preds)
#> [1] -0.6594194 -0.7698511 -0.6601775 -0.6487316 -0.6288337 -0.6847972

We can also obtain a summary of the results.

sl_fit_summary <- sl_fit$print()
#> [1] "SuperLearner:"
#> List of 2
#>  $ : chr "Pipeline(Lrnr_screener_randomForest_5_20->Stack)"
#>  $ : chr "Stack"
#> [1] "Lrnr_solnp_TRUE_TRUE_FALSE_1e-05"
#> $pars
#>  [1] 6.184654e-05 1.720116e-05 5.544575e-05 5.898108e-05 2.106970e-01
#>  [6] 2.915191e-01 1.720116e-05 1.178057e-01 3.793167e-01 4.507556e-04
#> 
#> $convergence
#> [1] 0
#> 
#> $values
#> [1] 1.020595 1.010955 1.010951
#> 
#> $lagrange
#>             [,1]
#> [1,] -0.04374293
#> 
#> $hessian
#>            [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
#>  [1,] 1.6979665 0.9035073 0.8210602 0.8982773 0.8667451 1.0239159 0.9035073
#>  [2,] 0.9035073 1.5706620 0.9105314 0.9739251 0.7008069 0.8278794 0.5706620
#>  [3,] 0.8210602 0.9105314 1.7324722 0.8192425 0.8895203 1.0519616 0.9105314
#>  [4,] 0.8982773 0.9739251 0.8192425 1.8711894 0.9460662 1.1077563 0.9739251
#>  [5,] 0.8667451 0.7008069 0.8895203 0.9460662 0.8181850 0.9358299 0.7008069
#>  [6,] 1.0239159 0.8278794 1.0519616 1.1077563 0.9358299 1.4877842 0.8278794
#>  [7,] 0.9035073 0.5706620 0.9105314 0.9739251 0.7008069 0.8278794 1.5706620
#>  [8,] 0.9823009 0.8668316 1.0268335 1.0794212 0.9382398 0.7654115 0.8668316
#>  [9,] 0.9608695 0.7878954 0.9542916 1.0178275 0.5926686 0.5945269 0.7878954
#> [10,] 0.7806872 0.6019418 0.7778830 0.8412361 0.4721807 0.6097128 0.6019418
#>            [,8]      [,9]     [,10]
#>  [1,] 0.9823009 0.9608695 0.7806872
#>  [2,] 0.8668316 0.7878954 0.6019418
#>  [3,] 1.0268335 0.9542916 0.7778830
#>  [4,] 1.0794212 1.0178275 0.8412361
#>  [5,] 0.9382398 0.5926686 0.4721807
#>  [6,] 0.7654115 0.5945269 0.6097128
#>  [7,] 0.8668316 0.7878954 0.6019418
#>  [8,] 1.8276642 0.9540252 0.4851260
#>  [9,] 0.9540252 1.2662454 0.9510900
#> [10,] 0.4851260 0.9510900 0.8333466
#> 
#> $ineqx0
#> NULL
#> 
#> $nfuneval
#> [1] 263
#> 
#> $outer.iter
#> [1] 2
#> 
#> $elapsed
#> Time difference of 0.1036305 secs
#> 
#> $vscale
#>  [1] 1.010955 0.000010 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
#>  [9] 1.000000 1.000000 1.000000 1.000000
#> 
#> $coefficients
#>                           Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glm_TRUE 
#>                                                                             0.0000000000 
#>                               Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_mean 
#>                                                                             0.0000000000 
#> Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE 
#>                                                                             0.0000000000 
#> Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE 
#>                                                                             0.0000000000 
#>                 Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_xgboost_20_1_4_0.1 
#>                                                                             0.2107414178 
#>                                                                      Stack_Lrnr_glm_TRUE 
#>                                                                             0.2915805478 
#>                                                                          Stack_Lrnr_mean 
#>                                                                             0.0000000000 
#>                                            Stack_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE 
#>                                                                             0.1178305533 
#>                                            Stack_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE 
#>                                                                             0.3793966305 
#>                                                            Stack_Lrnr_xgboost_20_1_4_0.1 
#>                                                                             0.0004508506 
#> 
#> $training_offset
#> [1] FALSE
#> 
#> $name
#> [1] "solnp"
#> 
#> [1] "Cross-validated risk (MSE, squared error loss):"
#>                                                                                      learner
#>  1:                           Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glm_TRUE
#>  2:                               Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_mean
#>  3: Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE
#>  4: Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE
#>  5:                 Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_xgboost_20_1_4_0.1
#>  6:                                                                      Stack_Lrnr_glm_TRUE
#>  7:                                                                          Stack_Lrnr_mean
#>  8:                                            Stack_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE
#>  9:                                            Stack_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE
#> 10:                                                            Stack_Lrnr_xgboost_20_1_4_0.1
#> 11:                                                                             SuperLearner
#>     coefficients mean_risk    SE_risk    fold_SD fold_min_risk fold_max_risk
#>  1: 0.0000000000  1.036483 0.02445142 0.07264280     0.8893079      1.131805
#>  2: 0.0000000000  1.065371 0.02503479 0.06410509     0.9338939      1.135933
#>  3: 0.0000000000  1.036478 0.02445751 0.07234017     0.8898660      1.130969
#>  4: 0.0000000000  1.036446 0.02445051 0.07260733     0.8894397      1.131734
#>  5: 0.2107414178  1.047139 0.02402321 0.07080623     0.9154444      1.128126
#>  6: 0.2915805478  1.020204 0.02395530 0.06749990     0.8944174      1.120024
#>  7: 0.0000000000  1.065371 0.02503479 0.06410509     0.9338939      1.135933
#>  8: 0.1178305533  1.015819 0.02373426 0.06555599     0.8862805      1.109816
#>  9: 0.3793966305  1.013498 0.02361186 0.06559624     0.8914167      1.108439
#> 10: 0.0004508506  1.031917 0.02351705 0.06444276     0.9104842      1.114357
#> 11:           NA  1.010944 0.02353629 0.06688407     0.8844468      1.103548

From the table of the printed Super Learner fit, we note that the Super Learner had a mean risk of 1.0109438 and that this ensemble weighted the ranger and glmnet learners highest while not weighting the mean learner highly.

We can also see that the glmnet learner had the lowest cross-validated mean risk, thus making it the cross-validated selector (or the discrete Super Learner). The mean risk of the Super Learner is calculated using the hold-out set that we visualized in the dt_sl plot.

5.5 Cross-validated Super Learner

We can cross-validate the Super Learner to see how well the Super Learner performs on unseen data, and obtain an estimate of the cross-validated risk of the Super Learner.

This estimation procedure requires an “external” layer of cross-validation, also called nested cross-validation, which involves setting aside a separate holdout sample that we don’t use to fit the Super Learner. This external cross validation procedure may also incorporate 10 folds, which is the default in sl3. However, we will incorporate 2 outer/external folds of cross-validation for computational efficiency.

We also need to specify a loss function to evaluate Super Learner. Documentation for the available loss functions can be found in the sl3 Loss Function Reference.

washb_task_new <- make_sl3_Task(
  data = washb_data,
  covariates = covars,
  outcome = outcome,
  folds = origami::make_folds(washb_data, fold_fun = folds_vfold, V = 2)
)
#> Warning in process_data(data, nodes, column_names = column_names, flag = flag, :
#> Missing covariate data detected: imputing covariates.
CVsl <- CV_lrnr_sl(sl_fit, washb_task_new, loss_squared_error)
CVsl %>%
  kable(digits = 4) %>%
  kableExtra:::kable_styling(fixed_thead = T) %>%
  scroll_box(width = "100%", height = "300px")
learner coefficients mean_risk SE_risk fold_SD fold_min_risk fold_max_risk
Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glm_TRUE 0.0709 1.0348 0.0244 0.0126 1.0259 1.0437
Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_mean 0.0000 1.0656 0.0250 0.0184 1.0525 1.0786
Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE 0.0013 1.0350 0.0245 0.0130 1.0257 1.0442
Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE 0.0000 1.0353 0.0245 0.0133 1.0259 1.0447
Pipeline(Lrnr_screener_randomForest_5_20->Stack)_Lrnr_xgboost_20_1_4_0.1 0.0799 1.0551 0.0242 0.0066 1.0504 1.0597
Stack_Lrnr_glm_TRUE 0.0482 1.0350 0.0273 0.0411 1.0060 1.0641
Stack_Lrnr_mean 0.0000 1.0656 0.0250 0.0184 1.0525 1.0786
Stack_Lrnr_glmnet_NULL_deviance_10_0_100_TRUE 0.1845 1.0188 0.0238 0.0243 1.0016 1.0360
Stack_Lrnr_glmnet_NULL_deviance_10_1_100_TRUE 0.5232 1.0145 0.0236 0.0236 0.9978 1.0311
Stack_Lrnr_xgboost_20_1_4_0.1 0.0920 1.0417 0.0235 0.0159 1.0304 1.0529
SuperLearner NA 1.0144 0.0234 0.0226 0.9984 1.0304

5.6 Variable Importance Measures with sl3

Variable importance can be interesting and informative. It can also be contradictory and confusing. Nevertheless, we like it, and so do collaborators, so we created a variable importance function in sl3! The sl3 varimp function returns a table with variables listed in decreasing order of importance (i.e. most important on the first row).

The measure of importance in sl3 is based on a risk difference between the learner fit with a permuted covariate and the learner fit with the true covariate, across all covariates. In this manner, the larger the risk difference, the more important the variable is in the prediction.

The intuition of this measure is that it calculates the risk (in terms of the average loss in predictive accuracy) of losing one covariate, while keeping everything else fixed, and compares it to the risk if the covariate was not lost. If this risk difference is zero then losing that covariate had no impact, and is thus not important by this measure. We do this across all of the covariates. As stated above, we don’t actually remove the covariate, we just permute/shuffle it, but the idea is that this shuffling distorts potentially meaningful information that was present in the covariate. This idea of permuting instead of removing saves a lot of time, and is also incorporated in the randomForest variable importance measures.

Let’s explore the sl3 variable importance measurements for the washb data.

washb_varimp <- varimp(sl_fit, loss_squared_error)
washb_varimp %>%
  kable(digits = 4) %>%
  kableExtra:::kable_styling(fixed_thead = T) %>%
  scroll_box(width = "100%", height = "300px")
X risk_diff
aged 0.0371
momedu 0.0079
asset_refrig 0.0049
month 0.0038
tr 0.0035
momheight 0.0032
Nlt18 0.0028
asset_chair 0.0028
asset_table 0.0022
elec 0.0018
floor 0.0015
walls 0.0011
momage 0.0009
asset_mobile 0.0005
delta_momheight 0.0003
asset_sewmach 0.0002
asset_bike 0.0002
fracode 0.0002
asset_moto 0.0001
asset_chouki 0.0001
asset_wardrobe 0.0000
hfiacat -0.0001
sex -0.0001
delta_momage -0.0002
roof -0.0002
Ncomp -0.0003
asset_tv -0.0003
asset_khat -0.0004
watmin -0.0008
# plot variable importance
washb_varimp %>%
  mutate(name = forcats::fct_reorder(X, risk_diff)) %>%
  ggplot(aes(x = risk_diff, y = name)) +
  geom_dotplot(binaxis = "y") +
  labs(x = "Risk Difference", y = "Covariate", 
       title = "sl3 Variable Importance for WASH Benefits Example Data")

5.7 Exercises

5.7.1 Predicting Myocardial Infarction with sl3

Follow the steps below to predict myocardial infarction (mi) using the available covariate data. We thank Prof. David Benkeser at Emory University for making the this Cardiovascular Health Study (CHS) data accessible.

# load the data set
db_data <-
 url("https://raw.githubusercontent.com/benkeser/sllecture/master/chspred.csv")
chspred <- read_csv(file = db_data, col_names = TRUE)
# take a quick peek
head(chspred) %>%
  kable(digits = 4) %>%
  kableExtra:::kable_styling(fixed_thead = T) %>%
  scroll_box(width = "100%", height = "300px")
waist alcoh hdl beta smoke ace ldl bmi aspirin gend age estrgn glu ins cysgfr dm fetuina whr hsed race logcystat logtrig logcrp logcre health logkcal sysbp mi
110.1642 0.0000 66.4974 0 0 1 114.2162 27.9975 0 0 73.5179 0 159.9314 70.3343 75.0078 1 0.1752 1.1690 1 1 -0.3420 5.4063 2.0126 -0.6739 0 4.3926 177.1345 0
89.9763 0.0000 50.0652 0 0 0 103.7766 20.8931 0 0 61.7723 0 153.3888 33.9695 82.7433 1 0.5717 0.9011 0 0 -0.0847 4.8592 3.2933 -0.5551 1 6.2071 136.3742 0
106.1941 8.4174 40.5059 0 0 0 165.7158 28.4554 1 1 72.9312 0 121.7145 -17.3017 74.6989 0 0.3517 1.1797 0 1 -0.4451 4.5088 0.3013 -0.0115 0 6.7320 135.1993 0
90.0566 0.0000 36.1750 0 0 0 45.2035 23.9608 0 0 79.1191 0 53.9691 11.7315 95.7823 0 0.5439 1.1360 0 0 -0.4807 5.1832 3.0243 -0.5751 1 7.3972 139.0182 0
78.6143 2.9790 71.0642 0 1 0 131.3121 10.9656 0 1 69.0179 0 94.3153 9.7112 72.7109 0 0.4916 1.1028 1 0 0.3121 4.2190 -0.7057 0.0053 1 8.2779 88.0470 0
91.6593 0.0000 59.4963 0 0 0 171.1872 29.1317 0 1 81.8346 0 212.9066 -28.2269 69.2184 1 0.4621 0.9529 1 0 -0.2872 5.1773 0.9705 0.2127 1 5.9942 69.5943 0
  1. Create an sl3 task, setting myocardial infarction mi as the outcome and using all available covariate data.
  2. Make a library of seven relatively fast base learning algorithms (i.e., do not consider BART or HAL). Customize hyperparameters for one of your learners. Feel free to use learners from sl3 or SuperLearner. You may use the same base learning library that is presented above.
  3. Incorporate feature selection with the screener screen.corP.
  4. Fit the metalearning step with the default metalearner.
  5. With the metalearner and base learners, make the Super Learner and train it on the task.
  6. Print your Super Learner fit by calling print() with $.
  7. Cross-validate your Super Learner fit to see how well it performs on unseen data. Specify loss_squared_error as the loss function to evaluate the Super Learner.
  8. Use the varimp() function to identify the most important predictor of myocardial infarction.

5.7.2 Predicting Recurrent Ischemic Stroke in an RCT with sl3

For this exercise, we will work with a random sample of 5,000 patients who participated in the International Stroke Trial (IST). This data is described in Chapter 3.2 of the tlverse handbook.

  1. Train a Super Learner to predict recurrent stroke DRSISC with the available covariate data (the 25 other variables). Of course, you can consider feature selection in the machine learning algorithms. In this data, the outcome is occasionally missing, so be sure to specify drop_missing_outcome = TRUE when defining the task.
  2. Use the SL-based predictions to calculate the area under the ROC curve (AUC).
  3. Calculate the cross-validated AUC to evaluate the performance of the Super Learner on unseen data.
  4. Which covariates are the most predictive of 14-day recurrent stroke, according to sl3 variable importance measures?
ist_data <- paste0("https://raw.githubusercontent.com/tlverse/",
                   "tlverse-handbook/master/data/ist_sample.csv") %>% fread()

# number 3 help
ist_task_CVsl <- make_sl3_Task(
  data = ist_data,
  outcome = "DRSISC",
  covariates = colnames(ist_data)[-which(names(ist_data) == "DRSISC")],
  drop_missing_outcome = TRUE,
  folds = origami::make_folds(
    n = sum(!is.na(ist_data$DRSISC)), 
    fold_fun = folds_vfold,
    V = 5
    )
  )

5.8 Concluding Remarks

  • The general ensemble learning approach of Super Learner can be applied to a diversity of estimation and prediction problems that can be defined by a loss function.

  • We just discussed conditional mean estimation, outcome prediction and variable importance. In future updates of this chapter, we will delve into prediction of a conditional density, and the optimal individualized treatment rule.

  • If we plug in the estimator returned by super learner into the target parameter mapping, then we would end up with an estimator that has the same bias as what we plugged in, and would not be asymptotically linear. It also would not be a plug-in estimator or efficient.

    • An asymptotically linear estimator is important to have, since they converge to the estimand at \(\frac{1}{\sqrt{n}}\) rate, and thereby permit formal statistical inference (i.e. confidence intervals and \(p\)-values).
    • Plug-in estimators of the estimand are desirable because they respect both the local and global constraints of the statistical model (e.g. bounds), and have they have better finite-sample properties.
    • An efficient estimator is optimal in the sense that it has the lowest possible variance, and is thus the most precise. An estimator is efficient if and only if is asymptotically linear with influence curve equal to the canonical gradient. The canonical gradient is a mathematical object that is specific to the target estimand, and it provides information on the level of difficulty of the estimation problem. The canonical gradient is shown in the chapters that follow. Practitioner’s do not need to know how to calculate a canonical gradient in order to understand efficiency and use Targeted Maximum Likelihood Estimation (TMLE). Metaphorically, you do not need to be Yoda in order to be a Jedi.
  • TMLE is a general strategy that succeeds in constructing efficient and asymptotically linear plug-in estimators.

  • Super Learner is fantastic for pure prediction, and for obtaining an initial estimate in the first step of TMLE, but we need the second step of TMLE to have the desirable statistical properties mentioned above.

  • In the chapters that follow, we focus on the targeted maximum likelihood estimator and the targeted minimum loss-based estimator, both referred to as TMLE.

5.9 Appendix

5.9.1 Exercise 1 Solution

Here is a potential solution to the sl3 Exercise 1 – Predicting Myocardial Infarction with sl3.

library(sl3)
library(origami)
library(data.table)
library(tidyverse)
library(ggplot2)

db_data <-
 url("https://raw.githubusercontent.com/benkeser/sllecture/master/chspred.csv")
chspred <- read_csv(file = db_data, col_names = TRUE)

# make task
chspred_task <- make_sl3_Task(
  data = chspred,
  covariates = head(colnames(chspred), -1),
  outcome = "mi"
  )

# make learners
glm_learner <- Lrnr_glm$new()
lasso_learner <- Lrnr_glmnet$new(alpha = 1)
ridge_learner <- Lrnr_glmnet$new(alpha = 0)
enet_learner <- Lrnr_glmnet$new(alpha = 0.5)
# curated_glm_learner uses formula = "mi ~ smoke + beta + waist"
curated_glm_learner <- Lrnr_glm_fast$new(covariates = c("smoke, beta, waist"))
mean_learner <- Lrnr_mean$new() # That is one mean learner!
glm_fast_learner <- Lrnr_glm_fast$new()
ranger_learner <- Lrnr_ranger$new()
svm_learner <- Lrnr_svm$new()
xgb_learner <- Lrnr_xgboost$new()

# screening
screen_cor <- make_learner(Lrnr_screener_corP)
glm_pipeline <- make_learner(Pipeline, screen_cor, glm_learner)

# stack learners together
stack <- make_learner(
  Stack,
  glm_pipeline, glm_learner,
  lasso_learner, ridge_learner, enet_learner,
  curated_glm_learner, mean_learner, glm_fast_learner,
  ranger_learner, svm_learner, xgb_learner
  )

# make and train super learner
sl <- Lrnr_sl$new(
  learners = stack
  )
sl_fit <- sl$train(chspred_task)
sl_fit$print()

CVsl <- CV_lrnr_sl(sl_fit, chspred_task, loss_squared_error)
CVsl

importance <- varimp(sl_fit, loss_squared_error)
importance %>%
  mutate(name = forcats::fct_reorder(X, risk_diff)) %>%
  ggplot(aes(x = risk_diff, y = name)) +
  geom_dotplot(binaxis = "y") +
  labs(x = "Risk Difference", y = "Covariate", 
       title = "sl3 Variable Importance for Myocardian Infarction Prediction")

5.9.2 Exercise 2 Solution

Here is a potential solution to sl3 Exercise 2 – Predicting Recurrent Ischemic Stroke in an RCT with sl3.

library(sl3)
library(origami)
library(data.table)
library(tidyverse)
library(ROCR) # for AUC calculation
library(ggplot2)

ist_data <- paste0("https://raw.githubusercontent.com/tlverse/",
                   "tlverse-handbook/master/data/ist_sample.csv") %>% fread()

# stack
ist_task <- make_sl3_Task(
  data = ist_data,
  outcome = "DRSISC",
  covariates = colnames(ist_data)[-which(names(ist_data) == "DRSISC")],
  drop_missing_outcome = TRUE
  )

# learner library
lrnr_glm <- Lrnr_glm$new()
lrnr_lasso <- Lrnr_glmnet$new(alpha = 1)
lrnr_ridge <- Lrnr_glmnet$new(alpha = 0)
lrnr_enet <- Lrnr_glmnet$new(alpha = 0.5)
lrnr_mean <- Lrnr_mean$new() 
lrnr_ranger <- Lrnr_ranger$new()
lrnr_svm <- Lrnr_svm$new()
# xgboost grid
grid_params <- list(max_depth = c(2, 5, 8),
                    eta = c(0.01, 0.15, 0.3))
grid <- expand.grid(grid_params, KEEP.OUT.ATTRS = FALSE)
params_default <- list(nthread = getOption("sl.cores.learners", 1))
xgb_learners <- apply(grid, MARGIN = 1, function(params_tune) {
  do.call(Lrnr_xgboost$new, c(params_default, as.list(params_tune)))})
learners <- unlist(list(xgb_learners, lrnr_ridge, lrnr_mean, lrnr_lasso,
                        lrnr_glm, lrnr_enet, lrnr_ranger, lrnr_svm),
                   recursive = TRUE)

# super learner
sl <- Lrnr_sl$new(learners)
sl_fit <- sl$train(ist_task)

# AUC
preds <- sl_fit$predict()
obs <- c(na.omit(ist_data$DRSISC))
AUC <- performance(prediction(sl_preds, obs), measure = "auc")@y.values[[1]]
plot(performance(prediction(sl_preds, obs), "tpr", "fpr"))

# CVsl
ist_task_CVsl <- make_sl3_Task(
  data = ist_data,
  outcome = "DRSISC",
  covariates = colnames(ist_data)[-which(names(ist_data) == "DRSISC")],
  drop_missing_outcome = TRUE,
  folds = origami::make_folds(
    n = sum(!is.na(ist_data$DRSISC)), 
    fold_fun = folds_vfold,
    V = 5
    )
  )
CVsl <- CV_lrnr_sl(sl_fit, ist_task_CVsl, loss_loglik_binomial)
CVsl

# sl3 variable importance plot
importance <- varimp(sl_fit, loss_loglik_binomial)

importance %>%
  mutate(name = forcats::fct_reorder(X, risk_diff)) %>%
  ggplot(aes(x = risk_diff, y = name)) +
  geom_dotplot(binaxis = "y") +
  labs(x = "Risk Difference", y = "Covariate", 
       title = "Variable Importance for Predicting Recurrent Ischemic Stroke")