David Benkeser
Introduction
This package can be used to learn the convex combination of a multivariate outcome that maximizes cross-validated R-squared of a Super Learner-based prediction. Super Learner is an ensemble machine learning method with an associated R package written by Eric Polley. The SuperLearner package provides a flexible interface for utilizing user-written wrappers to perform ensemble learning. A short tutorial on use of the Super Learner package can be found here.
In addition to estimating the best convex combination of outcomes for the sake of prediction, the r2weight package simultaneously estimates a function for predicting this learned convex combination and can compute a cross-validated estimate of the R-squared of this prediction algorithm for predicting the convex combination of outcomes. The package also computes influence function-based estimates of the standard error of the estimated performance metric that are used to construct confidence intervals and hypotheses tests. Finally, the package includes functions for computing variable importance measures for each variable.
Installation
The package can be installed directly from GitHub as follows:
# install/load devtools package
if(!("devtools" %in% row.names(installed.packages()))){
install.packages("devtools")
}
library("devtools")
# install/load r2weight from GitHub
if(!("r2weight" %in% row.names(installed.packages()))){
install_github("benkeser/r2weight")
}
library("r2weight")Use
The basic workflow of the package is to call the function optWeight, which estimates the optimal weights for the combined outcome. The function r2_optWeight can then be called using the optWeight object to obtain a cross-validated estimate of the R-squared for predicting the combined outcome.
Here we illustrate the method using simulated data.
# sample size
n <- 500
# set the seed
set.seed(12345)
# simulate nine covariate predictors
x1 <- runif(n,0,4)
x2 <- runif(n,0,4)
x3 <- runif(n,0,4)
x4 <- rbinom(n,1,0.75)
x5 <- rbinom(n,1,0.25)
x6 <- rbinom(n,1,0.5)
x7 <- runif(n,0,4)
x8 <- runif(n,0,4)
x9 <- runif(n,0,4)
# put all predictors in single data.frame
X <- data.frame(x1=x1, x2=x2, x3=x3, x4=x4, x5=x5,
x6=x6, x7=x7, x8=x8, x9=x9)
# simulate three outcomes
y1 <- x1 + 2*x2 + 4*x3 + x4 + 2*x5 + 4*x6 + 2*x7 + rnorm(n, 0, 5)
y2 <- x1 + 2*x2 + 4*x3 + x4 + 2*x5 + 4*x6 + 2*x8 + rnorm(n, 0, 5)
y3 <- x1 + 2*x2 + 4*x3 + x4 + 2*x5 + 4*x6 + 2*x9 + rnorm(n, 0, 5)
# put all outcomes in single data.frame
Y <- data.frame(y1 = y1, y2 = y2, y3 = y3)
# call optWeight using simple Super Learner library
out1 <- optWeight(Y = Y, X = X, SL.library = c("SL.mean","SL.glm","SL.step"))
# print the object
out1##
## Optimal weights for prediction with SuperLearner :
## y1 : 0.371
## y2 : 0.314
## y3 : 0.315
##
##
## R-squared for each outcome with SuperLearner :
## R2 CI.l CI.h
## y1 0.643 0.589 0.689
## y2 0.613 0.559 0.660
## y3 0.625 0.573 0.671The estimated optimal weights are shown (note that the true weights in this example are 1/3, 1/3, 1/3) along with the estimated cross validated R-squared for predicting each outcome using the Super Learner. There is an S3-method for predicting the combined outcome on new data.
# generate new data
newX <- data.frame(x1=1, x2=1, x3=1, x4=1, x5=1,
x6=1, x7=1, x8=1, x9=1)
# get prediction of combined outcome
predict(out1, newdata = newX)## [,1]
## [1,] 15.80969We can call r2_optWeight to get an estimate of the cross-validated R-squared for predicting the combined outcome with Super Learner.
# cross-validated R-squared
# set verbose = TRUE to see a progress bar
r2.out1 <- r2_optWeight(out1, Y = Y, X = X)
# print the output
r2.out1##
##
## R-squared for each outcome with SuperLearner :
## R2 CI.l CI.h
## y1 0.643 0.589 0.689
## y2 0.613 0.559 0.660
## y3 0.625 0.573 0.671
##
##
##
## R-squared for combined outcome with SuperLearner :
## R2 CI.l CI.h
## weighted combination 0.815 0.784 0.841The R-squared for each individual outcome is shown, as well as for the combined outcome. In this example, the true R-squared for individual outcomes is about 0.6 and for the combined outcome about 0.8.
Variable importance
A measure of variable importance for a particular variable can be defined as the difference in R-squared for the combined outcome when including and excluding that variable. These measures can be estimated using the r2_diff function.
# measure importance of x9
out2 <- optWeight(Y = Y, X = X[,1:8], SL.library = c("SL.glm","SL.mean","SL.step"))
# print output
out2 ##
## Optimal weights for prediction with SuperLearner :
## y1 : 0.404
## y2 : 0.349
## y3 : 0.248
##
##
## R-squared for each outcome with SuperLearner :
## R2 CI.l CI.h
## y1 0.643 0.590 0.690
## y2 0.614 0.561 0.661
## y3 0.525 0.463 0.581# compare to full fit to get importance for each individual outcome
outDiff <- r2_diff(out1, out2)
# difference in R-squared for first outcome
# with confidence interval and p-value for two-sided test that
# the difference equals 0
outDiff$y1$diff## est CI.l CI.h p
## 1 -0.0006708281 -0.002014265 0.0006726093 0.3277368# for the third outcome
outDiff$y3$diff## est CI.l CI.h p
## 1 0.09986118 0.06798041 0.131742 8.290725e-10# get R-squared for combined outcome excluding x9
r2.out2 <- r2_optWeight(out2, Y = Y, X = X[,1:8])
# print output, notice change in weights
r2.out2##
##
## R-squared for each outcome with SuperLearner :
## R2 CI.l CI.h
## y1 0.643 0.590 0.690
## y2 0.614 0.561 0.661
## y3 0.525 0.463 0.581
##
##
##
## R-squared for combined outcome with SuperLearner :
## R2 CI.l CI.h
## weighted combination 0.804 0.772 0.832# compare to full fit to get importance for combined outcome
outDiff.r2 <- r2_diff(r2.out1, r2.out2)
# print output
outDiff.r2## $diff
## est CI.l CI.h p
## 1 0.01067995 0.002252427 0.01910748 0.01299872
##
## $ratio
## est CI.l CI.h p
## 1 0.9454212 0.9049312 0.9877229 0.01196758
##
## $type
## [1] "r2_optWeight"
##
## $Ynames
## [1] "y1" "y2" "y3"
##
## attr(,"class")
## [1] "r2_diff"License
© 2016-2017 David C. Benkeser
The contents of this repository are distributed under the MIT license. See below for details:
The MIT License (MIT)
Copyright (c) 2016-2017 David C. Benkeser
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