In wuziyueemory/twostageSL: Two-Stage Super Learner Prediction Function

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twostageSL: Two-stage Super Learner for predicting healthcare cost with zero-inflation

Tools for analyzing healthcare cost data with zero inflation and skewed distribution via two stage super learner

Author: Ziyue Wu

Motivation

The study of healthcare expenditures has become an important area in epidemiological and public health researches as healthcare utilization and associated costs have increased rapidly in recent years. Estimating healthcare expenditures is challenging due to heavily skewed distributions and zero-inflation. Myriad methods have been developed for analyzing cost data, however, a priori determination of an appropriate method is difficult. Super-learning, a technique that considers an ensemble of methods for cost estimation, provides an interesting alternative for modeling healthcare expenditures as it addresses the difficulties in choosing the correct model form. The super learner has shown benefits over a single method in recent studies. Nevertheless, it is notable that none of the healthcare cost studies have explored the use of super learner in the case of zero-inflated cost data. Here we proposed a new method called the two-stage super learner that implemented the super-learning in a two-part model framework. The two-stage super learner had natural support for cases of zero healthcare utilization and extended the standard super learner by providing an additional layer of the ensemble, thus allowing for greater flexibility and improving the cost prediction.

Two-stage Super Learner

To deal with zero-inflation in the positively skewed data, we designed a method that implemented the super learner under the Two-part model framework (Two-stage Super Learner). Using $E[Y|X] = Pr(Y>0|X)E[Y|Y>0,X]$, the objective of estimating $E[Y|X]$ could be broken into two separate pieces: estimating $Pr(Y>0|X)$ and $E[Y|Y>0,X]$. This is done by adding an additional layer of the ensemble to tackle the problem of a point mass at zero. Rather than specifying a single library of prediction algorithms as in standard super learner, we specify two separate libraries of prediction algorithms, one for each of the two stages. Specifically, we specify the first stage library by positing a collection of different estimators predicting the probability of the outcome being positive and specify the second stage library by positing another collection of estimators predicting the mean of positive outcomes. Assuming the first stage library includes $K1$ methods and the second stage library includes $K2$ methods, then the two-stage super learner’s “whole library” would contain $K1*K2$ candidate estimators with each one representing a specific combination of methods from the first stage and second stage. The two-stage super learner is also an ensembling method built through a combination of algorithms that provide the best fit to the data.

Features

twostageSL is an R package that provides tools for analyzing health care cost data useing the super learning technique in the two stage framework.

• Automatic optimal predictor ensembling via cross-validation with one line of code.
• Dozens of algorithms: XGBoost, Random Forest, GBM, Lasso, SVM, BART, KNN, Decision Trees, Neural Networks, and additional algorithms specifically for heavily skewed zero-inflation data: Tweedie, Tobit, ZIP, Cox Hazard, Adaptive Hazard, Quantile Regression, Acclerated Failture Time.
• Two options for conducting either a two stage super learner or a standard superlearner. Include two seperate libraries for prediction algorithms at two stages, accompanied with another library with prediction algorithms for standard superlearner.
• Screen variables (feature selection) based on univariate association, Random Forest, Elastic Net, et al. or custom screening algorithms.
• Includes framework to quickly add custom algorithms to the ensemble.

Installation

twostageSL can be installed from Github using the following command:

# install.packages("devtools")

library(devtools)
devtools::install_github("wuziyueemory/twostageSL")

library(twostageSL)

Example

To show an example on how to use this package, we simulated an example with training data containing 1000 observations, 5 predictors and testing data containing 100 observations, 5 predictions, all following a normal distribution. The outcome is continuous and non-negative with relatively high proportion of zeros.

Generate training and testing set.

library(twostageSL)
set.seed(121)
## training set
n <- 1000
p <- 5
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
colnames(X) <- paste("X", 1:p, sep="")
X <- data.frame(X)
Y <- rep(NA,n)
## probability of outcome being zero
prob <- plogis(1 + X[,1] + X[,2] + X[,1]*X[,2])
g <- rbinom(n,1,prob)
## assign zero outcome
ind <- g==0
Y[ind] <- 0
## assign non-zero outcome
ind <- g==1
Y[ind] <- 10 + X[ind, 1] + sqrt(abs(X[ind, 2] * X[ind, 3])) + X[ind, 2] - X[ind, 3] + rnorm(sum(ind))

## test set
m <- 100
newX <- matrix(rnorm(m*p), nrow = m, ncol = p)
colnames(newX) <- paste("X", 1:p, sep="")
newX <- data.frame(newX)
newY <- rep(NA,m)
## probability of outcome being zero
newprob <- plogis(1 + newX[,1] + newX[,2] + newX[,1]*newX[,2])
newg <- rbinom(n,1,newprob)
## assign zero outcome
newind <- newg==0
newY[newind] <- 0
## assign non-zero outcome
newind <- g==1
newY[newind] <- 10 + newX[newind, 1] + sqrt(abs(newX[newind, 2] * newX[newind, 3])) + newX[newind, 2] - X[newind, 3] + rnorm(sum(newind))

The training data looks like

head(X)

The proportion of zeros in outcome Y.

mean(Y==0)

The distribution of outcome Y, which is zero-inflated and heavily skewed.

hist(Y,breaks = 100,freq = FALSE,main="Distribution of Y")

twostageSL is the core function to fit the two stage super learner. At a minimum for implementation, you need to specify the predictor matrix X, outcome variable Y, library of prediction algorithms at two stages, boolean for two stage superlearner or standard superlearner, distribution at two stages and number of folds for cross-validation. The package incldues all the prediction algorithms from package SuperLearner and also includes additional prediction algorithms that specifically used at stage 2 to deal with heavily skewed data.

Generate the library and run the two stage super learner

## generate the Library
twostage.library <- list(stage1=c("SL.mean","SL.glm","SL.earth"),
                       stage2=c("SL.mean","SL.glm","SL.earth","SL.coxph"))

onestage.library <- c("SL.mean","SL.glm","SL.earth")


## run the twostage super learner
two <- twostageSL(Y=Y,
                   X=X,
                   newX = newX,
                   library.2stage <- twostage.library,
                   library.1stage <- onestage.library,
                   twostage = TRUE,
                   cvControl = list(V = 5))
two

When setting twostage to FALSE, we get the result from standard super learner

## run the standard super learner
one <- twostageSL(Y=Y,
                  X=X,
                  newX = newX,
                  library.2stage <- twostage.library,
                  library.1stage <- onestage.library,
                  twostage = FALSE,
                  cvControl = list(V = 5))
one

We could also use CV.twostageSL to get v-fold cross validated risk estimate for two stage super learner, standard super learner, and all prediction algorithms in the library. In this case we set numbert of folds to be 2.

two_cv <- CV.twostageSL(
  Y = Y, X = X,
  library.2stage = twostage.library,
  library.1stage = onestage.library,
  cvControl = list(V = 2),
  innerCvControl = list(list(V = 5),
                        list(V = 5))
)
two_cv

The summary and plots are shown below, with the two stage super learner outperforming all other prediction algorithms.

summary(two_cv)
plot(two_cv)

We could use predict to obtain predictions on a new dataset from the twostageSL fit. The onlySL argument controls whether to compute predictions for all algorithms or for algorithms with non-zero coefficients in the two stage super learner fit.

newdata <- newX[c(1:50),]
prediction <- predict(two,newdata = newX, onlySL = TRUE)

This returns a list with predictions from two stage super learner (pred) and predictions for each algorithm in library (library.predict).

wuziyueemory/twostageSL documentation built on Oct. 19, 2020, 3:45 p.m.