Detailed guide
In this section, we provide a fuller example of estimating \(R^2\)-based variable importance in the context of assessing the importance of amino acid sequence features in predicting the neutralization sensitivity of the HIV virus to the broadly neutralizing antibody VRC01. For more information about this study, see Magaret, Benkeser, Williamson, et al. (2019).
Often when working with data we attempt to estimate the conditional mean of the outcome \(Y\) given features \(X\), defined as \(\mu_P(x) = E_P(Y \mid X = x)\).
There are many tools for estimating this conditional mean. We might choose a classical parametric tool such as linear regression. We might also want to be model-agnostic and use a more nonparametric approach to estimate the conditional mean. However,
- This involves using some nonparametric smoothing technique, which requires: (1) choosing a technique, and (2) selecting tuning parameters
- Naive optimal tuning balances out the bias and variance of the smoothing estimator. Is this the correct trade-off for estimating the conditional mean?
Once we have a good estimate of the conditional mean, it is often of scientific interest to understand which features contribute the most to the variation in \(\mu_P\). Specifically, we might consider \[\mu_{P, s}(x) = E_P(Y \mid X_{(-s)} = x_{(-s)}),\] where for a vector \(v\) and a set of indices \(s\), \(v_{-(s)}\) denotes the elements of \(v\) with index not in \(s\). By comparing \(\mu_{P, s}\) to \(\mu_P\) we can evaluate the importance of the \(s\)th element (or group of elements).
Assume that our data are generated according to the mechanism \(P_0\). We define the population \(R^2\) value of a given regression function \(\mu\) as \(R^2(\mu, P_0) = 1 - \frac{E_{P_0}\{Y - \mu(X)\}^2}{var_{P_0}(Y)}\), where the numerator of this expression is the population mean squared error and the denominator is the population variance. We can then define a nonparametric measure of variable importance, \[\psi_{0, s} = R^2(\mu_{P_0}, P_0) - R^2(\mu_{P_0,s}, P_0),\] which is the proportion of the variability in the outcome explained by including \(X_j\) in our chosen estimation technique.
This document introduces you to the basic tools in vimp
and how to apply them to a dataset. I will explore one method for
obtaining variable estimates using vimp: you only specify a
library of candidate estimators for the conditional means \(\mu_{P_0}\) and \(\mu_{P_0, s}\); you allow vimp
to obtain the optimal estimates of these quantities using the
SuperLearner (Laan, Polley, and
Hubbard 2007), and use these estimates to obtain variable
importance estimates. A second method (using precomputed estimates of
the regression functions) exists and is described in “Using precomputed regression
function estimates in vimp”.
A look at the VRC01 data
Throughout this document I will use the VRC01 data (Magaret, Benkeser, Williamson, et al. 2019), a subset of the data freely available from the Los Alamos National Laboratory’s Compile, Neutralize, and Tally Neutralizing Antibody Panels database. Information about these data is available here.
# read in the data
data("vrc01")While there are several outcomes of interest in these data
(continuous measures of neutralization and binary measures of
resistance), we will focus on the binary measure of resistance to VRC01
given by ic50.censored. This variable is a binary indicator
that the concentration of VRC01 necessary to neutralize 50% of viral
replicates in a sample (IC-50) was right-censored; since higher values
of IC-50 imply a more resistant virus, this indicator is a proxy for
viral resistance. In addition to the outcome of interest, there are
measurements on several groups of variables: viral subtype, geographic
region of origin (a potential confounding variable), amino acid sequence
features (further grouped into the CD4 binding sites, VRC01 binding
footprint, sites with sufficient exposed surface area, sites identified
as important for glycosylation, sites with residues that covary with the
VRC01 binding footprint, sites associated with VRC01-specific potential
N-linked glycosylation (PNGS) effects, sites in gp41 associated with
VRC01 neutralization or sensitivity, sites for indicating N-linked
glycosylation), region-specific counts of PNGS, viral geometry, cysteine
counts, and steric bulk at critical locations.
For the sake of simplicity, we will focus here on only three groups of features: viral subtype, geographic region of origin, and viral geometry features.
library("dplyr")##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary("tidyselect")## Warning: package 'tidyselect' was built under R version 4.0.5# retain only the columns of interest for this analysis
y <- vrc01$ic50.censored
X <- vrc01 %>%
select(starts_with("geog"), starts_with("subtype"), starts_with("length"))Since there are 17 features and two groups, it is of interest to determine variable importance both for the individual features separately and for the two groups of features (since the geographic variables are potential confounders).
A first approach: linear regression
Suppose that I believe that a linear model truly describes the relationship between the outcome and the covariates in these data. In that case, I would be justified in only fitting a linear regression to estimate the conditional means; this means that in my importance analysis, I should also use only linear regression. The analysis is achieved by the following:
geog_indx <- max(which(grepl("geog", names(X))))
set.seed(1234)
for (i in seq_len(ncol(X) - geog_indx)) {
# note that we're using a small number of cross-fitting folds for speed
lm_vim <- vim(Y = y, X = X, indx = geog_indx + i, run_regression = TRUE, SL.library = "SL.lm", type = "r_squared", cvControl = list(V = 2))
if (i == 1) {
lm_mat <- lm_vim
} else {
lm_mat <- merge_vim(lm_mat, lm_vim)
}
}
# print out the importance
lm_mat## Variable importance estimates:
## Estimate SE 95% CI VIMP > 0 p-value
## s = 5 0.13438499 0.06330276 [0.01031386, 0.2584561] TRUE 0.01688141
## s = 11 0.09569336 0.06240703 [0.00000000, 0.2180089] FALSE 0.06259177
## s = 18 0.07087508 0.05996798 [0.00000000, 0.1884102] FALSE 0.11862626
## s = 13 0.06857330 0.06312482 [0.00000000, 0.1922957] FALSE 0.13867032
## s = 8 0.05317778 0.06370201 [0.00000000, 0.1780314] FALSE 0.20191808
## s = 12 0.03756380 0.06238020 [0.00000000, 0.1598267] FALSE 0.27352878
## s = 21 0.02864616 0.06110488 [0.00000000, 0.1484095] FALSE 0.31960515
## s = 20 0.01252676 0.06701578 [0.00000000, 0.1438753] FALSE 0.42586066
## s = 16 0.01146941 0.06511882 [0.00000000, 0.1391000] FALSE 0.43009570
## s = 6 0.01018668 0.06250895 [0.00000000, 0.1327020] FALSE 0.43527356
## s = 7 0.00000000 0.06565987 [0.00000000, 0.1286910] FALSE 0.56052497
## s = 9 0.00000000 0.06131146 [0.00000000, 0.1201682] FALSE 0.85483413
## s = 10 0.00000000 0.06182744 [0.00000000, 0.1211796] FALSE 0.74269077
## s = 14 0.00000000 0.06162074 [0.00000000, 0.1207744] FALSE 0.63428683
## s = 15 0.00000000 0.06549287 [0.00000000, 0.1283637] FALSE 0.62766542
## s = 17 0.00000000 0.06045251 [0.00000000, 0.1184847] FALSE 0.86887592
## s = 19 0.00000000 0.06174183 [0.00000000, 0.1210118] FALSE 0.78557864Building a library of learners
In general, we don’t believe that a linear model truly holds.
Thinking about potential model misspecification leads us to consider
other algorithms. Suppose that I prefer to use generalized additive
models (Hastie and Tibshirani 1990) to
estimate \(\mu_{P_0}\) and \(\mu_{P_0, s}\), so I am planning on using
the gam package. Suppose that you prefer to use the elastic
net (Zou and Hastie 2005), and are
planning to use the glmnet package.
The choice of either method is somewhat subjective, and I also will have to use a technique like cross-validation to determine an optimal tuning parameter in each case. It is also possible that neither additive models nor the elastic net will do a good job estimating the true conditional means!
This motivates using SuperLearner to allow the data to
determine the optimal combination of base learners from a
library that I define. These base learners are a combination of
different methods (e.g., generalized additive models and elastic net)
and instances of the same method with different tuning parameter values
(e.g., additive models with 3 and 4 degrees of freedom). The Super
Learner is an example of model stacking, or model aggregation — these
approaches use a data-adaptive combination of base learners to make
predictions.
For instance, my library could include the elastic net, random forests (Breiman 2001), and gradient boosted trees (Friedman 2001) as follows:
# create a function for boosted stumps
SL.gbm.1 <- function(..., interaction.depth = 1) SL.gbm(..., interaction.depth = interaction.depth)
# create GAMs with different degrees of freedom
SL.gam.3 <- function(..., deg.gam = 3) SL.gam(..., deg.gam = deg.gam)
SL.gam.4 <- function(..., deg.gam = 4) SL.gam(..., deg.gam = deg.gam)
SL.gam.5 <- function(..., deg.gam = 5) SL.gam(..., deg.gam = deg.gam)
# add more levels of alpha for glmnet
create.SL.glmnet <- function(alpha = c(0.25, 0.5, 0.75)) {
for (mm in seq(length(alpha))) {
eval(parse(file = "", text = paste('SL.glmnet.', alpha[mm], '<- function(..., alpha = ', alpha[mm], ') SL.glmnet(..., alpha = alpha)', sep = '')), envir = .GlobalEnv)
}
invisible(TRUE)
}
create.SL.glmnet()
# add tuning parameters for randomForest
create.SL.randomForest <- function(tune = list(mtry = c(1, 5, 7), nodesize = c(1, 5, 10))) {
tuneGrid <- expand.grid(tune, stringsAsFactors = FALSE)
for (mm in seq(nrow(tuneGrid))) {
eval(parse(file = "", text = paste("SL.randomForest.", mm, "<- function(..., mtry = ", tuneGrid[mm, 1], ", nodesize = ", tuneGrid[mm, 2], ") SL.randomForest(..., mtry = mtry, nodesize = nodesize)", sep = "")), envir = .GlobalEnv)
}
invisible(TRUE)
}
create.SL.randomForest()
# create the library
learners <- c("SL.glmnet", "SL.glmnet.0.25", "SL.glmnet.0.5", "SL.glmnet.0.75",
"SL.randomForest", "SL.randomForest.1", "SL.randomForest.2", "SL.randomForest.3",
"SL.randomForest.4", "SL.randomForest.5", "SL.randomForest.6", "SL.randomForest.7",
"SL.randomForest.8", "SL.randomForest.9",
"SL.gbm.1")Now that I have created the library of learners, I can move on to estimating variable importance.
Estimating variable importance for a single variable
The main function for R-squared-based variable importance in the
vimp package is the vimp_rsquared() function.
There are five main arguments to vimp_rsquared():
Y, the outcomeX, the covariatesindx, which determines the feature I want to estimate variable importance forSL.library, the library of candidate learnersV, the number of cross-fitting folds (also referred to as cross-validation folds) to use for computing variable importance
The main arguments differ if precomputed regression function
estimates are used; please see “Using precomputed regression
function estimates in vimp” for further discussion of
this case.
Suppose that the first feature that I want to estimate variable
importance for is whether the viral subtype is 01_AE,
subtype.is.01_AE. Then supplying
vimp_rsquared() with
Y = yX = Xindx = 5SL.library = learnersV = 5
means that:
- I want to use
SuperLearner()to estimate the conditional means \(\mu_{P_0}\) and \(\mu_{P_0,s}\), and my candidate library islearners - I want to estimate variable importance for the fifth column of the
VRC01 covariates, which is
subtype.is.01_AE - I want to use five-fold cross-fitting to estimate importance
The call to vimp_rsquared() looks like this:
vimp_rsquared(Y = y, X = X,
indx = 5, run_regression = TRUE, SL.library = learners, V = 5)While this is the preferred method for estimating variable importance, using a large library of learners may cause the function to take time to run. Usually this is okay — in general, you took a long time to collect the data, so letting an algorithm run for a few hours should not be an issue.
However, for the sake of illustration, I can estimate varibable importance for 01_AE subtype only using only using a small library, a small number of cross-validation folds in the Super Learner, and a small number of cross-fitting folds as follows (again, I suggest using a larger number of folds and a larger library in practice):
# small learners library
learners.2 <- c("SL.ranger")
# small number of cross-fitting folds
V <- 2
# small number of CV folds for Super Learner
sl_cvcontrol <- list(V = 2)
# now estimate variable importance
set.seed(5678)
start_time <- Sys.time()
subtype_01_AE_vim <- vimp_rsquared(Y = y, X = X, indx = 5, SL.library = learners.2, na.rm = TRUE, env = environment(), V = V, cvControl = sl_cvcontrol, family = binomial())
end_time <- Sys.time()This code takes approximately 0.44 seconds to run on a (not very fast) PC. I can display these estimates:
subtype_01_AE_vim## Variable importance estimates:
## Estimate SE 95% CI VIMP > 0 p-value
## s = 5 0.09419998 0.05592714 [0, 0.2038152] TRUE 0.04605858The object returned by vimp_rsquared() also contains
lists of fitted values from using SuperLearner(); I access
these using $full_fit and $red_fit. For
example,
head(subtype_01_AE_vim$full_fit[[1]])## [1] 0.06375393head(subtype_01_AE_vim$red_fit[[1]])## [1] 0.05942255I can obtain estimates for the remaining individual features in the same way (again using only using a small library for illustration):
ests <- subtype_01_AE_vim
set.seed(1234)
for (i in seq_len(ncol(X) - geog_indx - 1)) {
# note that we're using a small number of cross-fitting folds for speed
this_vim <- vimp_rsquared(Y = y, X = X, indx = geog_indx + i + 1, run_regression = TRUE, SL.library = learners.2, V = V, cvControl = sl_cvcontrol, family = binomial())
ests <- merge_vim(ests, this_vim)
}Now that I have estimates of each of individual feature’s variable importance, I can view them all simultaneously by plotting:
library("ggplot2")
library("cowplot")## Warning: package 'cowplot' was built under R version 4.0.5theme_set(theme_cowplot())
all_vars <- c(paste0("Subtype is ", c("01_AE", "02_AG", "07_BC", "A1", "A1C", "A1D",
"B", "C", "D", "O", "Other")),
paste0("Length of ", c("Env", "gp120", "V5", "V5 outliers", "Loop E",
"Loop E outliers")))
est_plot_tib <- ests$mat %>%
mutate(
var_fct = rev(factor(s, levels = ests$mat$s,
labels = all_vars[as.numeric(ests$mat$s) - geog_indx],
ordered = TRUE))
)
# plot
est_plot_tib %>%
ggplot(aes(x = est, y = var_fct)) +
geom_point() +
geom_errorbarh(aes(xmin = cil, xmax = ciu)) +
xlab(expression(paste("Variable importance estimates: ", R^2, sep = ""))) +
ylab("") +
ggtitle("Estimated individual feature importance") +
labs(subtitle = "in the VRC01 data (considering only geographic confounders, subtype, and viral geometry)")
