R/vimp_rsquared.R
vimp_rsquared.Rd
Compute estimates of and confidence intervals for nonparametric $R^2$based
intrinsic variable importance. This is a wrapper function for cv_vim
,
with type = "r_squared"
.
vimp_rsquared( Y = NULL, X = NULL, cross_fitted_f1 = NULL, cross_fitted_f2 = NULL, f1 = NULL, f2 = NULL, indx = 1, V = 10, run_regression = TRUE, SL.library = c("SL.glmnet", "SL.xgboost", "SL.mean"), alpha = 0.05, delta = 0, na.rm = FALSE, cross_fitting_folds = NULL, sample_splitting_folds = NULL, stratified = FALSE, C = rep(1, length(Y)), Z = NULL, ipc_weights = rep(1, length(Y)), scale = "identity", ipc_est_type = "aipw", scale_est = TRUE, cross_fitted_se = TRUE, ... )
Y  the outcome. 

X  the covariates. 
cross_fitted_f1  the predicted values on validation data from a flexible estimation technique regressing Y on X in the training data. Provided as either (a) a list of length V, where each element in the list is a set of predictions on the corresponding validation data fold; or (b) a vector, where each element is the predicted value when that observation is part of the validation fold. If samplesplitting is requested, then these must be estimated specially; see Details. 
cross_fitted_f2  the predicted values on validation data from a
flexible estimation technique regressing either (a) the fitted values in

f1  the fitted values from a flexible estimation technique
regressing Y on X. If samplesplitting is requested, then these must be
estimated specially; see Details. If 
f2  the fitted values from a flexible estimation technique
regressing either (a) 
indx  the indices of the covariate(s) to calculate variable importance for; defaults to 1. 
V  the number of folds for crossfitting, defaults to 5. If

run_regression  if outcome Y and covariates X are passed to

SL.library  a character vector of learners to pass to

alpha  the level to compute the confidence interval at. Defaults to 0.05, corresponding to a 95% confidence interval. 
delta  the value of the \(\delta\)null (i.e., testing if importance < \(\delta\)); defaults to 0. 
na.rm  should we remove NAs in the outcome and fitted values
in computation? (defaults to 
cross_fitting_folds  the folds for crossfitting. Only used if

sample_splitting_folds  the folds used for samplesplitting;
these identify the observations that should be used to evaluate
predictiveness based on the full and reduced sets of covariates, respectively.
Only used if 
stratified  if run_regression = TRUE, then should the generated folds be stratified based on the outcome (helps to ensure class balance across crossvalidation folds) 
C  the indicator of coarsening (1 denotes observed, 0 denotes unobserved). 
Z  either (i) NULL (the default, in which case the argument

ipc_weights  weights for the computed influence curve (i.e., inverse probability weights for coarsenedatrandom settings). Assumed to be already inverted (i.e., ipc_weights = 1 / [estimated probability weights]). 
scale  should CIs be computed on original ("identity") or logit ("logit") scale? 
ipc_est_type  the type of procedure used for coarsenedatrandom
settings; options are "ipw" (for inverse probability weighting) or
"aipw" (for augmented inverse probability weighting).
Only used if 
scale_est  should the point estimate be scaled to be greater than 0?
Defaults to 
cross_fitted_se  should we use crossfitting to estimate the standard
errors ( 
...  other arguments to the estimation tool, see "See also". 
An object of classes vim
and vim_rsquared
.
See Details for more information.
We define the population variable importance measure (VIM) for the group of features (or single feature) \(s\) with respect to the predictiveness measure \(V\) by $$\psi_{0,s} := V(f_0, P_0)  V(f_{0,s}, P_0),$$ where \(f_0\) is the population predictiveness maximizing function, \(f_{0,s}\) is the population predictiveness maximizing function that is only allowed to access the features with index not in \(s\), and \(P_0\) is the true datagenerating distribution.
Crossfitted VIM estimates are computed differently if samplesplitting is requested versus if it is not. We recommend using samplesplitting in most cases, since only in this case will inferences be valid if the variable(s) of interest have truly zero population importance. The purpose of crossfitting is to estimate \(f_0\) and \(f_{0,s}\) on independent data from estimating \(P_0\); this can result in improved performance, especially when using flexible learning algorithms. The purpose of samplesplitting is to estimate \(f_0\) and \(f_{0,s}\) on independent data; this allows valid inference under the null hypothesis of zero importance.
Without samplesplitting, crossfitted VIM estimates are obtained by first splitting the data into \(K\) folds; then using each fold in turn as a holdout set, constructing estimators \(f_{n,k}\) and \(f_{n,k,s}\) of \(f_0\) and \(f_{0,s}\), respectively on the training data and estimator \(P_{n,k}\) of \(P_0\) using the test data; and finally, computing $$\psi_{n,s} := K^{(1)}\sum_{k=1}^K \{V(f_{n,k},P_{n,k})  V(f_{n,k,s}, P_{n,k})\}.$$
With samplesplitting, crossfitted VIM estimates are obtained by first splitting the data into \(2K\) folds. These folds are further divided into 2 groups of folds. Then, for each fold \(k\) in the first group, estimator \(f_{n,k}\) of \(f_0\) is constructed using all data besides the kth fold in the group (i.e., \((2K  1)/(2K)\) of the data) and estimator \(P_{n,k}\) of \(P_0\) is constructed using the heldout data (i.e., \(1/2K\) of the data); then, computing $$v_{n,k} = V(f_{n,k},P_{n,k}).$$ Similarly, for each fold \(k\) in the second group, estimator \(f_{n,k,s}\) of \(f_{0,s}\) is constructed using all data besides the kth fold in the group (i.e., \((2K  1)/(2K)\) of the data) and estimator \(P_{n,k}\) of \(P_0\) is constructed using the heldout data (i.e., \(1/2K\) of the data); then, computing $$v_{n,k,s} = V(f_{n,k,s},P_{n,k}).$$ Finally, $$\psi_{n,s} := K^{(1)}\sum_{k=1}^K \{v_{n,k}  v_{n,k,s}\}.$$
See the paper by Williamson, Gilbert, Simon, and Carone for more
details on the mathematics behind the cv_vim
function, and the
validity of the confidence intervals.
In the interest of transparency, we return most of the calculations
within the vim
object. This results in a list including:
the column(s) to calculate variable importance for
the library of learners passed to SuperLearner
the fitted values of the chosen method fit to the full data (a list, for train and test data)
the fitted values of the chosen method fit to the reduced data (a list, for train and test data)
the estimated variable importance
the naive estimator of variable importance
the estimated efficient influence function
the estimated efficient influence function for the full regression
the estimated efficient influence function for the reduced regression
the standard error for the estimated variable importance
the \((1\alpha) \times 100\)% confidence interval for the variable importance estimate
a decision to either reject (TRUE) or not reject (FALSE) the null hypothesis, based on a conservative test
a pvalue based on the same test as test
the object returned by the estimation procedure for the full data regression (if applicable)
the object returned by the estimation procedure for the reduced data regression (if applicable)
the level, for confidence interval calculation
the folds used for hypothesis testing
the folds used for crossfitting
the outcome
the weights
a tibble with the estimate, SE, CI, hypothesis testing decision, and pvalue
SuperLearner
for specific usage of the
SuperLearner
function and package.
# generate the data # generate X p < 2 n < 100 x < data.frame(replicate(p, stats::runif(n, 5, 5))) # apply the function to the x's smooth < (x[,1]/5)^2*(x[,1]+7)/5 + (x[,2]/3)^2 # generate Y ~ Normal (smooth, 1) y < smooth + stats::rnorm(n, 0, 1) # set up a library for SuperLearner; note simple library for speed library("SuperLearner") learners < c("SL.glm", "SL.mean") # estimate (with a small number of folds, for illustration only) est < vimp_rsquared(y, x, indx = 2, alpha = 0.05, run_regression = TRUE, SL.library = learners, V = 2, cvControl = list(V = 2))#> Warning: Original estimate < 0; returning zero.