vimp_regression: Nonparametric Intrinsic Variable Importance Estimates: ANOVA

Description

Compute estimates of and confidence intervals for nonparametric ANOVA-based intrinsic variable importance. This is a wrapper function for cv_vim, with type = "anova". This function is deprecated in vimp version 2.0.0.

Usage

vimp_regression(
  Y = NULL,
  X = NULL,
  cross_fitted_f1 = NULL,
  cross_fitted_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,
  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,
  ...
)

Value

An object of classes vim and vim_regression. See Details for more information.

Arguments

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; a list of length V, where each object is a set of predictions on the validation data. If sample-splitting 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 cross_fitted_f1, or (b) Y, on X withholding the columns in indx; a list of length V, where each object is a set of predictions on the validation data. If sample-splitting is requested, then these must be estimated specially; see Details.
indx: the indices of the covariate(s) to calculate variable importance for; defaults to 1.
V: the number of folds for cross-fitting, defaults to 5. If sample_splitting = TRUE, then a special type of V-fold cross-fitting is done. See Details for a more detailed explanation.
run_regression: if outcome Y and covariates X are passed to cv_vim, and run_regression is TRUE, then Super Learner will be used; otherwise, variable importance will be computed using the inputted fitted values.
SL.library: a character vector of learners to pass to SuperLearner, if f1 and f2 are Y and X, respectively. Defaults to SL.glmnet, SL.xgboost, and SL.mean.
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 NA's in the outcome and fitted values in computation? (defaults to FALSE)
cross_fitting_folds: the folds for cross-fitting. Only used if run_regression = FALSE.
stratified: if run_regression = TRUE, then should the generated folds be stratified based on the outcome (helps to ensure class balance across cross-fitting folds)
C: the indicator of coarsening (1 denotes observed, 0 denotes unobserved).
Z: either (i) NULL (the default, in which case the argument C above must be all ones), or (ii) a character vector specifying the variable(s) among Y and X that are thought to play a role in the coarsening mechanism.
ipc_weights: weights for the computed influence curve (i.e., inverse probability weights for coarsened-at-random settings). Assumed to be already inverted (i.e., ipc_weights = 1 / [estimated probability weights]).
scale: should CIs be computed on original ("identity", default) or logit ("logit") scale?
ipc_est_type: the type of procedure used for coarsened-at-random settings; options are "ipw" (for inverse probability weighting) or "aipw" (for augmented inverse probability weighting). Only used if C is not all equal to 1.
scale_est: should the point estimate be scaled to be greater than 0? Defaults to TRUE.
cross_fitted_se: should we use cross-fitting to estimate the standard errors (TRUE, the default) or not (FALSE)?
...: other arguments to the estimation tool, see "See also".

Details

We define the population ANOVA parameter for the group of features (or single feature) $s$ by $$\psi_{0,s} := E_0\{f_0(X) - f_{0,s}(X)\}^2/var_0(Y),$$ where $f_0$ is the population conditional mean using all features, $f_{0,s}$ is the population conditional mean using the features with index not in $s$, and $E_0$ and $var_0$ denote expectation and variance under the true data-generating distribution, respectively.

Cross-fitted ANOVA estimates are computed by first splitting the data into $K$ folds; then using each fold in turn as a hold-out 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 $E_{n,k}$ of $E_0$ using the test data; and finally, computing $$\psi_{n,s} := K^{(-1)}\sum_{k=1}^K E_{n,k}\{f_{n,k}(X) - f_{n,k,s}(X)\}^2/var_n(Y),$$ where $var_n$ is the empirical variance. See the paper by Williamson, Gilbert, Simon, and Carone for more details on the mathematics behind this function.

Examples

Run this code

# 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_regression(y, x, indx = 2,
           alpha = 0.05, run_regression = TRUE,
           SL.library = learners, V = 2, cvControl = list(V = 2))