vimp_anova {vimp}

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 type has limited functionality compared to other types; in particular, null hypothesis tests are not possible using type = "anova". If you want to do null hypothesis testing on an equivalent population parameter, use vimp_rsquared instead.

Usage

vimp_anova(
  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 = "logit",
  ipc_est_type = "aipw",
  scale_est = TRUE,
  cross_fitted_se = TRUE,
  ...
)

Arguments

Y

the outcome.

X

the covariates. If type = "average_value", then the exposure variable should be part of X, with its name provided in exposure_name.

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 vector, where each element is the predicted value when that observation is part of the validation fold; or (b) a list of length V, where each element in the list is a set of predictions on the corresponding validation data fold. If sample-splitting is requested, then these must be estimated specially; see Details. However, the resulting vector should be the same length as Y; if using a list, then the summed length of each element across the list should be the same length as Y (i.e., each observation is included in the predictions).

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. Provided as either (a) a vector, where each element is the predicted value when that observation is part of the validation fold; or (b) a list of length V, where each element in the list is a set of predictions on the corresponding validation data fold. If sample-splitting is requested, then these must be estimated specially; see Details. However, the resulting vector should be the same length as Y; if using a list, then the summed length of each element across the list should be the same length as Y (i.e., each observation is included in the predictions).

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 vimp_accuracy, 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 NAs 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-validation 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. To specify the outcome, use "Y"; to specify covariates, use a character number corresponding to the desired position in X (e.g., "1").

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") or another scale? (options are "log" and "logit")

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 or equal to 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.

Value

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

See Also

SuperLearner for specific usage of the SuperLearner function and package.

Examples

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