sp_vim {vimp}

Shapley Population Variable Importance Measure (SPVIM) Estimates and Inference

Description

Compute estimates and confidence intervals for the SPVIMs, using cross-fitting.

Usage

sp_vim(
  Y = NULL,
  X = NULL,
  V = 5,
  type = "r_squared",
  SL.library = c("SL.glmnet", "SL.xgboost", "SL.mean"),
  univariate_SL.library = NULL,
  gamma = 1,
  alpha = 0.05,
  delta = 0,
  na.rm = FALSE,
  stratified = FALSE,
  verbose = FALSE,
  sample_splitting = TRUE,
  final_point_estimate = "split",
  C = rep(1, length(Y)),
  Z = NULL,
  ipc_scale = "identity",
  ipc_weights = rep(1, length(Y)),
  ipc_est_type = "aipw",
  scale = "identity",
  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.

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.

type

the type of importance to compute; defaults to r_squared, but other supported options are auc, accuracy, deviance, and anova.

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.

univariate_SL.library

(optional) a character vector of learners to pass to SuperLearner for estimating univariate regression functions. Defaults to SL.polymars

gamma

the fraction of the sample size to use when sampling subsets (e.g., gamma = 1 samples the same number of subsets as the sample size)

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)

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)

verbose

should sp_vim and SuperLearner print out progress? (defaults to FALSE)

sample_splitting

should we use sample-splitting to estimate the full and reduced predictiveness? Defaults to TRUE, since inferences made using sample_splitting = FALSE will be invalid for variables with truly zero importance.

final_point_estimate

if sample splitting is used, should the final point estimates be based on only the sample-split folds used for inference ("split", the default), or should they instead be based on the full dataset ("full") or the average across the point estimates from each sample split ("average")? All three options result in valid point estimates – sample-splitting is only required for valid inference.

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_scale

what scale should the inverse probability weight correction be applied on (if any)? Defaults to "identity". (other options are "log" and "logit")

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]).

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

should CIs be computed on original ("identity") or another scale? (options are "log" and "logit")

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 SPVIM as the weighted average of the population difference in predictiveness over all subsets of features not containing feature j.

This is equivalent to finding the solution to a population weighted least squares problem. This key fact allows us to estimate the SPVIM using weighted least squares, where we first sample subsets from the power set of all possible features using the Shapley sampling distribution; then use cross-fitting to obtain estimators of the predictiveness of each sampled subset; and finally, solve the least squares problem given in Williamson and Feng (2020).

See the paper by Williamson and Feng (2020) for more details on the mathematics behind this 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 containing:

SL.library

the library of learners passed to SuperLearner

v

the estimated predictiveness measure for each sampled subset

fit_lst

the fitted values on the entire dataset from the chosen method for each sampled subset

preds_lst

the cross-fitted predicted values from the chosen method for each sampled subset

est

the estimated SPVIM value for each feature

ics

the influence functions for each sampled subset

var_v_contribs

the contibutions to the variance from estimating predictiveness

var_s_contribs

the contributions to the variance from sampling subsets

ic_lst

a list of the SPVIM influence function contributions

se

the standard errors for the estimated variable importance

ci

the (1-\alpha) \times 100% confidence intervals based on the variable importance estimates

p_value

p-values for the null hypothesis test of zero importance for each variable

test_statistic

the test statistic for each null hypothesis test of zero importance

test

a hypothesis testing decision for each null hypothesis test (for each variable having zero importance)

gamma

the fraction of the sample size used when sampling subsets

alpha

the level, for confidence interval calculation

delta

the delta value used for hypothesis testing

y

the outcome

ipc_weights

the weights

scale

the scale on which CIs were computed

mat

- a tibble with the estimates, SEs, CIs, hypothesis testing decisions, and p-values

Value

An object of class vim. See Details for more information.

See Also

SuperLearner for specific usage of the SuperLearner function and package.

Examples

n <- 100
p <- 2
# generate the data
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 <- as.matrix(smooth + stats::rnorm(n, 0, 1))

# set up a library for SuperLearner; note simple library for speed
library("SuperLearner")
learners <- c("SL.glm")

# -----------------------------------------
# using Super Learner (with a small number of CV folds,
# for illustration only)
# -----------------------------------------
set.seed(4747)
est <- sp_vim(Y = y, X = x, V = 2, type = "r_squared",
SL.library = learners, alpha = 0.05)