VIM: Variance-based Importance Measures in linear model

In sensitivity: Global Sensitivity Analysis of Model Outputs and Importance Measures

Variance-based Importance Measures in linear model

Description

VIM summarizes some linear variance-based importance measures useful in data analysis/machine learning context (dependent inputs' case): VIF (i.e. variance inflation factor which is a multicollinearity metric), squared SRC, squared PCC, LMG and PMVD, as well as the R2 and Q2 of the linear regression model

Usage

VIM(X, y, logistic = FALSE, nboot = 0, 
    conf = 0.95, max.iter = 1000, parl = NULL)
## S3 method for class 'VIM'
print(x, ...)
## S3 method for class 'VIM'
plot(x, ylim = c(0,1), ...)
## S3 method for class 'VIM'
ggplot(data, mapping = aes(), ..., ylim = c(0,1), 
  environment = parent.frame())

Arguments

X	a matrix or data frame containing the observed covariates (i.e., features, input variables...).
y	a numeric vector containing the observed outcomes (i.e., dependent variable). If logistic=TRUE, can be a numeric vector of zeros and ones, or a logical vector, or a factor.
logistic	logical. If TRUE, the analysis is done via a logistic regression(binomial GLM).
nboot	the number of bootstrap replicates for the computation of confidence intervals.
conf	the confidence level of the bootstrap confidence intervals.
max.iter	if logistic=TRUE, the maximum number of iterative optimization steps allowed for the logistic regression. Default is 1000.
parl	number of cores on which to parallelize the computation. If NULL, then no parallelization is done.
x	the object returned by VIM.
data	the object returned by VIM.
ylim	the y-coordinate limits of the plot.
mapping	Default list of aesthetic mappings to use for plot. If not specified, must be supplied in each layer added to the plot.
environment	[Deprecated] Used prior to tidy evaluation.
...	arguments to be passed to methods, such as graphical parameters (see par).

Details

This function cannot be used with categorical inputs.

For logistic regression (logistic=TRUE), the R^2 value is equal to:

R^2 = 1-\frac{\textrm{model deviance}}{\textrm{null deviance}}

If too many cores for the machine are passed on to the parl argument, the chosen number of cores is defaulted to the available cores minus one.

Value

VIM returns a list of class "VIM", containing the following components:

call	the matched call.
R2	a data frame containing the estimations of the R2.
Q2	a data frame containing the estimations of the Q2.
VIF	a data frame containing the estimations of the VIF.
SRC2	a data frame containing the estimations of the squared SRC.
PCC2	a data frame containing the estimations of the squared PCC.
LMG	a data frame containing the estimations of the LMG.
PMVD	a data frame containing the estimations of the PMVD.
X	the observed covariates.
y	the observed outcomes.
logistic	logical. TRUE if the analysis has been made by logistic regression.
nboot	number of bootstrap replicates.
max.iter	if logistic=TRUE, the maximum number of iterative optimization steps allowed for the logistic regression. Default is 1000.
parl	number of chosen cores for the computation.
conf	level for the confidence intervals by bootstrap.

Author(s)

Bertrand Iooss

References

L. Clouvel, B. Iooss, V. Chabridon, M. Il Idrissi and F. Robin, 2024, An overview of variance-based importance measures in the linear regression context: comparative analyses and numerical tests, Socio-Environmental Systems Modelling, vol. 7, 18681, 2025, doi:10.18174/sesmo.1868. https://hal.science/hal-04102053

See Also

src, pcc, src, lmg, pmvd

Examples

library(parallel)
library(boot)
library(car)

library(mvtnorm)

set.seed(1234)
n <- 100
sigma<-matrix(c(1,0,0,0.9, 0,1,-0.8,0, 0,-0.8,1,0, 0.9,0,0,1), nr=4, nc=4)

############################
# Gaussian correlated inputs

X <- as.data.frame(rmvnorm(n, rep(0,4), sigma))
colnames(X) <- c("X1","X2","X3","X4")

#############################
# Linear Model with small noise, two correlated inputs (X2 and X3) and 
# one dummy input (X4) correlated with another (X1)
epsilon <- rnorm(n,0,0.1)
y <- with(X, X1 - X2 + 0.5 * X3 + epsilon)

# Without Bootstrap confidence intervals
x <- VIM(X, y)
print(x)
plot(x)
library(ggplot2) ; ggplot(x)

# With Boostrap confidence intervals
x <- VIM(X, y, nboot=100, conf=0.9)
print(x)
plot(x)
library(ggplot2) ; ggplot(x)

############################
# Logistic Regression (same regression model)

epsilon <- rnorm(n,0,0.1)
y <- with(X, X1 - X2 + 0.5 * X3 + epsilon > 0)

x <- VIM(X, y, logistic = TRUE)
print(x)
plot(x)
library(ggplot2) ; ggplot(x)

sensitivity documentation built on Feb. 28, 2026, 1:06 a.m.