Individual Variable Priority (iVarPro): Case-Specific Variable Importance

Individual Variable Priority (iVarPro) computes case-specific (individual-level) variable importance scores. For each observation in the data and for each predictor identified by the VarPro analysis, iVarPro returns a local gradient-based priority measure that quantifies how sensitive that case's prediction is to changes in that variable.

ivarpro(object,
        adaptive = TRUE,
        cut = NULL,
        cut.max = 1,
        ncut = 51,
        nmin = 20, nmax = 150,
        y.external = NULL,
        noise.na = TRUE,
        max.rules.tree = NULL,
        max.tree = NULL,
        use.loo = TRUE,
        use.abs = FALSE,
        path.store.membership = TRUE,
        keep.data = TRUE)

Arguments

object

varpro object from a previous call to varpro, or a rfsrc object.

adaptive

Logical. If FALSE and cut is not supplied, the cut grid is constructed as seq(0, cut.max, length.out = ncut). If TRUE (default) and cut is not supplied, a data-adaptive upper bound for the neighborhood scale is computed from the sample size using a simple bandwidth-style rule-of-thumb, and cut is constructed as a sequence from 0 to this data-adaptive maximum (subject to cut.max). This provides a convenient way to automatically sharpen the local neighborhood for case-specific gradients when the sample size is moderate to large.

cut

Optional user-supplied sequence of \(\lambda\) values used to relax the constraint region in the local linear regression model. For continuous release variables, each value in cut is calibrated so that cut = 1 corresponds to one standard deviation of the release coordinate. If cut is supplied, it is used as-is and the arguments cut.max, ncut, and adaptive are ignored. For binary or one-hot encoded release variables, the full released region is used and cut does not control neighborhood size.

cut.max

Maximum value of the \(\lambda\) grid used to define the local neighborhood for continuous release variables when cut is not supplied. By default, cut is constructed as seq(0, cut.max, length.out = ncut) (or up to a data-adaptive value if adaptive = TRUE). Smaller values of cut.max yield more local, sharper case-specific gradients, while larger values yield smoother, more global behavior.

ncut

Length of the cut grid when cut is not supplied. The grid is constructed as seq(0, cut.max, length.out = ncut) (or up to an adaptively chosen maximum if adaptive = TRUE).

nmin

Minimum number of observations required for fitting a local linear model.

nmax

Maximum number of observations allowed for fitting a local linear model. Internally, nmax is capped at 10% of the sample size.

y.external

Optional user-supplied response vector or matrix to use as the dependent variable in the local linear regression. Must have the same number of rows as the feature matrix and match the dimension and type expected for the outcome family.

noise.na

Logical. If TRUE (default), gradients for noisy or non-signal variables are set to NA; if FALSE, they are set to zero.

max.rules.tree

Maximum number of rules per tree. If unspecified, the value from the varpro object is used, while for rfsrc objects, a default value is used.

max.tree

Maximum number of trees used to extract rules. If unspecified, the value from the varpro object is used, while for rfsrc objects, a default value is used.

use.loo

Logical. If TRUE (default), leave-one-out cross-validation is used to select the best neighborhood size (i.e., the best value in cut) for each rule and release variable. If FALSE, the neighborhood is chosen to use the largest available sample that satisfies nmin and nmax.

use.abs

Use the absolute gradient for individual importance? Default is FALSE which uses the actual gradient.

path.store.membership

Store the rule membership indices (OOB case IDs) in the returned object for later ladder/band calculations? Setting FALSE can substantially reduce memory usage when the number of rules is large, but disables ladder-based bands in partial.ivarpro() and prevents ivarpro_band() from being used. Default is TRUE.

keep.data

Save the x and y data (default is TRUE)? Used for downstream plots.

Details

Understanding individual-level (case-specific) variable importance is important in applications where decisions are made at the level of a single person, unit, or record. A predictor may have only a modest average effect, yet be highly influential for certain cases, or the direction of its effect may differ across individuals.

The VarPro framework summarizes population-level importance by defining feature-space regions using rule-based splitting and computing importance using only observed data. iVarPro (Lu and Ishwaran, 2025) extends this idea to the individual level by quantifying how sensitive each case's prediction is to small changes in a predictor identified by the VarPro rule set.

For each VarPro rule, iVarPro considers the corresponding rule-defined region and then releases the rule along the rule's release coordinate. Intuitively, releasing a region means keeping the other rule constraints in place while allowing additional variation in the released variable, which provides the information needed to estimate a local directional effect. A simple local linear regression is then fit on this released region, and the resulting slope is used as a local, gradient-based priority score. Case-specific scores are obtained by aggregating the relevant rule-level gradients over the rules that apply to each case.

Neighborhood size and cut.max. For continuous release variables, the size of the local neighborhood used for slope estimation is controlled by cut (constructed from cut.max and ncut when not supplied). Smaller neighborhoods produce more local behavior and can better reflect sharp changes, while larger neighborhoods produce smoother, more global behavior. When use.loo = TRUE, the neighborhood size is chosen in a data-driven way using a leave-one-out criterion; when use.loo = FALSE, the choice is based on meeting the requested sample-size bounds nmin and nmax. When adaptive = TRUE and cut is not supplied, an additional sample-size based rule is used to limit the maximum neighborhood scale (subject to cut.max).

For binary or one-hot encoded release variables, iVarPro interprets the local effect as a scaled finite difference between the two levels (0 and 1), conditional on the other rule constraints; in this case cut does not control the neighborhood along the binary coordinate.

Cut.max ladder (neighborhood sensitivity). Because the choice of neighborhood scale can affect the estimated local gradients, iVarPro also records a ladder of rule-level gradient estimates across the candidate neighborhood sizes defined by the cut grid. These ladder values can be summarized (e.g., ranges or quantiles) and used to visualize how sensitive case-specific gradients are to the neighborhood choice, without repeatedly refitting iVarPro for many different cut.max values. Ladder-based case summaries require rule membership information; set path.store.membership = TRUE to enable ladder bands and related summaries, or leave it FALSE to reduce memory usage when the number of rules is very large. See examples below.

Settings that are currently handled. The flexibility of this framework makes it suitable for quantifying case-specific variable importance in regression, classification, and survival settings. Currently, multivariate forests are not handled.

Value

For univariate outcomes (and two-class classification treated as a single score), a numeric data.frame of dimension \(n \times p\) containing case-specific (individual-level) variable priority values, where \(n\) is the number of observations and \(p\) is the number of predictors in object$xvar.names.

  • Each row corresponds to a case (observation) in the original data.

  • Each column corresponds to a predictor variable in object$xvar.names.

The entry in row \(i\) and column \(j\) is the iVarPro importance score for variable \(j\) for case \(i\), measuring the local sensitivity of that case's prediction to changes in that variable. Predictors that are never used as release variables in the VarPro rule set may appear with constant NA values (when noise.na = TRUE) or constant zero values (when noise.na = FALSE).

Ladder/path information. The returned object carries an attribute "ivarpro.path" containing additional information used for ladder-based summaries and plotting. In particular:

cut

The full cut grid used to evaluate candidate local neighborhoods.

cut.ladder

The interior values of cut (excluding the endpoints) used for the cut.max ladder path.

rule.imp.ladder

A numeric matrix of dimension \(R \times L\) storing rule-level gradients selected under each ladder truncation, where \(R\) is the number of retained rules and \(L = length(cut.ladder)\).

rule.variable

Integer vector of length \(R\) giving the release-variable index for each retained rule.

oobMembership

Optional list (length \(R\)) giving the OOB membership indices for each retained rule; included only when path.store.membership = TRUE.

Additional tuning flags and rule metadata (e.g., use.loo, adaptive, and tree/branch identifiers) may also be included for diagnostics.

Author

Min Lu and Hemant Ishwaran

References

Lu, M. and Ishwaran, H. (2025). Individual variable priority: a model-independent local gradient method for variable importance. Artificial Intelligence Review, 58:407.

See also

varpro

Examples

# \donttest{

## ------------------------------------------------------------
##
## survival example with shap-like plot
##
## ------------------------------------------------------------

data(peakVO2, package = "randomForestSRC")
o <- varpro(Surv(ttodead, died)~., peakVO2, ntree = 50)

## canonical standard analysis
imp1 <- ivarpro(o)
shap.ivarpro(imp1)

## non-adaptive analysis
imp2 <- ivarpro(o, adaptive = FALSE)
shap.ivarpro(imp2)

## non-adaptive using a small cut.max
imp3 <- ivarpro(o, cut.max = 0.5, adaptive = FALSE)
shap.ivarpro(imp3)


## ------------------------------------------------------------
##
## synthetic regression example with partial plot
##
## ------------------------------------------------------------

## true regression function
true.function <- function(which.simulation) {
  if (which.simulation == 1) {
    function(x1, x2) { 1 * (x2 <= .25) +
      15 * x2 * (x1 <= .5 & x2 > .25) +
      (7 * x1 + 7 * x2) * (x1 > .5 & x2 > .25) }
  }
  else if (which.simulation == 2) {
    function(x1, x2) { r <- x1^2 + x2^2; 5 * r * (r <= .5) }
  }
  else {
    function(x1, x2) { 6 * x1 * x2 }
  }
}

## simulation function
simfunction <- function(n = 1000, true.function, d = 20, sd = 1) {
  d <- max(2, d)
  X <- matrix(runif(n * d, 0, 1), ncol = d)
  dta <- data.frame(list(
    x = X,
    y = true.function(X[, 1], X[, 2]) + rnorm(n, sd = sd)
  ))
  colnames(dta)[1:d] <- paste("x", 1:d, sep = "")
  dta
}

## simulate the data
which.simulation <- 1
df <- simfunction(n = 500, true.function(which.simulation))

## varpro analysis
vp <- varpro(y ~ ., df)

## ivarpro analysis 
imp <- ivarpro(vp)

## partial plot of x2 
partial.ivarpro(imp, var="x2")

## partial plot of x2 without ladder band
partial.ivarpro(imp, var="x2", ladder=FALSE)

## optional: use only a subset of ladder cuts
partial.ivarpro(imp, var="x2", ladder=TRUE, ladder.cuts=1:8)

## partial plot with color/size using x1 (color) and y (size)
partial.ivarpro(imp, var="x2", col.var="x1", size.var="y")

## ------------------------------------------------------------
##
## survival example with partial plot
##
## ------------------------------------------------------------

data(peakVO2, package = "randomForestSRC")

## varpro/importance call
i.pv <- ivarpro(varpro(Surv(ttodead, died)~., peakVO2))

## partial plot of peak vo2
## color displays interval (a measure of exercise time)
## size displays "y" which is predicted mortality in survival
partial.ivarpro(i.pv, var="peak.vo2", col.var="interval", size.var="y")

## same but using beta blockers for color
partial.ivarpro(i.pv, var="peak.vo2", col.var="betablok", size.var="y")

# }