Simulation Settings for Feature Importance Methods

Introduction

The xplainfi package provides several data generating processes (DGPs) designed to illustrate specific strengths and weaknesses of different feature importance methods. Each DGP focuses on one primary challenge to make the differences between methods clear.

This article provides a comprehensive overview of all simulation settings, including their mathematical formulations and causal structures visualized as directed acyclic graphs (DAGs).

Overview of Simulation Settings

Overview of simulation settings and expected method behavior
DGP	Challenge	PFI Behavior	CFI Behavior
sim_dgp_correlated	Spurious correlation	High for spurious x2	Low for spurious x2
sim_dgp_mediated	Mediation effects	Shows total effects	Shows direct effects
sim_dgp_confounded	Confounding	Biased upward	Less biased
sim_dgp_interactions	Interaction effects	Low (no main effects)	High (captures interactions)
sim_dgp_independent	Baseline (no challenges)	Accurate	Accurate
sim_dgp_ewald	Mixed effects	Mixed	Mixed

1. Correlated Features DGP

This DGP creates a highly correlated spurious predictor to illustrate the fundamental difference between marginal and conditional importance methods.

Mathematical Model

\[(X_1, X_2)^T \sim \text{MVN}(0, \Sigma)\]

where \(\Sigma\) is a \(2 \times 2\) covariance matrix with 1 on the diagonal and correlation \(r\) (default 0.9) on the off-diagonal.

\[X_3 \sim N(0,1), \quad X_4 \sim N(0,1)\] \[Y = 2 \cdot X_1 + X_3 + \varepsilon\]

where \(\varepsilon \sim N(0, 0.2^2)\).

Causal Structure

DAG for correlated features DGP

Usage Example

set.seed(123)
task <- sim_dgp_correlated(n = 500)

# Check correlation between X1 and X2
cor(task$data()[, c("x1", "x2")])
#>          x1       x2
#> x1 1.000000 0.885718
#> x2 0.885718 1.000000

# True coefficients: x1=2.0, x2=0, x3=1.0, x4=0
# Note: x2 is highly correlated with x1 but has NO causal effect!

Expected Behavior

Marginal methods (PFI, Marginal SAGE): Will falsely assign high importance to x2 because permuting it breaks the correlation with x1, creating unrealistic data that confuses the model
Conditional methods (CFI, Conditional SAGE): Should correctly assign near-zero importance to x2 because conditional sampling preserves the correlation, revealing that x2 adds no information beyond what x1 provides
Key insight: x2 is a spurious predictor - it appears predictive due to correlation with x1 but has no causal effect on y

2. Mediated Effects DGP

This DGP demonstrates the difference between total and direct causal effects. Some features affect the outcome only through mediators.

Mathematical Model

\[\text{exposure} \sim N(0,1), \quad \text{direct} \sim N(0,1)\] \[\text{mediator} = 0.8 \cdot \text{exposure} + 0.6 \cdot \text{direct} + \varepsilon_m\] \[Y = 1.5 \cdot \text{mediator} + 0.5 \cdot \text{direct} + \varepsilon\]

where \(\varepsilon_m \sim N(0, 0.3^2)\) and \(\varepsilon \sim N(0, 0.2^2)\).

Causal Structure

DAG for mediated effects DGP

Usage Example

set.seed(123)
task <- sim_dgp_mediated(n = 500)

# Calculate total effect of exposure
# Total effect = 0.8 * 1.5 = 1.2 (through mediator)
# Direct effect = 0 (no direct path to Y)

Expected Behavior

PFI: Shows total effects (exposure appears important with effect approximately 1.2)
CFI: Shows direct effects (exposure appears unimportant when conditioning on mediator)
RFI with mediator: Should show direct effects similar to CFI

3. Confounding DGP

This DGP includes a confounder that affects both a feature and the outcome.

Mathematical Model

\[H \sim N(0,1) \quad \text{(confounder)}\] \[X_1 = H + \varepsilon_1\] \[\text{proxy} = H + \varepsilon_p, \quad \text{independent} \sim N(0,1)\] \[Y = H + X_1 + \text{independent} + \varepsilon\]

where all \(\varepsilon \sim N(0, 0.5^2)\) independently.

Causal Structure

DAG for confounding DGP

Usage Example

set.seed(123)
# Hidden confounder scenario (default)
task_hidden <- sim_dgp_confounded(n = 500, hidden = TRUE)
task_hidden$feature_names # proxy available but not confounder
#> [1] "independent" "proxy"       "x1"

# Observable confounder scenario
task_observed <- sim_dgp_confounded(n = 500, hidden = FALSE)
task_observed$feature_names # both confounder and proxy available
#> [1] "confounder"  "independent" "proxy"       "x1"

Expected Behavior

PFI: Will show inflated importance for x1 due to confounding (the confounder H contributes to x1’s predictive power beyond its direct effect)
CFI: Should partially account for confounding through conditional sampling
RFI conditioning on proxy: Should reduce confounding bias by conditioning on the noisy measurement of the confounder

4. Interaction Effects DGP

This DGP demonstrates a pure interaction effect where features have no main effects.

Mathematical Model

\[Y = 2 \cdot X_1 \cdot X_2 + X_3 + \varepsilon\]

where \(X_j \sim N(0,1)\) independently and \(\varepsilon \sim N(0, 0.5^2)\).

Causal Structure

DAG for interaction effects DGP

Usage Example

set.seed(123)
task <- sim_dgp_interactions(n = 500)

# Note: X1 and X2 have NO main effects
# Their importance comes ONLY through their interaction

Expected Behavior

PFI: Should assign near-zero importance to x1 and x2 (no marginal effect)
CFI: Should capture the interaction and assign high importance to x1 and x2
LOCO: May show high importance for x1 and x2 (removing either breaks the interaction)
LOCI: Should show near-zero importance for x1 and x2 (individually useless)

5. Independent Features DGP (Baseline)

This is a baseline scenario where all features are independent and their effects are additive. All importance methods should give similar results.

Mathematical Model

\[Y = 2.0 \cdot X_1 + 1.0 \cdot X_2 + 0.5 \cdot X_3 + \varepsilon\]

where \(X_j \sim N(0,1)\) independently and \(\varepsilon \sim N(0, 0.2^2)\).

Causal Structure

DAG for independent features DGP

Usage Example

set.seed(123)
task <- sim_dgp_independent(n = 500)

# All methods should rank features consistently:
# important1 > important2 > important3 > unimportant1,2 (approx. 0)

Expected Behavior

All methods: Should rank features consistently by their true effect sizes
Ground truth: important1 (2.0) > important2 (1.0) > important3 (0.5) > unimportant1,2 (0)

6. Ewald et al. (2024) DGP

Reproduces the data generating process from Ewald et al. (2024) for benchmarking feature importance methods. Includes correlated features and interaction effects.

Mathematical Model

\[X_1, X_3, X_5 \sim \text{Uniform}(0,1)\] \[X_2 = X_1 + \varepsilon_2, \quad \varepsilon_2 \sim N(0, 0.001)\] \[X_4 = X_3 + \varepsilon_4, \quad \varepsilon_4 \sim N(0, 0.1)\] \[Y = X_4 + X_5 + X_4 \cdot X_5 + \varepsilon, \quad \varepsilon \sim N(0, 0.1)\]

Causal Structure

DAG for Ewald et al. (2024) DGP

Usage Example

sim_dgp_ewald(n = 500)
#> 
#> ── <TaskRegr> (500x6) ──────────────────────────────────────────────────────────
#> • Target: y
#> • Properties: -
#> • Features (5):
#>   • dbl (5): x1, x2, x3, x4, x5