Analysis

Variance Inflation Factor (VIF): Detects multicollinearity among variables to ensure independent inputs for clustering.

Correlation Analysis: Identifies highly correlated variables that may distort clustering outcomes.

Principal Component Analysis (PCA): Explores data structure, reduces dimensionality, and supports cluster visualization.

K-Means Clustering: Groups states into distinct regulatory typologies based on multiple variables.

Elbow Method (WCSS): Determines the optimal number of clusters by evaluating model fit.

One-Way ANOVA / Kruskal-Wallis Test: Tests whether continuous variables significantly differ across clusters.

Chi-Square Test of Independence: Assesses whether categorical variables vary significantly across clusters.

Welch’s t-tests with Bonferroni Correction: Conducts pairwise comparisons between clusters and controls for Type I error.

Effect Size Metrics (Eta-squared, Cohen’s d, Cramér’s V): Measures the magnitude of differences across clusters.

Linear Discriminant Analysis (LDA): Identifies which variables best separate clusters using linear functions.

Quadratic Discriminant Analysis (QDA): Checks for nonlinear boundaries and validates cluster separability.

Decision Trees: Reveals how combinations of variables predict cluster membership; shows interaction and threshold effects.

Random Forests: Ranks variable importance and handles complex, nonlinear patterns across features.

Permutation Importance: Measures how much each variable contributes to model prediction accuracy.

Boxplots and Density Plots: Visualize distributions and differences of variables across clusters.

Energy Justice Index (EJI): Constructs a composite outcome variable representing recognition, procedural, and distributional justice using standardized and scaled indicators.

Z-score Standardization + Percentile Ranking: Normalizes variables and translates scores into a 0–10 scale for comparability across utilities.

Geometric Mean Indexing: Aggregates justice dimensions into a single Energy Justice Index score, preventing overcompensation across dimensions.

Decision Trees (CART): Provides rule-based splits to identify key predictors of justice outcomes and uncover interaction effects.

Random Forests: Ranks variable importance using permutation and impurity metrics, robust to non-linearities and high-dimensional data.

NOTEARS (Structure Learning): Estimates directed acyclic graphs (DAGs) to reveal linear, conditional dependencies among explanatory variables and justice outcomes.

Causal Additive Models (CAM): Captures non-linear and additive relationships where NOTEARS is too restrictive due to its linear assumptions.

Bootstrapped Stability Selection: Assesses consistency of relationships across subsamples to improve model robustness.

Comparative Modeling by Regulatory Environment: Estimates separate models for each of four regulatory typologies to detect context-specific pathways.

Cross-sectional Analysis Design: Enables population-wide comparison of 166 IOUs without relying on sampling, supporting generalizability.

Structural Equation Modeling (SEM): Tests the hypothesized relationships between regulatory, institutional, and market factors and energy justice outcomes using latent constructs.

Latent Variable Modeling: Captures unobservable concepts (e.g., regulatory environment, institutional capacity) using multiple indicators to reduce measurement error and enhance construct validity.

Maximum Likelihood (ML) & Robust Estimation: Estimates SEM parameters while adjusting for potential non-normality or heteroskedasticity in the data.

Model Fit Indices (RMSEA, CFI, TLI, SRMR): Assesses how well the SEM fits the observed data and guides model refinement.

Direct, Indirect, and Total Effect Estimation: Decomposes effects within SEM to reveal causal pathways and mediating mechanisms.

Sensitivity Analysis: Tests the robustness of results across different ownership types, regions, and model specifications.

Expanded SEM Model (All Utility Types): Assesses whether the validated pathways among IOUs generalize to municipal and cooperative utilities.

Multi-Group SEM: Evaluates moderation by ownership type by estimating structural models separately for IOUs, municipals, and cooperatives.

Nested Model Comparison (Chi-square Difference Test): Tests whether constraining paths across groups significantly worsens model fit—indicating moderation by ownership type.

Fit Index Difference Testing (ΔCFI, ΔRMSEA, ΔSRMR): Assesses meaningful differences between constrained and unconstrained models to detect ownership-based moderation.

Full-Population Analysis (n = 1,233 utilities): Enables broad generalizability and avoids sampling bias in comparing IOUs, municipals, and cooperatives.