Core Methodological Insight
RWA addresses multicollinearity by attributing shared
(collinear) variance between predictors to those predictors in
proportion to their structure with the outcome. Unlike traditional
regression coefficients that reflect only unique contributions, RWA
considers both direct and indirect (shared)
effects.
The Johnson (2000) algorithm performs an
eigenvalue decomposition of the predictor correlation
matrix to create uncorrelated principal components, then regresses the
outcome on these components. This procedure yields weights virtually
identical to what would be obtained by averaging a predictor’s
incremental R² contribution over all possible subsets of
predictors (the logic used in dominance analysis), but without
brute-force search over subsets, making it computationally
efficient.
Demonstrating RWA’s Theoretical Properties
Let’s demonstrate this with a controlled example where we know the
true relationships:
# Create controlled scenario to demonstrate RWA's theoretical properties
set.seed(123)
n <- 200
# Generate predictors with known correlation structure
x1 <- rnorm(n)
x2 <- 0.7 * x1 + 0.3 * rnorm(n) # r ≈ 0.7 with x1
x3 <- 0.5 * x1 + 0.8 * rnorm(n) # r ≈ 0.5 with x1
x4 <- rnorm(n) # Independent
# True population model with known coefficients
y <- 0.6 * x1 + 0.4 * x2 + 0.3 * x3 + 0.2 * x4 + rnorm(n, sd = 0.5)
theory_data <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3, x4 = x4)
# Compare traditional regression vs RWA
lm_theory <- lm(y ~ x1 + x2 + x3 + x4, data = theory_data)
rwa_theory <- rwa(theory_data, "y", c("x1", "x2", "x3", "x4"))
# Show how multicollinearity affects traditional coefficients
cat("True population contributions (designed into simulation):\n")
#> True population contributions (designed into simulation):
true_contributions <- c(0.6, 0.4, 0.3, 0.2)
names(true_contributions) <- c("x1", "x2", "x3", "x4")
print(true_contributions)
#> x1 x2 x3 x4
#> 0.6 0.4 0.3 0.2
cat("\nStandardized regression coefficients (distorted by multicollinearity):\n")
#>
#> Standardized regression coefficients (distorted by multicollinearity):
std_betas <- summary(lm_theory)$coefficients[2:5, "Estimate"]
names(std_betas) <- c("x1", "x2", "x3", "x4")
print(round(std_betas, 3))
#> x1 x2 x3 x4
#> 0.449 0.586 0.285 0.164
cat("\nRWA weights (better reflect true importance despite correlations):\n")
#>
#> RWA weights (better reflect true importance despite correlations):
rwa_weights_theory <- rwa_theory$result$Raw.RelWeight
names(rwa_weights_theory) <- rwa_theory$result$Predictors
print(round(rwa_weights_theory, 3))
#> [1] 0.306 0.304 0.155 0.021
# Calculate correlation between methods and true values
cor_with_true <- data.frame(
Method = c("Std_Betas", "RWA_Weights"),
Correlation_with_True = c(
cor(abs(std_betas), true_contributions),
cor(rwa_weights_theory[names(true_contributions)], true_contributions)
)
)
print("Correlation with true population values:")
#> [1] "Correlation with true population values:"
print(cor_with_true)
#> Method Correlation_with_True
#> 1 Std_Betas 0.6924877
#> 2 RWA_Weights NA
Current Validity: Why RWA Remains Relevant
RWA remains a widely accepted and useful method for
evaluating predictor importance under multicollinearity. Since
its introduction, numerous studies have validated that RWA provides a
fair assessment of each variable’s importance even in high-correlation
settings.
Evidence from research practice: -
Organizational Psychology: Determining which employee
factors predict performance when intercorrelated - Marketing
Research: Key driver analysis where product attributes tend to
move together
- Educational Research: Assessing cognitive abilities
that naturally correlate - Healthcare Analytics:
Understanding risk factors that often co-occur
Methodological validation: 1. RWA reveals
important predictors that regression beta weights obscure due
to multicollinearity 2. Rankings from RWA correlate >
0.99 with dominance analysis in most scenarios 3.
Bootstrap confidence intervals provide robust
significance testing 4. Computational efficiency allows
analysis of large predictor sets
