---
title: "Introduction to Relative Weights Analysis with the rwa Package"
author: "Martin Chan"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to Relative Weights Analysis with the rwa Package}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%",
  error = FALSE,
  warning = FALSE,
  message = FALSE
)
```

```{r setup}
library(rwa)
library(dplyr)
library(ggplot2)
```

## Introduction

**Relative Weights Analysis (RWA)** is a powerful method for determining the relative importance of predictor variables in multiple regression models, particularly when dealing with multicollinearity. This vignette provides a comprehensive guide to using the `rwa` package, explaining the methodology, interpreting results, exploring advanced features, and evaluating the theoretical foundations and validity of the method.

## Background and Methodology

### What is Relative Weights Analysis?

Relative Weights Analysis addresses a fundamental problem in multiple regression: when predictor variables are correlated with each other (multicollinearity), it becomes difficult to determine the true contribution of each variable to the outcome. Traditional regression coefficients can be misleading, unstable, or difficult to interpret in the presence of multicollinearity.

RWA decomposes the total variance predicted in a regression model (R²) into weights that accurately reflect the proportional contribution of the various predictor variables. The method implemented in this package is based on **Tonidandel and LeBreton (2015)**, with its original roots in **Johnson (2000)**.

### How RWA Works

The core insight of RWA is to create a set of orthogonal (uncorrelated) variables that are maximally related to the original predictors, then use these in a regression model. The process involves:

1. **Eigenvalue decomposition** of the predictor correlation matrix
2. **Transformation** of predictors into orthogonal components
3. **Regression** using the transformed predictors
4. **Decomposition** of R² into relative weights

This approach allows us to determine each variable's unique contribution to the explained variance, even when predictors are highly correlated.

### When to Use RWA

RWA is particularly useful when:

- **Multicollinearity** is present among predictors
- You need to **rank variables** by importance
- **Traditional regression coefficients** are unstable or difficult to interpret
- You want to understand **proportional contributions** to explained variance
- Conducting **key drivers analysis** in market research or business analytics

## Theoretical Foundation and Methodological Evaluation

### 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:

```{r theoretical-demonstration}
# 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_contributions <- c(0.6, 0.4, 0.3, 0.2)
names(true_contributions) <- c("x1", "x2", "x3", "x4")
print(true_contributions)

cat("\nStandardized regression coefficients (distorted by multicollinearity):\n")
std_betas <- summary(lm_theory)$coefficients[2:5, "Estimate"]
names(std_betas) <- c("x1", "x2", "x3", "x4")
print(round(std_betas, 3))

cat("\nRWA weights (better reflect true importance despite correlations):\n")
rwa_weights_theory <- rwa_theory$result$Raw.RelWeight
names(rwa_weights_theory) <- rwa_theory$result$Predictors
print(round(rwa_weights_theory, 3))

# 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:")
print(cor_with_true)
```

### 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

## Basic Usage

### Simple Example with mtcars

Let's start with a basic example using the built-in `mtcars` dataset to predict fuel efficiency (`mpg`):

```{r basic-example}
# Basic RWA
result_basic <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp", "gear"))

# View the results
result_basic$result
```

### Understanding the Output

The basic output includes several key components:

```{r output-explanation}
# Predictor variables used
result_basic$predictors

# Model R-squared
result_basic$rsquare

# Number of complete observations
result_basic$n

# Correlation matrices (for advanced users)
str(result_basic$RXX)  # Predictor correlation matrix
str(result_basic$RXY)  # Predictor-outcome correlations
```

### Interpreting the Results Table

The main results table contains:

- **Variables**: Names of predictor variables
- **Raw.RelWeight**: Raw relative weights (sum to R²)
- **Rescaled.RelWeight**: Rescaled weights as percentages (sum to 100%)
- **Sign**: Direction of relationship with outcome (+/-)

By default, results are automatically sorted by rescaled relative weights in descending order, making it easy to identify the most important predictors:

```{r interpret-results}
# Results are sorted by default (most important first)
result_basic$result

# Raw weights sum to R-squared
sum(result_basic$result$Raw.RelWeight)
result_basic$rsquare

# Rescaled weights sum to 100%
sum(result_basic$result$Rescaled.RelWeight)
```

### Controlling Output Sorting

You can control whether results are sorted using the `sort` parameter:

```{r sorting-example}
# Default behavior: sorted by importance (descending)
result_sorted <- mtcars %>%
  rwa(outcome = "mpg", predictors = c("cyl", "disp", "hp", "gear"))

result_sorted$result

# Preserve original predictor order
result_unsorted <- mtcars %>%
  rwa(outcome = "mpg", predictors = c("cyl", "disp", "hp", "gear"), sort = FALSE)

result_unsorted$result
```

## Advanced Features

### Adding Signs to Weights

The `applysigns` parameter adds directional information to help interpret whether variables positively or negatively influence the outcome:

```{r signs-example}
result_signs <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp", "gear"),
      applysigns = TRUE)

result_signs$result
```

### Visualization

The package allows you to visualize the relative importance by piping the results to `plot_rwa()`:

```{r visualization, fig.width=8, fig.height=5}
# Generate RWA results 
rwa_result <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp", "gear", "wt"))

# Create plot
rwa_result %>% plot_rwa()

# The rescaled relative weights
rwa_result$result
```

## Bootstrap Confidence Intervals

The `rwa` package includes powerful bootstrap functionality for determining the statistical significance of relative weights. For detailed coverage of bootstrap methods, see the dedicated vignette:

```{r eval=FALSE}
vignette("bootstrap-confidence-intervals", package = "rwa")
```

### Quick Bootstrap Example

```{r bootstrap-example}
# Basic bootstrap analysis
bootstrap_result <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp"),
      bootstrap = TRUE,
      n_bootstrap = 500)  # Reduced for speed

# View significant predictors
bootstrap_result$result %>%
  filter(Raw.Significant == TRUE) %>%
  select(Variables, Rescaled.RelWeight, Raw.RelWeight.CI.Lower, Raw.RelWeight.CI.Upper)
```

For comprehensive coverage of bootstrap methods, advanced features, and best practices, consult the bootstrap vignette.

## Real-World Example: Diamond Price Analysis

Let's explore a more complex example using the `diamonds` dataset:

```{r diamonds-example}
# Analyze diamond price drivers
diamonds_subset <- diamonds %>%
  select(price, carat, depth, table, x, y, z) %>%
  sample_n(1000)  # Sample for faster computation

diamond_rwa <- diamonds_subset %>%
  rwa(outcome = "price",
      predictors = c("carat", "depth", "table", "x", "y", "z"),
      applysigns = TRUE)

diamond_rwa$result
```

For bootstrap analysis of this example with confidence intervals, see:
```{r eval=FALSE}
vignette("bootstrap-confidence-intervals", package = "rwa")
```

## Comparison with Traditional Regression

Let's compare RWA results with traditional multiple regression to highlight the differences:

```{r regression-comparison}
# Traditional regression
lm_model <- lm(mpg ~ cyl + disp + hp + gear, data = mtcars)
lm_summary <- summary(lm_model)

# Display regression summary
print(lm_summary)

# RWA results
rwa_model <- mtcars %>%
  rwa(outcome = "mpg", predictors = c("cyl", "disp", "hp", "gear"))

# Compare importance rankings
comparison <- data.frame(
  Variable = rwa_model$predictors,
  RWA_Rescaled = rwa_model$result$Rescaled.RelWeight,
  RWA_Rank = rank(-rwa_model$result$Rescaled.RelWeight)
)

print(comparison)
```

### Methodological Advantages of RWA

#### 1. Superior to Standard Regression Under Multicollinearity

Traditional standardized beta coefficients reflect **unique contribution holding others constant**, which severely underestimates variables that share variance with others. RWA gives credit to predictors for variance they share with the outcome **both individually and jointly** with other variables.

#### 2. Avoids Stepwise Selection Problems

Stepwise methods arbitrarily pick one variable from a collinear set and drop others, potentially leading to incorrect conclusions about importance. Unlike stepwise selection, **RWA does not throw away information** – it shows how much each predictor contributes given the complete set.

#### 3. More Interpretable than Principal Components

While PCA addresses multicollinearity, it **buries the identity of individual predictors** in composite factors. RWA uses orthogonalization behind the scenes but **returns results in terms of original predictors**, maintaining interpretability.

#### 4. Computational Efficiency vs. Dominance Analysis

RWA's greatest achievement is producing **almost identical results to dominance analysis but far more efficiently**. Research shows:

- **Correlation > 0.99** between RWA and dominance analysis rankings
- **Exponential time savings**: dominance analysis requires 2^p models; RWA solves equations directly
- **Enhanced capabilities**: easier to provide confidence intervals and significance tests

```{r efficiency-demonstration}
# Demonstrate computational considerations
predictors <- c("cyl", "disp", "hp", "gear")
n_predictors <- length(predictors)

cat("Number of predictors:", n_predictors, "\n")
cat("Dominance analysis would require", 2^n_predictors, "subset models\n")
cat("RWA solves this in a single matrix operation\n")

# Show RWA speed (for demonstration)
start_time <- Sys.time()
rwa_speed_test <- mtcars %>% rwa(outcome = "mpg", predictors = predictors)
end_time <- Sys.time()
cat("RWA computation time:", round(as.numeric(end_time - start_time, units = "secs"), 4), "seconds\n")
```

## Critical Limitations and When to Exercise Caution

### 1. Not a Replacement for Theory

**RWA is fundamentally a descriptive, variance-partitioning tool.** It tells us "how much prediction was attributable to X" but **does not imply causation**. As Tonidandel & LeBreton emphasize: relative weights **"are not causal indicators and thus do not necessarily dictate a course of action."**

RWA should enhance interpretation of regression models but decisions must still be guided by substantive theory.

### 2. The Redundancy Problem

**RWA is not a magic bullet for extreme multicollinearity stemming from redundant predictors.** When predictors measure essentially the same construct, RWA will split the contribution between them, potentially making each appear less important individually.

```{r redundancy-demonstration}
# Demonstrate the redundancy limitation
set.seed(456)
x1_orig <- rnorm(100)
x1_dup <- x1_orig + rnorm(100, sd = 0.05)  # Nearly identical (r ≈ 0.99)
y_simple <- 0.8 * x1_orig + rnorm(100, sd = 0.5)

redundant_data <- data.frame(y = y_simple, x1_original = x1_orig, 
                           x1_duplicate = x1_dup)

cat("Correlation between 'different' predictors:", cor(x1_orig, x1_dup), "\n")

# RWA correctly splits redundant variance
redundant_rwa <- rwa(redundant_data, "y", c("x1_original", "x1_duplicate"))
print("RWA with redundant predictors:")
print(redundant_rwa$result)

cat("\nEach variable appears less important individually,")
cat("\nbut together they account for most variance.\n")
cat("Combined contribution:", 
    sum(redundant_rwa$result$Raw.RelWeight), "\n")
```

This is **statistically appropriate** – it reflects that one variable alone could do the job of both. Users must recognize that **the pair as a whole may be very important even if each alone has a modest weight.**

### 3. Sample Size and Stability

RWA shares multiple regression's sensitivity to **sample size and predictor-to-observation ratios.** Bootstrap confidence intervals help quantify uncertainty, but "bootstrapping will not overcome the inherent limitations associated with a small sample size."

## Best Practices and Recommendations

### 1. Sample Size Considerations

```{r sample-size}
# Check your sample size
n_obs <- mtcars %>% 
  select(mpg, cyl, disp, hp, gear) %>% 
  na.omit() %>% 
  nrow()

cat("Sample size:", n_obs)
cat("\nRule of thumb: At least 5-10 observations per predictor")
```

### 2. Variable Selection and Model Specification

RWA works best with:
- **Continuous variables** (though categorical can be used with caution)
- **Theoretically meaningful predictors**  
- **Reasonable predictor-to-observation ratios**
- **Predictors that are distinct but may be correlated**

### 3. When to Use RWA vs. Alternatives

**Use RWA when:**
1. **Predictors are theoretically distinct but empirically correlated**
2. **Understanding relative contribution (not just prediction) is the goal**
3. **Sample size supports stable multiple regression**
4. **Linear relationships are reasonable assumptions**

**Consider alternatives when:**
1. **Predictors are essentially redundant measures**
2. **Sample size is very small relative to predictors**
3. **Primary goal is variable selection rather than interpretation**
4. **Non-linear effects are suspected**

### 4. Reporting Results

When reporting RWA results, include:
- **Sample size** and missing data handling
- **Raw and rescaled weights**
- **Model R-squared**
- **Confidence intervals** for key predictors (when using bootstrap)

For bootstrap confidence intervals and advanced statistical considerations, see:
```{r eval=FALSE}
vignette("bootstrap-confidence-intervals", package = "rwa")
```

## Troubleshooting Common Issues

### Multicollinearity Warnings

If you encounter extreme multicollinearity:

```{r multicollinearity-check}
# Check correlation matrix
cor_matrix <- mtcars %>%
  select(cyl, disp, hp, gear) %>%
  cor()

# Look for high correlations (>0.9)
high_cor <- which(abs(cor_matrix) > 0.9 & cor_matrix != 1, arr.ind = TRUE)
if(nrow(high_cor) > 0) {
  cat("High correlations detected between variables")
}
```

### Missing Data

RWA handles missing data through listwise deletion:

```{r missing-data}
# Check for missing data patterns
missing_summary <- mtcars %>%
  select(mpg, cyl, disp, hp, gear) %>%
  summarise_all(~sum(is.na(.)))

print(missing_summary)
```

## References

This package and methodology are based on the following key references:

**Primary Sources:**

- **Johnson, J. W. (2000)**. A heuristic method for estimating the relative weight of predictor variables in multiple regression. *Multivariate Behavioral Research*, 35(1), 1-19. DOI: 10.1207/S15327906MBR3501_1

- **Tonidandel, S., & LeBreton, J. M. (2015)**. RWA Web: A free, comprehensive, web-based, and user-friendly tool for relative weight analyses. *Journal of Business and Psychology*, 30(2), 207-216. DOI: 10.1007/s10869-014-9351-z

**Additional Reading:**

- Azen, R., & Budescu, D. V. (2003). The dominance analysis approach for comparing predictors in multiple regression. *Psychological Methods*, 8(2), 129-148.

- Grömping, U. (2006). Relative importance for linear regression in R: The package relaimpo. *Journal of Statistical Software*, 17(1), 1-27.

- Johnson, J. W., & LeBreton, J. M. (2004). History and use of relative importance indices in organizational research. *Organizational Research Methods*, 7(3), 238-257.

For bootstrap methods and statistical significance testing, see the dedicated bootstrap vignette and its references.

For the latest methodological developments: https://www.scotttonidandel.com/rwa-web/

## Conclusion

The `rwa` package provides a user-friendly implementation of Relative Weights Analysis in R with seamless tidyverse integration. This vignette covered both practical implementation and theoretical foundations.

### Key Takeaways

**Methodological Strengths:**

- **Handles multicollinearity effectively** by partitioning shared variance appropriately
- **Provides interpretable results** as percentages of explained variance  
- **Computationally efficient** compared to dominance analysis
- **Statistically robust** with bootstrap confidence intervals

**Important Limitations:**

- **Not a replacement for theory** - results are descriptive, not causal
- **Redundant predictors** will split variance, potentially appearing less important individually
- **Sample size matters** - adequate observations needed for stable estimates
- **Linear model assumptions** apply

### When to Use RWA

RWA is particularly valuable for:

- **Key drivers analysis** in market research and business analytics  
- **Multicollinear predictors** in scientific research
- **Situations requiring interpretable importance measures**
- **Linear models where traditional regression coefficients are misleading**

### The Bottom Line

RWA provides **clear, quantifiable answers to questions about predictor importance in a multicollinear world** – answers that are often obscured when using standard regression approaches. With proper understanding of its assumptions and limitations, RWA remains an essential tool for researchers and practitioners dealing with correlated predictors.

Whether you're conducting key drivers analysis in market research, exploring variable importance in predictive modeling, or addressing multicollinearity in academic research, RWA provides valuable insights that complement traditional regression approaches.

For advanced bootstrap methods and statistical significance testing, see:
```{r eval=FALSE}
vignette("bootstrap-confidence-intervals", package = "rwa")
```