Bootstrap Confidence Intervals for Relative Weights Analysis

Why Bootstrap for RWA?

The Statistical Challenge

As noted by Tonidandel et al. (2009):

“The difficulty in determining the statistical significance of relative weights stems from the fact that the exact (or small sample) sampling distribution of relative weights is unknown.”

Traditional RWA provides point estimates of relative importance but lacks a framework for statistical inference. Bootstrap methods solve this by empirically estimating the sampling distribution of relative weights.

Bootstrap Solution

Bootstrap resampling: 1. Creates multiple samples from your original data 2. Calculates RWA for each bootstrap sample
3. Estimates confidence intervals from the distribution of bootstrap results 4. Enables significance testing by comparing each predictor against a randomly generated variable

How significance is assessed

A relative weight cannot be tested against zero. Raw relative weights are non-negative, so a predictor with no real relationship to the outcome still receives a small positive weight, and an interval around that weight will almost always exclude zero.

Following Tonidandel, LeBreton and Johnson (2009) — who suggest comparing a weight against that of a randomly generated variable to judge whether it exceeds what chance alone would produce (see also the discussion in vignette("evaluating-rwa-method-reference")) — significance is instead assessed by adding a randomly generated variable to the model and bootstrapping the difference between each predictor’s weight and the random variable’s weight.

The directional cutoff applied by this package is: a predictor is flagged significant only when the lower bound of that difference interval is above zero, meaning it explains meaningfully more variance than noise would. An interval lying entirely below zero means the predictor performed worse than the random variable, which is equally not evidence of importance, so it is not flagged significant either.

This means the package reports two distinct intervals:

Columns	Purpose
`Raw.RelWeight.CI.Lower` / `.Upper`	Descriptive interval around the weight itself. Not a significance test.
`Random.Diff.CI.Lower` / `.Upper`	Difference from a random variable’s weight. `Raw.Significant` is `TRUE` when `Random.Diff.CI.Lower > 0`.

Basic Bootstrap Analysis

Sampling scope

The implementation uses independent, identically distributed (iid) individual-row resampling. For observation-weighted RWA, each sampled row carries its original weight; clusters, strata, and replicate-weight survey designs are not supported. Outcome-missing rows are removed before resampling, while other missing-data filters are applied within each sample. Invalid samples stop with an error rather than dropping predictors or retrying. See vignette("weighted-missing-data") for the filtering contract and examples.

Simple Bootstrap Example

# Bootstrap analysis with 1000 samples
result_bootstrap <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp", "gear"),
      bootstrap = TRUE,
      n_bootstrap = 1000,
      conf_level = 0.95)

# View results with confidence intervals
result_bootstrap$result
#>   Variables Raw.RelWeight Rescaled.RelWeight Sign Raw.RelWeight.CI.Lower
#> 1        hp     0.2321744           29.79691    -             0.17994305
#> 2       cyl     0.2284797           29.32274    -             0.17978211
#> 3      disp     0.2221469           28.50999    -             0.15733870
#> 4      gear     0.0963886           12.37037    +             0.04021568
#>   Raw.RelWeight.CI.Upper Random.Diff.CI.Lower Random.Diff.CI.Upper
#> 1              0.2776723           0.18500702            0.3029704
#> 2              0.2817795           0.15651347            0.3036993
#> 3              0.2806779           0.15031200            0.2954721
#> 4              0.1725336           0.02443039            0.1955195
#>   Raw.Significant
#> 1            TRUE
#> 2            TRUE
#> 3            TRUE
#> 4            TRUE

Understanding Bootstrap Output

The bootstrap analysis enhances the standard RWA output with:

Raw.RelWeight.CI.Lower/Upper: confidence intervals around each raw weight, for description rather than testing
Random.Diff.CI.Lower/Upper: confidence intervals for the difference from a randomly generated variable’s weight
Raw.Significant: significance flag, TRUE when the random-comparison interval excludes zero

# Bootstrap-specific information
cat("Bootstrap samples used:", result_bootstrap$bootstrap$n_bootstrap, "\n")
#> Bootstrap samples used: 1000

# Detailed CI information
print(result_bootstrap$bootstrap$ci_results$raw_weights)
#> # A tibble: 4 × 6
#>   variable weight_index ci_lower ci_upper ci_method ci_type
#>   <chr>           <int>    <dbl>    <dbl> <chr>     <chr>  
#> 1 cyl                 1   0.180     0.282 bca       raw    
#> 2 disp                2   0.157     0.281 bca       raw    
#> 3 hp                  3   0.180     0.278 bca       raw    
#> 4 gear                4   0.0402    0.173 bca       raw

# Identify significant predictors
significant_vars <- result_bootstrap$result %>%
  filter(Raw.Significant == TRUE) %>%
  pull(Variables)

cat("Significant predictors:", paste(significant_vars, collapse = ", "))
#> Significant predictors: hp, cyl, disp, gear

Advanced Bootstrap Features

Comprehensive Bootstrap Analysis

For detailed analysis including focal variable comparisons:

# Comprehensive bootstrap with focal variable comparison
result_comprehensive <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp", "gear", "wt"),
      bootstrap = TRUE,
      comprehensive = TRUE,
      focal = "wt",  # Compare other variables to weight
      n_bootstrap = 500)  # Fewer samples for speed

# Access all bootstrap results
names(result_comprehensive$bootstrap$ci_results)
#> [1] "raw_weights"       "random_comparison" "focal_comparison"

Bootstrap Parameters

Key parameters for bootstrap analysis:

n_bootstrap: Number of bootstrap samples (default: 1000)
conf_level: Confidence level (default: 0.95)
focal: Focal variable for comparative analysis
comprehensive: Enable additional bootstrap tests

# Example with different parameters
custom_bootstrap <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp"),
      bootstrap = TRUE,
      n_bootstrap = 2000,  # More samples for precision
      conf_level = 0.99)   # 99% confidence intervals

custom_bootstrap$result
#>   Variables Raw.RelWeight Rescaled.RelWeight Sign Raw.RelWeight.CI.Lower
#> 1       cyl     0.3837012           50.51586    -              0.2813497
#> 2      disp     0.3758646           49.48414    -              0.2309071
#>   Raw.RelWeight.CI.Upper Random.Diff.CI.Lower Random.Diff.CI.Upper
#> 1              0.4568403            0.1748707            0.4470895
#> 2              0.4637872            0.2073237            0.4656827
#>   Raw.Significant
#> 1            TRUE
#> 2            TRUE

Rescaled Weight Confidence Intervals

Important Considerations

Rescaled weight confidence intervals should be interpreted with caution due to compositional data constraints. They are not recommended for formal statistical inference.

# Rescaled CIs (use with caution)
result_rescaled_ci <- mtcars %>%
  rwa(outcome = "mpg",
      predictors = c("cyl", "disp", "hp"),
      bootstrap = TRUE,
      include_rescaled_ci = TRUE,
      n_bootstrap = 500)

# Note the warning message about interpretation
result_rescaled_ci$result
#>   Variables Raw.RelWeight Rescaled.RelWeight Sign Raw.RelWeight.CI.Lower
#> 1      disp     0.2793550           36.37966    -              0.2124971
#> 2       cyl     0.2723144           35.46279    -              0.2133034
#> 3        hp     0.2162184           28.15755    -              0.1438879
#>   Raw.RelWeight.CI.Upper Random.Diff.CI.Lower Random.Diff.CI.Upper
#> 1              0.3456126           0.15780643            0.3218527
#> 2              0.3322052           0.11788124            0.2895544
#> 3              0.2622216           0.02978695            0.2415767
#>   Raw.Significant Rescaled.RelWeight.CI.Lower Rescaled.RelWeight.CI.Upper
#> 1            TRUE                    30.10095                    42.87168
#> 2            TRUE                    30.20278                    43.36890
#> 3            TRUE                    20.80491                    35.82311

Why Rescaled CIs Are Problematic

Rescaled weights are compositional data (they sum to 100%), which creates dependencies between variables. This violates assumptions needed for independent confidence intervals.

Recommendation: Focus on raw weight confidence intervals for statistical inference.

Real-World Applications

Diamond Price Analysis

# 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"),
      bootstrap = TRUE,
      applysigns = TRUE,
      n_bootstrap = 500)

print(diamond_rwa$result)
#>   Variables Raw.RelWeight Rescaled.RelWeight Sign Sign.Rescaled.RelWeight
#> 1     carat  0.2827671839        33.61747478    +             33.61747478
#> 2         y  0.2410353741        28.65608554    +             28.65608554
#> 3         x  0.2390627389        28.42156395    +             28.42156395
#> 4         z  0.0721865633         8.58207780    +              8.58207780
#> 5     table  0.0053869575         0.64044174    +              0.64044174
#> 6     depth  0.0006927239         0.08235619    -             -0.08235619
#>   Raw.RelWeight.CI.Lower Raw.RelWeight.CI.Upper Random.Diff.CI.Lower
#> 1           0.2540326046           0.3432575285          0.255325134
#> 2           0.2227669146           0.2868371992          0.221011017
#> 3           0.2206290118           0.2847261569          0.218437901
#> 4          -0.0603570553           0.1169361261         -0.063730598
#> 5           0.0008142242           0.0086470381         -0.002033129
#> 6          -0.0020080833           0.0008013738         -0.003736398
#>   Random.Diff.CI.Upper Raw.Significant
#> 1         0.3379597540            TRUE
#> 2         0.2852931602            TRUE
#> 3         0.2827117062            TRUE
#> 4         0.1153072073           FALSE
#> 5         0.0071742124           FALSE
#> 6        -0.0002043083           FALSE

Interpreting Results

# Focus on significant predictors (results are already sorted by importance)
significant_drivers <- diamond_rwa$result %>%
  filter(Raw.Significant == TRUE) %>%
  select(Variables, Rescaled.RelWeight, Sign.Rescaled.RelWeight)

cat("Significant diamond price drivers (sorted by importance):\n")
#> Significant diamond price drivers (sorted by importance):
print(significant_drivers)
#>   Variables Rescaled.RelWeight Sign.Rescaled.RelWeight
#> 1     carat           33.61747                33.61747
#> 2         y           28.65609                28.65609
#> 3         x           28.42156                28.42156

cat("\nModel R-squared:", round(diamond_rwa$rsquare, 3))
#> 
#> Model R-squared: 0.841

Best Practices

1. Sample Size Guidelines

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

cat("Sample size:", n_obs)
#> Sample size: 32
cat("\nRecommended bootstrap samples:", min(2000, n_obs * 10))
#> 
#> Recommended bootstrap samples: 320

# Rule of thumb: At least 1000 bootstrap samples, more for smaller datasets

2. Confidence Interval Interpretation

The intervals around the raw weights describe precision, not significance. Use them to judge how tightly each weight is estimated; use Raw.Significant (from the random-variable comparison) to judge importance.

# Examine CI characteristics
ci_data <- result_bootstrap$bootstrap$ci_results$raw_weights
print(head(ci_data))
#> # A tibble: 4 × 6
#>   variable weight_index ci_lower ci_upper ci_method ci_type
#>   <chr>           <int>    <dbl>    <dbl> <chr>     <chr>  
#> 1 cyl                 1   0.180     0.282 bca       raw    
#> 2 disp                2   0.157     0.281 bca       raw    
#> 3 hp                  3   0.180     0.278 bca       raw    
#> 4 gear                4   0.0402    0.173 bca       raw

# Assess precision
ci_analysis <- ci_data %>%
  mutate(
    ci_width = ci_upper - ci_lower,
    precision = case_when(
      ci_width < 0.05 ~ "High precision",
      ci_width < 0.15 ~ "Medium precision", 
      TRUE ~ "Low precision"
    )
  )

print(ci_analysis)
#> # A tibble: 4 × 8
#>   variable weight_index ci_lower ci_upper ci_method ci_type ci_width precision  
#>   <chr>           <int>    <dbl>    <dbl> <chr>     <chr>      <dbl> <chr>      
#> 1 cyl                 1   0.180     0.282 bca       raw       0.102  Medium pre…
#> 2 disp                2   0.157     0.281 bca       raw       0.123  Medium pre…
#> 3 hp                  3   0.180     0.278 bca       raw       0.0977 Medium pre…
#> 4 gear                4   0.0402    0.173 bca       raw       0.132  Medium pre…

3. Bootstrap Method Selection

The package automatically selects the best available bootstrap CI method:

BCA (Bias-Corrected and Accelerated) - Preferred when possible
Percentile - Fallback if BCA fails
Basic bootstrap - Final fallback option

# Check which methods were used
ci_methods <- result_bootstrap$bootstrap$ci_results$raw_weights %>%
  count(ci_method)

print(ci_methods)
#> # A tibble: 1 × 2
#>   ci_method     n
#>   <chr>     <int>
#> 1 bca           4

Performance Considerations

Bootstrap Speed Tips

# For large datasets or many predictors, consider:

# 1. Reduce bootstrap samples for initial exploration
quick_result <- mtcars %>%
  rwa(outcome = "mpg", 
      predictors = c("cyl", "disp"), 
      bootstrap = TRUE, 
      n_bootstrap = 500)  # Faster

# 2. Use comprehensive analysis only when needed
# comprehensive = TRUE adds computational overhead

# 3. Consider parallel processing for very large analyses
# (not currently implemented but could be future enhancement)

Memory Usage

# Bootstrap objects can be large - access specific components
str(result_bootstrap$bootstrap, max.level = 1)
#> List of 7
#>  $ boot_object       :List of 11
#>   ..- attr(*, "class")= chr "boot"
#>   ..- attr(*, "boot_type")= chr "boot"
#>  $ boot_object_random:List of 11
#>   ..- attr(*, "class")= chr "boot"
#>   ..- attr(*, "boot_type")= chr "boot"
#>  $ ci_results        :List of 2
#>  $ n_bootstrap       : num 1000
#>  $ conf_level        : num 0.95
#>  $ comprehensive     : logi FALSE
#>  $ focal             : NULL

# For memory efficiency, extract only needed results
ci_summary <- result_bootstrap$bootstrap$ci_results$raw_weights %>%
  select(variable, ci_lower, ci_upper, ci_method)

print(ci_summary)
#> # A tibble: 4 × 4
#>   variable ci_lower ci_upper ci_method
#>   <chr>       <dbl>    <dbl> <chr>    
#> 1 cyl        0.180     0.282 bca      
#> 2 disp       0.157     0.281 bca      
#> 3 hp         0.180     0.278 bca      
#> 4 gear       0.0402    0.173 bca

Troubleshooting

Common Bootstrap Issues

# 1. Check for perfect multicollinearity
cor_check <- mtcars %>%
  select(cyl, disp, hp, gear) %>%
  cor()

# Look for correlations = 1.0 (excluding diagonal)
perfect_cor <- which(abs(cor_check) == 1 & cor_check != diag(diag(cor_check)), arr.ind = TRUE)

if(length(perfect_cor) > 0) {
  cat("Perfect multicollinearity detected - remove redundant variables")
} else {
  cat("No perfect multicollinearity detected")
}
#> No perfect multicollinearity detected

# 2. Ensure adequate sample size
min_sample_size <- 5 * length(c("cyl", "disp", "hp", "gear"))  # 5 obs per predictor
actual_sample_size <- nrow(na.omit(mtcars[c("mpg", "cyl", "disp", "hp", "gear")]))

cat("\nMinimum recommended sample size:", min_sample_size)
#> 
#> Minimum recommended sample size: 20
cat("\nActual sample size:", actual_sample_size)
#> 
#> Actual sample size: 32

Reporting Bootstrap Results

Standard Reporting Format

When reporting bootstrap RWA results, include:

Sample size and missing data handling
Bootstrap parameters (number of samples, confidence level)
CI method used (BCA, percentile, basic)
Significant predictors with confidence intervals
Model fit (R-squared)

Example Report

# Generate a summary report
report_data <- result_bootstrap$result %>%
  filter(Raw.Significant == TRUE) %>%
  arrange(desc(Rescaled.RelWeight)) %>%
  select(Variables, Rescaled.RelWeight, Raw.RelWeight.CI.Lower, Raw.RelWeight.CI.Upper)

cat("Relative Weights Analysis Results\n")
#> Relative Weights Analysis Results
cat("=================================\n")
#> =================================
cat("Sample size:", result_bootstrap$n, "\n")
#> Sample size: 32
cat("Bootstrap samples:", result_bootstrap$bootstrap$n_bootstrap, "\n")
#> Bootstrap samples: 1000
cat("Model R-squared:", round(result_bootstrap$rsquare, 3), "\n\n")
#> Model R-squared: 0.779
cat("Significant Predictors:\n")
#> Significant Predictors:
print(report_data)
#>   Variables Rescaled.RelWeight Raw.RelWeight.CI.Lower Raw.RelWeight.CI.Upper
#> 1        hp           29.79691             0.17994305              0.2776723
#> 2       cyl           29.32274             0.17978211              0.2817795
#> 3      disp           28.50999             0.15733870              0.2806779
#> 4      gear           12.37037             0.04021568              0.1725336

References

Bootstrap Methods in RWA:

Tonidandel, S., LeBreton, J. M., & Johnson, J. W. (2009). Determining the statistical significance of relative weights. Psychological Methods, 14(4), 387-399.

General Bootstrap Theory:

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman & Hall/CRC.

Compositional Data Analysis:

Aitchison, J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Conclusion

Bootstrap confidence intervals provide a robust solution for statistical inference in Relative Weights Analysis. By following the guidelines in this vignette, researchers can:

Determine statistical significance of predictor importance
Report confidence intervals with appropriate interpretations
Avoid common pitfalls in bootstrap analysis
Apply best practices for reliable results

The bootstrap functionality in the rwa package represents a significant advancement in making RWA a complete tool for both exploratory analysis and confirmatory research.