# Correlations

#### Aaron R. Caldwell

#### 2025-08-22

The TOSTER package provides several functions for calculating and analyzing correlations. These functions extend beyond traditional correlation tests by offering equivalence testing capabilities and robust correlation methods. The included functions are based on research by Goertzen and Cribbie (2010) (`z_cor_test` & `compare_cor`), and Wilcox (2011) (`boot_cor_test`)1.

# Simple Correlation Test

Basic tests of association can be performed with the `z_cor_test` function. This function is styled after R’s built-in `cor.test` function but uses Fisher’s z transformation as the basis for all significance tests (p-values). Despite this difference in methodology, the confidence intervals are typically very similar to those produced by `cor.test`.

    library(TOSTER)
    # Base R correlation test
    cor.test(mtcars$mpg, mtcars$qsec)

    ##
    ##  Pearson's product-moment correlation
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## t = 2.5252, df = 30, p-value = 0.01708
    ## alternative hypothesis: true correlation is not equal to 0
    ## 95 percent confidence interval:
    ##  0.08195487 0.66961864
    ## sample estimates:
    ##      cor
    ## 0.418684

    # TOSTER's z-transformed correlation test
    z_cor_test(mtcars$mpg, mtcars$qsec)

    ##
    ##  Pearson's product-moment correlation
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## z = 2.4023, N = 32, p-value = 0.01629
    ## alternative hypothesis: true correlation is not equal to 0
    ## 95 percent confidence interval:
    ##  0.08195487 0.66961864
    ## sample estimates:
    ##      cor
    ## 0.418684

Like `cor.test`, the `z_cor_test` function supports Spearman and Kendall correlation coefficients:

    # Spearman correlation
    z_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "spear") # Short form accepted; "spearman" also works

    ##
    ##  Spearman's rank correlation rho
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## z = 2.6474, N = 32, p-value = 0.008111
    ## alternative hypothesis: true rho is not equal to 0
    ## 95 percent confidence interval:
    ##  0.1306771 0.7068501
    ## sample estimates:
    ##       rho
    ## 0.4669358

    # Kendall correlation
    z_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "kendall")

    ##
    ##  Kendall's rank correlation tau
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## z = 2.6134, N = 32, p-value = 0.008964
    ## alternative hypothesis: true tau is not equal to 0
    ## 95 percent confidence interval:
    ##  0.08145572 0.51634821
    ## sample estimates:
    ##       tau
    ## 0.3153652

# Advantages of z_cor_test

The main advantage of `z_cor_test` over the standard `cor.test` is its ability to perform equivalence testing (TOST) or any hypothesis test where the null hypothesis isn’t zero. This makes it particularly useful for research questions focused on demonstrating practical equivalence or testing against specific correlation thresholds.

    # Equivalence test with null boundary of 0.4
    z_cor_test(mtcars$mpg,
               mtcars$qsec,
               alternative = "e", # e for equivalence
               null = .4)

    ##
    ##  Pearson's product-moment correlation
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## z = 0.12088, N = 32, p-value = 0.5481
    ## alternative hypothesis: equivalence
    ## null values:
    ## correlation correlation
    ##         0.4        -0.4
    ## 90 percent confidence interval:
    ##  0.1397334 0.6360650
    ## sample estimates:
    ##      cor
    ## 0.418684

In this example, we’re testing whether the correlation is equivalent to zero within the boundaries of ±0.4.

# Using Summary Statistics

A key advantage of TOSTER is the ability to perform correlation tests using only summary statistics, which is particularly useful when reviewing published literature or working with limited data access. The `corsum_test` function enables this functionality:

    # Testing a correlation of 0.121 from a sample of 105 paired observations
    corsum_test(r = .121,
                n = 105,
                alternative = "e",
                null = .4)

    ##
    ##  Pearson's product-moment correlation
    ##
    ## data:  x and y
    ## z = -3.0506, N = 105, p-value = 0.001142
    ## alternative hypothesis: equivalence
    ## null values:
    ## correlation correlation
    ##         0.4        -0.4
    ## 90 percent confidence interval:
    ##  -0.0412456  0.2770284
    ## sample estimates:
    ##   cor
    ## 0.121

This example tests whether a correlation of 0.121 from a sample of 105 paired observations is equivalent to zero within the boundaries of ±0.4.

# Bootstrapped Correlation Test

For more robust analyses when raw data is available, TOSTER provides the `boot_cor_test` function. This bootstrapping approach generally produces more reliable results than Fisher’s z-based tests, especially when outliers are present or distribution assumptions are violated.

    set.seed(993) # Setting seed for reproducibility
    boot_cor_test(mtcars$mpg,
               mtcars$qsec,
               alternative = "e",
               null = .4)

    ##
    ##  Bootstrapped Pearson's product-moment correlation
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## N = 32, p-value = 0.6088
    ## alternative hypothesis: equivalence
    ## null values:
    ## correlation correlation
    ##         0.4        -0.4
    ## 90 percent confidence interval:
    ##  0.2512992 0.5930089
    ## sample estimates:
    ##      cor
    ## 0.418684

    # Bootstrapped Spearman correlation
    boot_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "spear",
               alternative = "e",
               null = .4)

    ##
    ##  Bootstrapped Spearman's rank correlation rho
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## N = 32, p-value = 0.6713
    ## alternative hypothesis: equivalence
    ## null values:
    ##  rho  rho
    ##  0.4 -0.4
    ## 90 percent confidence interval:
    ##  0.2676957 0.7365611
    ## sample estimates:
    ##       rho
    ## 0.4669358

    # Bootstrapped Kendall correlation
    boot_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "ken", # Short form accepted
               alternative = "e",
               null = .4)

    ##
    ##  Bootstrapped Kendall's rank correlation tau
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## N = 32, p-value = 0.2276
    ## alternative hypothesis: equivalence
    ## null values:
    ##  tau  tau
    ##  0.4 -0.4
    ## 90 percent confidence interval:
    ##  0.1442440 0.5090649
    ## sample estimates:
    ##       tau
    ## 0.3153652

## Robust Correlation Methods

The `boot_cor_test` function also provides access to robust correlation methods that are less sensitive to outliers and violations of normality:

    # Winsorized correlation with 10% trimming
    boot_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "win",
               alternative = "e",
               null = .4,
               tr = .1) # Set trim amount (default is 0.2)

    ##
    ##  Bootstrapped Winsorized correlation wincor
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## N = 32, p-value = 0.6878
    ## alternative hypothesis: equivalence
    ## null values:
    ## wincor wincor
    ##    0.4   -0.4
    ## 90 percent confidence interval:
    ##  0.2651060 0.7125775
    ## sample estimates:
    ##   wincor
    ## 0.464062

    # Percentage bend correlation
    boot_cor_test(mtcars$mpg,
               mtcars$qsec,
               method = "bend",
               alternative = "e",
               null = .4,
               beta = .15) # Beta parameter controlling resistance to outliers

    ##
    ##  Bootstrapped percentage bend correlation pb
    ##
    ## data:  mtcars$mpg and mtcars$qsec
    ## N = 32, p-value = 0.6933
    ## alternative hypothesis: equivalence
    ## null values:
    ##   pb   pb
    ##  0.4 -0.4
    ## 90 percent confidence interval:
    ##  0.2509131 0.6634791
    ## sample estimates:
    ##        pb
    ## 0.4484488

The Winsorized correlation reduces the impact of outliers by replacing extreme values with less extreme values. The percentage bend correlation is another robust method that downweights the influence of outliers in the calculation.

# Comparing Correlations

TOSTER provides tools for comparing correlations between independent groups or studies. This is useful for testing differences in relationships across populations or for evaluating replication studies.

## Summary Statistics Approach

When only summary statistics are available, the `compare_cor` function can be used:

    # Comparing correlation r1=0.8 from n=40 with r2=0.2 from n=100
    compare_cor(r1 = .8,
                df1 = 38,  # df = n-2
                r2 = .2,
                df2 = 98)  # df = n-2

    ##
    ##  Difference between two independent correlations (Fisher's z transform)
    ##
    ## data:  Summary Statistics
    ## z = 4.6364, p-value = 3.545e-06
    ## alternative hypothesis: true difference between correlations is not equal to 0
    ## sample estimates:
    ## difference between correlations
    ##                             0.6

The `compare_cor` function supports different methods for comparing correlations:

    # Testing equivalence using Fisher's method
    compare_cor(r1 = .8,
                df1 = 38,
                r2 = .2,
                df2 = 98,
                null = .2,
                method = "f", # Fisher (can also use "fisher")
                alternative = "e") # Equivalence test

    ##
    ##  Difference between two independent correlations (Fisher's z transform)
    ##
    ## data:  Summary Statistics
    ## z = 0.69315, p-value = 0.9998
    ## alternative hypothesis: equivalence
    ## null values:
    ## difference between correlations difference between correlations
    ##                             0.2                            -0.2
    ## sample estimates:
    ## difference between correlations
    ##                             0.6

Available methods include:

- Fisher’s z transformation (`method = "fisher"` or `"f"`): Tests the difference between correlations on the z-transformed scale. This is generally recommended for most applications.
- Kraatz’s method (`method = "kraatz"` or `"k"`): Directly measures the difference between correlation coefficients.

While both methods are appropriate for general significance testing, they may have limited statistical power in some scenarios (Counsell and Cribbie 2015).

## Bootstrapped Comparison

When raw data is available for both correlations, the `boot_compare_cor` function offers a more robust approach through bootstrapping:

    set.seed(8922) # Setting seed for reproducibility
    # Generating example data
    x1 = rnorm(40)
    y1 = rnorm(40)

    x2 = rnorm(100)
    y2 = rnorm(100)

    # Bootstrap comparison with winsorized correlation
    boot_compare_cor(
      x1 = x1,
      x2 = x2,
      y1 = y1,
      y2 = y2,
      null = .2,
      alternative = "e", # Equivalence test
      method = "win" # Winsorized correlation
    )

    ##
    ##  Bootstrapped difference in Winsorized correlation wincor
    ##
    ## data:  x1 and y1 vs. x2 and y2
    ## n1 = 40, n2 = 100, p-value = 0.7739
    ## alternative hypothesis: true differnce in wincor is  0.2
    ## 90 percent confidence interval:
    ##  -0.2970547  0.3978333
    ## sample estimates:
    ##     wincor
    ## 0.06383164

This approach has several advantages:

- It does not rely on the Fisher’s z-transformation approximation
- It can incorporate robust correlation methods
- It can provide more accurate confidence intervals, especially when typical assumptions are violated

# Practical Recommendations

When choosing which correlation method to use in TOSTER:

1. If raw data is available:
   - For most cases, use `boot_cor_test` with Pearson, Spearman, or Kendall methods
   - When outliers or distribution assumptions are concerns, consider the robust methods (winsorized or percentage bend)
2. If only summary statistics are available:
   - Use `corsum_test` for single correlation analysis
   - Use `compare_cor` with the Fisher method for comparing correlations
3. For equivalence testing:
   - Carefully select meaningful boundaries (null values) based on your research context
   - Consider what effect size would be practically insignificant in your field

# Advanced Usage

## Custom Bootstrap Methods

The bootstrapped functions in TOSTER allow customization of the bootstrap procedure:

    # Customizing the bootstrap procedure
    boot_cor_test(
      x = mtcars$mpg,
      y = mtcars$qsec,
      method = "pearson",
      R = 2000,  # Increasing number of bootstrap samples
      alpha = 0.01,  # Using 99% confidence interval
      alternative = "t"  # Two-sided test
    )

## Working with Missing Data

By default, the correlation functions in TOSTER use pairwise complete observations:

    # Example with missing data
    x_with_na <- c(mtcars$mpg, NA, NA)
    y_with_na <- c(mtcars$qsec, 10, NA)

    # Default behavior handles NAs with pairwise deletion
    z_cor_test(x_with_na, y_with_na)

# References

Counsell, Alyssa, and Robert A Cribbie. 2015. “Equivalence Tests for Comparing Correlation and Regression Coefficients.” British Journal of Mathematical and Statistical Psychology 68 (2): 292–309. https://doi.org/10.1111/bmsp.12045.
Goertzen, Jason R, and Robert A Cribbie. 2010. “Detecting a Lack of Association: An Equivalence Testing Approach.” British Journal of Mathematical and Statistical Psychology 63 (3): 527–37. https://doi.org/10.1348/000711009X475853.
Wilcox, Rand R. 2011. Introduction to Robust Estimation and Hypothesis Testing. Academic press.

1. The bootstrapped functions were adapted from code posted by Rand Wilcox on his website, with modifications inspired by Guillaume Rousselet’s `bootcorci` R package, available on GitHub: https://github.com/GRousselet↩︎