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Bootstrapped Correlation Coefficients

boot_cor_test.Rd

[Stable]

A function for bootstrap-based correlation tests using various correlation coefficients including Pearson's, Kendall's, Spearman's, Winsorized, and percentage bend correlations. This function supports standard, equivalence, and minimal effect testing with robust bootstrap methods.

Usage

boot_cor_test(
  x,
  y,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman", "winsorized", "bendpercent"),
  alpha = 0.05,
  null = 0,
  boot_ci = c("basic", "perc"),
  R = 1999,
  ...
)

Arguments

x

a (non-empty) numeric vector of data values.

y

an optional (non-empty) numeric vector of data values.

alternative

a character string specifying the alternative hypothesis:

  • "two.sided": correlation is not equal to null (default)

  • "greater": correlation is greater than null

  • "less": correlation is less than null

  • "equivalence": correlation is within the equivalence bounds (TOST)

  • "minimal.effect": correlation is outside the equivalence bounds (TOST)

You can specify just the initial letter.

method

a character string indicating which correlation coefficient to use:

  • "pearson": standard Pearson product-moment correlation

  • "kendall": Kendall's tau rank correlation

  • "spearman": Spearman's rho rank correlation

  • "winsorized": Winsorized correlation (robust to outliers)

  • "bendpercent": percentage bend correlation (robust to marginal outliers)

Can be abbreviated.

alpha

alpha level (default = 0.05)

null

a number or vector indicating the null hypothesis value(s):

  • For standard tests: a single value (default = 0)

  • For equivalence/minimal effect tests: either a single value (symmetric bounds ±value will be created) or a vector of two values representing the lower and upper bounds

boot_ci

type of bootstrap confidence interval:

  • "basic": basic/empirical bootstrap CI

  • "perc": percentile bootstrap CI (default)

R

number of bootstrap replications (default = 1999).

...

additional arguments passed to correlation functions, such as:

  • tr: trim for Winsorized correlation (default = 0.2)

  • beta: for percentage bend correlation (default = 0.2)

Value

A list with class "htest" containing the following components:

  • p.value: the bootstrap p-value of the test.

  • parameter: the number of observations used in the test.

  • conf.int: a bootstrap confidence interval for the correlation coefficient.

  • estimate: the estimated correlation coefficient, with name "cor", "tau", "rho", "pb", or "wincor" corresponding to the method employed.

  • stderr: the bootstrap standard error of the correlation coefficient.

  • null.value: the value(s) of the correlation under the null hypothesis.

  • alternative: character string indicating the alternative hypothesis.

  • method: a character string indicating which bootstrapped correlation was measured.

  • data.name: a character string giving the names of the data.

  • boot_res: vector of bootstrap correlation estimates.

  • call: the matched call.

Details

This function uses bootstrap methods to calculate correlation coefficients and their confidence intervals. P-values are calculated from a re-sampled null distribution.

The bootstrap correlation methods in this package offer two robust correlations beyond the standard methods:

  1. Winsorized correlation: Replaces extreme values with less extreme values before calculating the correlation. The trim parameter (default: tr = 0.2) determines the proportion of data to be Winsorized.

  2. Percentage bend correlation: A robust correlation that downweights the influence of outliers. The beta parameter (default = 0.2) determines the bending constant.

These calculations are based on Rand Wilcox's R functions for his book (Wilcox, 2017), and adapted from their implementation in Guillaume Rousselet's R package "bootcorci".

The function supports both standard hypothesis testing and equivalence/minimal effect testing:

  • For standard tests (two.sided, less, greater), the function tests whether the correlation differs from the null value (typically 0).

  • For equivalence testing ("equivalence"), it determines whether the correlation falls within the specified bounds, which can be set asymmetrically.

  • For minimal effect testing ("minimal.effect"), it determines whether the correlation falls outside the specified bounds.

When performing equivalence or minimal effect testing:

  • If a single value is provided for null, symmetric bounds ±value will be used

  • If two values are provided for null, they will be used as the lower and upper bounds

See vignette("correlations") for more details.

References

Wilcox, R.R. (2009) Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. Communications in Statistics - Simulation and Computation, 38, 2220–2234.

Wilcox, R.R. (2017) Introduction to Robust Estimation and Hypothesis Testing, 4th edition. Academic Press.

See also

Other Correlations: corsum_test(), plot_cor(), power_z_cor(), z_cor_test()

Examples

# Example 1: Standard bootstrap test with Pearson correlation
x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
boot_cor_test(x, y, method = "pearson", alternative = "two.sided",
              R = 999) # Fewer replicates for example
#> 
#> 	Bootstrapped Pearson's product-moment correlation
#> 
#> data:  x and y
#> N = 9, p-value = 0.008008
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#>  0.1873998 0.9186527
#> sample estimates:
#>       cor 
#> 0.5711816 
#> 

# Example 2: Equivalence test with Spearman correlation
# Testing if correlation is equivalent to zero within ±0.3
boot_cor_test(x, y, method = "spearman", alternative = "equivalence",
             null = 0.3, R = 999)
#> 
#> 	Bootstrapped Spearman's rank correlation rho
#> 
#> data:  x and y
#> N = 9, p-value = 0.8188
#> alternative hypothesis: equivalence
#> null values:
#>  rho  rho 
#>  0.3 -0.3 
#> 90 percent confidence interval:
#>  0.2540541 1.1908257
#> sample estimates:
#> rho 
#> 0.6 
#> 

# Example 3: Using robust correlation methods
# Using Winsorized correlation with custom trim
boot_cor_test(x, y, method = "winsorized", tr = 0.1,
             R = 999)
#> 
#> 	Bootstrapped Winsorized correlation wincor
#> 
#> data:  x and y
#> N = 9, p-value = 0.006006
#> alternative hypothesis: true wincor is not equal to 0
#> 95 percent confidence interval:
#>  0.1930584 0.9217130
#> sample estimates:
#>    wincor 
#> 0.5711816 
#> 

# Example 4: Using percentage bend correlation
boot_cor_test(x, y, method = "bendpercent", beta = 0.2,
             R = 999)
#> 
#> 	Bootstrapped percentage bend correlation pb
#> 
#> data:  x and y
#> N = 9, p-value = 0.06006
#> alternative hypothesis: true pb is not equal to 0
#> 95 percent confidence interval:
#>  0.5442422 1.6082703
#> sample estimates:
#>        pb 
#> 0.7556884 
#> 

# Example 5: Minimal effect test with asymmetric bounds
# Testing if correlation is outside bounds of -0.1 and 0.4
boot_cor_test(x, y, method = "pearson", alternative = "minimal.effect",
             null = c(-0.1, 0.4), R = 999)
#> 
#> 	Bootstrapped Pearson's product-moment correlation
#> 
#> data:  x and y
#> N = 9, p-value = 0.1151
#> alternative hypothesis: minimal.effect
#> null values:
#> correlation correlation 
#>        -0.1         0.4 
#> 90 percent confidence interval:
#>  0.2049974 0.8543517
#> sample estimates:
#>       cor 
#> 0.5711816 
#>