cor_test_b: Test for Association/Correlation Between Paired Samples via...

In bayesics: Bayesian Analyses for One- and Two-Sample Inference and Regression Methods

Test for Association/Correlation Between Paired Samples via Kendall's tau

Description

Test for Association/Correlation Between Paired Samples via Kendall's tau

Usage

cor_test_b(x, ...)

## Default S3 method:
cor_test_b(
  x,
  y,
  tau = 0,
  ROPE,
  prior = "centered",
  prior_shapes,
  CI_level = 0.95,
  plot = TRUE,
  ...
)

## S3 method for class 'formula'
cor_test_b(
  formula,
  data,
  tau = 0,
  ROPE,
  prior = "centered",
  prior_shapes,
  CI_level = 0.95,
  plot = TRUE,
  ...
)

Arguments

x, y	numeric vectors of data values. x and y must have the same length.
...	optional arguments.
tau	If provided, cor_test_b will return the posterior probability that Kendall's tau is less than this value.
ROPE	If a single number, ROPE will be \tau\pm ROPE. If a vector of length 2, these will serve as the ROPE bounds. Defaults to \pm0.05.
prior	Beta prior used on phi (see details). Either "uniform' (Beta(1,1)), "centered' (Beta(2,2)), "positive" (Beta(3.9,2), putting 80% of the prior mass above 0.5), or "negative" (Beta(2,3.9), putting 80% of the prior mass below 0.5).
prior_shapes	Vector of length two, giving the shape parameters for the beta distribution that will act as the prior on \phi (see details).
CI_level	The posterior probability to be contained in the credible interval.
plot	logical. Should a plot be shown?
formula	ADD description!
data	ADD description!

Details

cor_test_b relies on the robust Kendall's tau, defined to be

\tau := \frac{(\# \text{concordant pairs}) - (\# \text{discordant pairs})}{(\# \text{concordant pairs}) - (\# \text{discordant pairs})},

where a concordant pair is a pair of points such that if the rank of the x values is higher for the first (second) point of the pair, so too the rank of the y value is higher for the first (second) point of the pair.

The Bayesian approach of Chechile (2020) puts a Beta prior on phi, the proportion of concordance, i.e.,

\phi := \frac{(\# \text{concordant pairs})}{(\# \text{concordant pairs}) - (\# \text{discordant pairs})}.

The relationship between the two, then, is \tau = 2\phi - 1, or equivalently \phi = (\tau + 1)/2.

For more information, see dfba_bivariate_concordance and vignette("dfba_bivariate_concordance",package = "DFBA").

Value

(returned invisible) A list with the following:

• posterior_mean: posterior mean of Kendall's tau

• CI: Credible interval bounds

• Pr_less_than_tau: Posterior probability that Kendall's tau is less than provided reference tau

• Pr_in_ROPE: Posterior probability that Kendall's tau is in the ROPE

• prob_plot: Posterior and prior plot

• posterior_parameters: The posterior for Kendall's tau is a location shift and scaled beta distribution to fall over the range -1 to 1, i.e., 0.5 * (\tau + 1.0) follows a beta with shape parameters given by posterior_parameters.

• dfba_bivariate_concordance_object: The underlying object from the DFBA package

References

Chechile, R.A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution_Free Statistics. Cambridge: MIT Press.

Chechile, R.A., & Barch, D.H. (2021). A distribution-free, Bayesian goodness-of-fit method for assessing similar scientific prediction equations. Journal of Mathematical Psychology. https://doi.org/10.1016/j.jmp.2021.102638

Lindley, D. V., & Phillips, L. D. (1976). Inference for a Bernoulli process (a Bayesian view). The American Statistician, 30, 112-119.

Barch DH, Chechile RA (2023). DFBA: Distribution-Free Bayesian Analysis. doi:10.32614/CRAN.package.DFBA

Examples

# Generate data
set.seed(2025)
N = 50
x = rnorm(N)
y = x + 4 * rnorm(N)

# Test for non-zero correlation
cor_test_b(x,y)

# Input can be in the form of formula and data
cor_test_b(~ asdf + qwer,
           data = data.frame(asdf = x,
                             qwer = y))

# Other priors can be used, also.  See help for details.
cor_test_b(x,y,
           prior = "uniform")
cor_test_b(x,y,
           prior = "negative")
cor_test_b(x,y,
           prior = "positive")
cor_test_b(x,y,
           prior_shapes = c(10,10))

bayesics documentation built on March 11, 2026, 5:07 p.m.