wilcoxon_test_b: Bayesian Wilcoxon Rank Sum (aka Mann-Whitney U) and Signed...

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

Bayesian Wilcoxon Rank Sum (aka Mann-Whitney U) and Signed Rank Analyses

Description

Bayesian Wilcoxon Rank Sum (aka Mann-Whitney U) and Signed Rank Analyses

Usage

wilcoxon_test_b(
  x,
  y,
  paired = FALSE,
  p = 0.5,
  ROPE,
  prior = "centered",
  prior_shapes,
  CI_level = 0.95,
  plot = TRUE,
  seed = 1
)

Arguments

x	numeric vector of data values. Non-finite (e.g., infinite or missing) values will be omitted.
y	an optional numeric vector of data values: as with x non-finite values will be omitted.
paired	if TRUE and y is supplied, x-y will be the input of the Bayesian Wilcoxon signed rank test.
p	numeric. Signed rank: wilcox_test_b will return the posterior probability that the population proportion of positive values (i.e., x>y) is greater than this value. Rank sum/Mann-Whitney U: wilcox_test_b will return the posterior probability that the \Omega_x (see details) is greater than this value.
ROPE	If a single number, ROPE will be p\pmROPE. If a vector of length 2, these will serve as the ROPE bounds. Defaults to \pm 0.05.
prior	Prior used on the probability that x > y. Either "uniform" (Beta(1,1)), or "centered" (Beta(2,2)). This is ignored if prior_shapes is provided.
prior_shapes	Vector of length two, giving the shape parameters for the beta distribution that will act as the prior on the population proportions.
CI_level	The posterior probability to be contained in the credible interval.
plot	logical. Should a plot be shown?
seed	Always set your seed! (Unused for \geq 20 observations.)

Details

Bayesian Wilcoxon signed rank analysis For a single input vector or paired data, the Bayesian signed rank analysis will be performed. The estimand is the proportion of (differenced) values that are positive. For more information, see dfba_wilcoxon and vignette("dfba_wilcoxon",package = "DFBA").

Bayesian Wilcoxon rank sum/Mann-Whitney analysis For unpaired x and y inputs, the Bayesian rank sum analysis will be performed. The estimand is \Omega_x:=\lim_{n\to\infty} \frac{U_x}{U_x + U_y}, where U_x is the number of pairs (i,j) such that x_i > y_j, and vice versa for U_y. That is, it is the population proportion of all untied pairs for which x > y. Larger values imply that x is stochastically larger than y. For more information, see dfba_mann_whitney and vignette("dfba_mann_whitney",package = "DFBA").

Value

(returned invisible) If signed rank analysis is implemented, a list with the following:

• posterior_mean: Posterior mean of the proportion of differences that are positive

• CI: Credible interval of the proportion of differences that are positive

• Pr_less_than_p: Probability proportion of differences that are positive is less than the argument p

• Pr_in_ROPE: Probability proportion of differences that are positive is in the ROPE

• prob_plot: Prior and posterior plot of differences that are positive

• posterior_parameters: Posterior beta shape parameters for the proportion of differences that are positive

• BF_for_phi_gr_onehalf_vs_phi_less_onehalf: Bayes factor giving evidence in favor of the proportion of differences that are positive being greater than one half vs. less than one half

• dfba_wilcoxon_object: Underlying DFBA object

If rank sum analysis is implemented, a list with the following:

• posterior_mean: Posterior mean of \Omega_x (see details)

• CI: Credible interval for \Omega_x

• Pr_less_than_p: Posterior probability \Omega_x is less than the argument p

• Pr_in_ROPE: Probability \Omega_x is in the ROPE

• prob_plot: Prior and posterior plot of \Omega_x

• posterior_parameters: Posterior beta shape parameters for \Omega_x

• BF_for_Omegax_gr_onehalf_vs_Omegax_less_onehalf: Bayes factor in favor of \Omega_x being greater than one half vs. less than one half

• dfba_wilcoxon_object: Underlying DFBA object

References

Chechile, R.A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction to Distribution-Free Methods. Cambridge: MIT Press.

Chechile, R. A. (2018) A Bayesian analysis for the Wilcoxon signed-rank statistic. Communications in Statistics - Theory and Methods, https://doi.org/10.1080/03610926.2017.1388402

Chechile, R.A. (2020). A Bayesian analysis for the Mann-Whitney statistic. Communications in Statistics – Theory and Methods 49(3): 670-696. https://doi.org/10.1080/03610926.2018.1549247.

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

Examples

# Signed rank analysis
## Generate data
N = 150
set.seed(2025)
test_data = 
  data.frame(x = rbeta(N,2,10),
             y = rbeta(N,5,10))

## input differenced data
wilcoxon_test_b(test_data$x - test_data$y)
## input paired data vectors individually
wilcoxon_test_b(test_data$x,
                test_data$y,
                paired = TRUE)

## Use different priors
wilcoxon_test_b(test_data$x - test_data$y,
                prior = "uniform")
wilcoxon_test_b(test_data$x - test_data$y,
                prior_shapes = c(5,5))

## Change ROPE bounds
wilcoxon_test_b(test_data$x - test_data$y,
                ROPE = 0.1)

# Rank sum analysis
## Generate data
set.seed(2025)
N = 150
x = rbeta(N,2,10)
y = rbeta(N + 1,5,10)

## Perform analysis
wilcoxon_test_b(x,y)

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