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Mann-Whitney Test

Includes: Exact p-values (n ≤ 50), Normal approximation (ties + continuity correction), Hodges–Lehmann estimate of shift + confidence interval at the selected level (optional).
Purpose: Nonparametric alternative to the two-sample t-test for comparing two independent groups.

Overview

The Mann–Whitney U test (also called the Wilcoxon rank-sum test) compares two independent samples by asking whether one group tends to produce larger values than the other.

  • It works for continuous or ordinal outcomes.
  • It does not assume normality.
  • If the two distributions have a similar shape/spread, the result is often interpreted as a difference in location (median/shift).

BESHStatNG reports:

  • U statistics (U₁ and U₂),
  • a two-sided p-value based on a normal approximation (with ties + continuity correction),
  • exact p-values when available (n ≤ 50),
  • optional Hodges–Lehmann shift estimate with confidence interval at the selected level.

Example dataset

In the screenshots, the first two columns from the dataset below are used as the two groups:

  • Column 1: Relaxation response and biofeedback
  • Column 2: Relaxation response alone

Download:

Only the first two columns are used for this Mann–Whitney example. (The third column is ignored here.)

Screenshots (BESHStatNG)

Input tab

Mann-Whitney – Input

Options tab

Mann-Whitney – Options

Results

Mann-Whitney – Results

When to use it

Use the Mann–Whitney test when:

  • you have two independent groups (different subjects in each group),
  • the outcome is numeric or at least ordinal,
  • you want a robust comparison when normality is questionable and/or outliers are present.

Key requirements / considerations:

  • Independence: observations are independent within and between groups.
  • Interpretation as “median difference”: this is most defensible when group distributions have a similar shape/spread; otherwise the test is better described as a test of stochastic dominance (one group tends to yield larger values).

Inputs in Excel

BESHStatNG uses the shared “Group by Column / Group by ID” dialog (also used for Unpaired t-test and ROC).

Option A: Group by Column (two ranges)

  • Data: Group 1: a single-column range for Group 1 values
  • Data: Group 2: a single-column range for Group 2 values

Example: two adjacent columns containing the two groups.

Option B: Group by ID (two columns: group + values)

  • Group ID: a column containing exactly two group labels (e.g., 1/2 or A/B)
  • Values: a column containing the numeric measurement

BESHStatNG splits the values by the unique group IDs and runs the test.

Output destination

  • Output range (current sheet)
  • New worksheet
  • New workbook

Options

On the Options tab you can add:

  • Full Descriptive Statistics — summary statistics per group
  • Box and Whiskers Plot — Tukey boxplot comparing the two groups
  • Hodges–Lehmann Estimate of Shift — pseudo-median location shift and confidence interval at the selected level
  • Alpha — two-sided significance level used for the Hodges–Lehmann confidence interval.
    Default: 0.05 (95% confidence interval)

Output and how to read it

Main results table

BESHStatNG writes a “Mann-Whitney test” table containing:

  • n (sample size per group)
  • Median, Q1, Q3 (quartiles used throughout BESHStatNG)
  • Test statistic U: reports U₁ (Group 1) and U₂ (Group 2)
  • Two sided p-value: normal approximation (ties + continuity corrected)
  • Exact p-values (when available): two-sided, low-side (“Group 1 smaller”), upper-side (“Group 1 larger”)

Example interpretation (from the screenshots):

  • U₁ = 9, U₂ = 27 (n₁ = n₂ = 6)
  • Normal-approx two-sided p = 0.171979
  • Exact two-sided p = 0.162338
  • The direction suggests Group 1 tends to be smaller, but p > 0.05 means the evidence is not strong at the 5% level.

Hodges–Lehmann shift table (optional)

When enabled, BESHStatNG adds a second table:

  • mean/median diff (CI at selected level)

This is the Hodges–Lehmann estimate of the location shift (Group 1 − Group 2).
In the example: −14 (−38 to 19), suggesting Group 1 is about 14 units lower, with a wide CI spanning 0.
The screenshot uses the default alpha = 0.05, so the displayed interval is 95%.

What it does (math and implementation details)

Let:

  • Group 1 values: \(x_1,\dots,x_{n_1}\)
  • Group 2 values: \(y_1,\dots,y_{n_2}\)
  • \(n = n_1+n_2\)

A) Rank-sum and U statistics

Combine all observations and assign ranks \(1,\dots,n\).
Ties are given the average rank (midranks).

Let \(R_1\) be the sum of ranks for Group 1.

\[ U_1 = R_1 - \frac{n_1(n_1+1)}{2}, \qquad U_2 = n_1n_2 - U_1 \]

BESHStatNG reports both \(U_1\) and \(U_2\), and uses:

\[ U_{\text{obs}} = \min(U_1, U_2) \]

Exact p-values (tie-aware; computed by the add-in)

The add-in can compute exact Mann–Whitney p-values even when ties are present.
For total sample size n = n₁ + n₂ ≤ 50, it uses a fast exact algorithm to evaluate the null distribution of the test statistic (based on the exact distribution of the rank-sum / U statistic under all permissible allocations of ranks, including ties).

Because ties slightly change the set of possible rank-sum outcomes (and their probabilities), some software falls back to an asymptotic approximation when ties occur. In contrast, the add-in remains exact (tie-aware) within the above sample-size limit.

BESHStatNG reports:

  • Exact low-side p-value: \(p_\text{low}=P(U \le U_{\text{obs}})\)
  • Exact upper-side p-value: \(p_\text{up}=P(U > U_{\text{obs}})\)
  • Exact two-sided p-value: \(p_{2\text{s}}=\min\{1, 2\min(p_\text{low},p_\text{up})\}\)

This matches the “low-side” and “upper-side” textual interpretations in the output table.

Normal approximation (standard; tie- and continuity-corrected)

The add-in uses the standard asymptotic normal approximation for the Mann–Whitney (Wilcoxon rank-sum) statistic, including the usual tie correction in the variance and an optional continuity correction.
Minor differences versus other software can occur due to different defaults for the continuity correction and slightly different two-sided p-value conventions.

BESHStatNG always reports an asymptotic (normal) p-value using:

  • Mean under \(H_0\): \(\mu_U = \frac{n_1n_2}{2}\)

  • Tie correction factor:

\[ C = \frac{\sum_k (t_k^3-t_k)}{n^3-n} \]

where \(t_k\) are the sizes of tied groups in the combined sample.

  • Variance:
\[ \sigma_U^2 = \frac{n_1n_2(n+1)}{12}\,(1-C) \]
  • Continuity-corrected Z (BESHStatNG uses \(U_{\text{obs}}\), so Z is negative):
\[ Z = \frac{U_{\text{obs}} - \mu_U + 0.5}{\sigma_U} \]

Two-sided p-value:

\[ p = 2\,\Phi(-|Z|) \]

D) Hodges–Lehmann estimate of shift (optional)

Define all pairwise differences:

\[ d_{ij} = x_i - y_j,\qquad i=1,\dots,n_1,\; j=1,\dots,n_2 \]

The Hodges–Lehmann estimate (pseudo-median shift) is:

\[ \hat{\Delta} = \mathrm{median}\{d_{ij}\} \]

Confidence interval at level \(1-\alpha\)

Let the sorted differences be:

\[ d_{(1)} \le d_{(2)} \le \dots \le d_{(n_1n_2)} \]

For \(n_1 \le 20\) and \(n_2 \le 20\), BESHStatNG uses an exact critical value \(W^*\) from Conover (1999), Table A7 only when \(\alpha = 0.05\), then converts it to:

\[ k = W^* - \frac{n_1(n_1+1)}{2} \]

and reports:

\[ \text{CI}_{1-\alpha} = \big[d_{(k)},\; d_{(n_1n_2-k+1)}\big] \]

For larger samples, or for other confidence levels, BESHStatNG uses a normal-approximation index based on \(z_{1-\alpha/2}\) and Excel-style percentiles for the bounds.

R code (reference using standard R functions)

R’s built-in stats::wilcox.test() performs the same test (it reports the rank-sum statistic W, which can be converted to U). It can also return the Hodges–Lehmann location shift and a confidence interval via conf.int = TRUE.

# Mann–Whitney / Wilcoxon rank-sum example (using stats::wilcox.test)
dat <- read.csv("020kruskalwalliscsv.csv")

# Group 1 = column 1, Group 2 = column 2
x <- dat[[1]]
y <- dat[[2]]

# Drop missing/non-finite values (BESHStatNG ignores invalid cells)
x <- x[is.finite(x)]
y <- y[is.finite(y)]

n1 <- length(x); n2 <- length(y)

# --- Descriptive stats (quartiles matching BESHStatNG: type = 5) ---
q1 <- quantile(x, probs = c(0.25, 0.5, 0.75), type = 5, names = FALSE)
q2 <- quantile(y, probs = c(0.25, 0.5, 0.75), type = 5, names = FALSE)

# --- Normal approximation p-value (ties + continuity correction) ---
# (forces asymptotic calculation, even when an exact p-value is available)
wt_norm <- wilcox.test(x, y,
                       alternative = "two.sided",
                       exact = FALSE,     # force normal approximation
                       correct = TRUE)    # continuity correction
p_norm_2s <- wt_norm$p.value

# --- Exact p-values (when R can compute them) ---
# Note: R only computes an *exact* p-value when there are no ties.
wt_2s  <- wilcox.test(x, y, alternative = "two.sided", exact = TRUE, correct = TRUE)
wt_low <- wilcox.test(x, y, alternative = "less",      exact = TRUE, correct = TRUE)    # Group 1 smaller
wt_up  <- wilcox.test(x, y, alternative = "greater",   exact = TRUE, correct = TRUE)    # Group 1 larger

p_exact_2s <- wt_2s$p.value
p_low      <- wt_low$p.value
p_up       <- wt_up$p.value

# --- Convert W (rank-sum) to U to match BESHStatNG output ---
W  <- as.numeric(wt_2s$statistic)               # Wilcoxon rank-sum statistic (sum of ranks for Group 1)
U1 <- W - n1 * (n1 + 1) / 2
U2 <- n1 * n2 - U1

# --- Hodges–Lehmann shift estimate + CI (package method) ---
alpha <- 0.05   # current dialog default; use 0.10 for a 90% CI

wt_hl <- wilcox.test(x, y,
                     alternative = "two.sided",
                     exact = TRUE,
                     correct = TRUE,
                     conf.int = TRUE,
                     conf.level = 1 - alpha)
hl    <- unname(wt_hl$estimate)                 # HL (pseudo-median) shift (Group 1 - Group 2)
hl_ci <- unname(wt_hl$conf.int)                 # CI for HL shift

Why R results may be slightly different from BESHStatNG

  • Exact p-values with ties: wilcox.test(..., exact = TRUE) computes an exact p-value only when there are no ties in the pooled data.
    If ties are present, R will typically issue a warning and return an asymptotic p-value instead, while BESHStatNG may still report its own “exact” result for small \(n\) (based on its internal algorithm).

  • Continuity correction details: both use a continuity correction, but implementations can differ subtly (e.g., whether the correction is applied to \(W\) vs \(U\), and how the sign is chosen for two-sided tests).
    These differences usually change p-values only in the third or later decimal place.

  • Hodges–Lehmann CI method: BESHStatNG’s small-sample CI uses Conover’s tabulated critical values (and an internal index conversion), whereas R inverts the Wilcoxon distribution as implemented in wilcox.test().
    The HL point estimate is typically the same, but the CI bounds can differ by one step in the ordered pairwise differences.

Quartiles in R

BESHStatNG uses the CDF method (SAS Method 5) for quartiles.
To match its \(Q_1\), Median and \(Q_3\) in R, use quantile(..., type = 5) (as shown above).

Notes

  • Invalid / missing cells: empty cells, text, logical values, and Excel error values are ignored during data import (only numeric values are used).
  • Exact results availability: BESHStatNG computes exact p-values when the total sample size \(n \le 50\). Otherwise it reports the normal approximation.
  • Effect size (optional reporting): a simple “common language” effect size is \(U_1/(n_1n_2)\), interpretable as the estimated probability that a random Group 1 observation exceeds a random Group 2 observation (with midrank handling for ties).

See also