Mood’s Median Test | Nonparametric Analysis

Mood's median test dichotomizes observations from two independent groups as above or below the pooled median and compares the proportions falling on each side.

Applicable Study Designs#

Design	Method
Two independent groups; proportions above and below the pooled median	Median test
Two independent groups; all rank information	Rank-sum / Mann–Whitney
One sample or paired data	Sign / Signed-rank

Hypotheses and the 2×2 Table#

H_0:\eta_1=\eta_2\qquad H_1:\eta_1\ne\eta_2

	Above the pooled median	Below	Total
Group 1	m₁	n₁−m₁	n₁
Group 2	m₂	n₂−m₂	n₂
Total	M	N−M	N

Use a prespecified rule for observations equal to the pooled median. They are often excluded, but when ties are numerous the result may depend on this rule, so report it explicitly.

Hypergeometric Probability#

assignments matching the table

\binom{n_1}{m_1}\binom{n_2}{m_2}

all assignments

\binom{n_1+n_2}{m_1+m_2}

table probability

P(M_1=m_1\mid M)=\frac{\binom{n_1}{m_1}\binom{n_2}{m_2}}{\binom{n_1+n_2}{m_1+m_2}}

This is the probability of one observed table, and the two-sided p-value sums the probability of all tables being equally or more inconsistent with H₀. This is the same conditional calculation as Fisher’s exact test.

Counts on Either Side of the Pooled Median#

N is even

m_1+m_2=\frac N2

If N is odd, exclude the median

m_1+m_2=\frac{N-1}{2}

Two-proportion approximation for large samples#

difference in proportions

\hat p_1-\hat p_2=\frac{m_1}{n_1}-\frac{m_2}{n_2}

pooled proportion

\hat p=\frac{m_1+m_2}{n_1+n_2},\quad\hat q=1-\hat p

Z statistic

Z=\frac{m_1/n_1-m_2/n_2}{\sqrt{\hat p\hat q(1/n_1+1/n_2)}}