Namespaces
Variants
Actions

Difference between revisions of "Mann-Whitney test"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Mann–Whitney test to Mann-Whitney test: ascii title)
m (tex encoded by computer)
Line 1: Line 1:
A statistical test for testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622401.png" /> of homogeneity of two samples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622403.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622404.png" /> elements of which are mutually independent and have continuous distributions. This test, suggested by H.B. Mann and D.R. Whitney [[#References|[1]]], is based on the statistic
+
<!--
 +
m0622401.png
 +
$#A+1 = 13 n = 0
 +
$#C+1 = 13 : ~/encyclopedia/old_files/data/M062/M.0602240 Mann\ANDWhitney test
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622405.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622406.png" /> is the statistic of the [[Wilcoxon test|Wilcoxon test]] intended for testing the same hypothesis, equal to the sum of the ranks of the elements of the second sample among the pooled order statistics (cf. [[Order statistic|Order statistic]]), and
+
A statistical test for testing the hypothesis $  H _ {0} $
 +
of homogeneity of two samples  $  X _ {1} \dots X _ {n} $
 +
and  $  Y _ {1} \dots Y _ {m} $,
 +
all  $  m + n $
 +
elements of which are mutually independent and have continuous distributions. This test, suggested by H.B. Mann and D.R. Whitney [[#References|[1]]], is based on the statistic
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622407.png" /></td> </tr></table>
+
$$
 +
= W -  
 +
\frac{1}{2}
 +
m ( m + 1 )  = \
 +
\sum _ { i= } 1 ^ { n }  \
 +
\sum _ { j= } 1 ^ { m }
 +
\delta _ {ij} ,
 +
$$
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622408.png" /> counts the number of cases when the elements of the second sample exceed elements of the first sample. It follows from the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m0622409.png" /> that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m06224010.png" /> is true, then
+
where  $  W $
 +
is the statistic of the [[Wilcoxon test|Wilcoxon test]] intended for testing the same hypothesis, equal to the sum of the ranks of the elements of the second sample among the pooled order statistics (cf. [[Order statistic|Order statistic]]), and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m06224011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
\delta _ {ij}  = \
 +
\left \{
  
and, in addition, this statistic has all the properties of the Wilcoxon statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m06224012.png" />, including asymptotic normality with parameters (*).
+
Thus,  $  U $
 +
counts the number of cases when the elements of the second sample exceed elements of the first sample. It follows from the definition of  $  U $
 +
that if  $  H _ {0} $
 +
is true, then
 +
 
 +
$$ \tag{* }
 +
{\mathsf E} U  = 
 +
\frac{nm}{2}
 +
,\ \
 +
{\mathsf D} U  = 
 +
\frac{n m ( n + m + 1 ) }{12}
 +
,
 +
$$
 +
 
 +
and, in addition, this statistic has all the properties of the Wilcoxon statistic $  W $,  
 +
including asymptotic normality with parameters (*).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.B. Mann,  D.R. Whitney,  "On a test whether one of two random variables is statistically larger than the other"  ''Ann. Math. Stat.'' , '''18'''  (1947)  pp. 50–60</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.B. Mann,  D.R. Whitney,  "On a test whether one of two random variables is statistically larger than the other"  ''Ann. Math. Stat.'' , '''18'''  (1947)  pp. 50–60</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Instead of Mann–Whitney test, the phrase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062240/m06224014.png" />-test is also used.
+
Instead of Mann–Whitney test, the phrase $  U $-
 +
test is also used.

Revision as of 07:59, 6 June 2020


A statistical test for testing the hypothesis $ H _ {0} $ of homogeneity of two samples $ X _ {1} \dots X _ {n} $ and $ Y _ {1} \dots Y _ {m} $, all $ m + n $ elements of which are mutually independent and have continuous distributions. This test, suggested by H.B. Mann and D.R. Whitney [1], is based on the statistic

$$ U = W - \frac{1}{2} m ( m + 1 ) = \ \sum _ { i= } 1 ^ { n } \ \sum _ { j= } 1 ^ { m } \delta _ {ij} , $$

where $ W $ is the statistic of the Wilcoxon test intended for testing the same hypothesis, equal to the sum of the ranks of the elements of the second sample among the pooled order statistics (cf. Order statistic), and

$$ \delta _ {ij} = \ \left \{ Thus, $ U $ counts the number of cases when the elements of the second sample exceed elements of the first sample. It follows from the definition of $ U $ that if $ H _ {0} $ is true, then $$ \tag{* } {\mathsf E} U = \frac{nm}{2}

,\ \ 

{\mathsf D} U = \frac{n m ( n + m + 1 ) }{12}

,

$$

and, in addition, this statistic has all the properties of the Wilcoxon statistic $ W $, including asymptotic normality with parameters (*).

References

[1] H.B. Mann, D.R. Whitney, "On a test whether one of two random variables is statistically larger than the other" Ann. Math. Stat. , 18 (1947) pp. 50–60

Comments

Instead of Mann–Whitney test, the phrase $ U $- test is also used.

How to Cite This Entry:
Mann-Whitney test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mann-Whitney_test&oldid=22789
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article