# kruskalwallis

Kruskal-Wallis test

## Syntax

## Description

returns
the `p`

= kruskalwallis(`x`

)*p*-value for the null hypothesis that the data in each column
of the matrix `x`

comes from the same distribution, using the
Kruskal-Wallis test. The
alternative hypothesis is that not all samples come from the same distribution. The
Kruskal-Wallis test provides a nonparametric alternative to a one-way ANOVA. For
more information, see Kruskal-Wallis Test.

returns
the `p`

= kruskalwallis(`x`

,`group`

,`displayopt`

)*p*-value of the test and lets you display or
suppress the ANOVA table and box plot.

## Examples

### Test Data Samples for the Same Distribution

Create two different normal probability distribution objects. The first distribution has `mu = 0`

and `sigma = 1`

, and the second distribution has `mu = 2`

and `sigma = 1`

.

pd1 = makedist('Normal'); pd2 = makedist('Normal','mu',2,'sigma',1);

Create a matrix of sample data by generating random numbers from these two distributions.

rng('default'); % for reproducibility x = [random(pd1,20,2),random(pd2,20,1)];

The first two columns of `x`

contain data generated from the first distribution, while the third column contains data generated from the second distribution.

Test the null hypothesis that the sample data from each column in `x`

comes from the same distribution.

p = kruskalwallis(x)

p = 3.6896e-06

The returned value of `p`

indicates that `kruskalwallis`

rejects the null hypothesis that all three data samples come from the same distribution at a 1% significance level. The ANOVA table provides additional test results, and the box plot visually presents the summary statistics for each column in `x`

.

### Conduct Follow-up Tests for Unequal Medians

Create two different normal probability distribution objects. The first distribution has `mu = 0`

and `sigma = 1`

. The second distribution has `mu = 2`

and `sigma = 1`

.

pd1 = makedist('Normal'); pd2 = makedist('Normal','mu',2,'sigma',1);

Create a matrix of sample data by generating random numbers from these two distributions.

rng('default'); % for reproducibility x = [random(pd1,20,2),random(pd2,20,1)];

The first two columns of `x`

contain data generated from the first distribution, while the third column contains data generated from the second distribution.

Test the null hypothesis that the sample data from each column in `x`

comes from the same distribution. Suppress the output displays, and generate the structure `stats`

to use in further testing.

`[p,tbl,stats] = kruskalwallis(x,[],'off')`

p = 3.6896e-06

`tbl=`*4×6 cell array*
{'Source' } {'SS' } {'df'} {'MS' } {'Chi-sq' } {'Prob>Chi-sq'}
{'Columns'} {[7.6311e+03]} {[ 2]} {[3.8155e+03]} {[ 25.0200]} {[ 3.6896e-06]}
{'Error' } {[1.0364e+04]} {[57]} {[ 181.8228]} {0x0 double} {0x0 double }
{'Total' } {[ 17995]} {[59]} {0x0 double } {0x0 double} {0x0 double }

`stats = `*struct with fields:*
gnames: [3x1 char]
n: [20 20 20]
source: 'kruskalwallis'
meanranks: [26.7500 18.9500 45.8000]
sumt: 0

The returned value of `p`

indicates that the test rejects the null hypothesis at the 1% significance level. You can use the structure `stats`

to perform additional follow-up testing. The cell array `tbl`

contains the same data as the graphical ANOVA table, including column and row labels.

Conduct a follow-up test to identify which data sample comes from a different distribution.

c = multcompare(stats);

Note: Intervals can be used for testing but are not simultaneous confidence intervals.

Display the multiple comparison results in a table.

tbl = array2table(c,"VariableNames", ... ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"])

`tbl=`*3×6 table*
Group A Group B Lower Limit A-B Upper Limit P-value
_______ _______ ___________ ______ ___________ __________
1 2 -5.1435 7.8 20.744 0.33446
1 3 -31.994 -19.05 -6.1065 0.0016282
2 3 -39.794 -26.85 -13.906 3.4768e-06

The results indicate that there is a significant difference between groups 1 and 3, so the test rejects the null hypothesis that the data in these two groups comes from the same distribution. The same is true for groups 2 and 3. However, there is not a significant difference between groups 1 and 2, so the test does not reject the null hypothesis that these two groups come from the same distribution. Therefore, these results suggest that the data in groups 1 and 2 come from the same distribution, and the data in group 3 comes from a different distribution.

### Test for the Same Distribution Across Groups

Create a vector, `strength`

, containing measurements of the strength of metal beams. Create a second vector, `alloy`

, indicating the type of metal alloy from which the corresponding beam is made.

strength = [82 86 79 83 84 85 86 87 74 82 ... 78 75 76 77 79 79 77 78 82 79]; alloy = {'st','st','st','st','st','st','st','st',... 'al1','al1','al1','al1','al1','al1',... 'al2','al2','al2','al2','al2','al2'};

Test the null hypothesis that the beam strength measurements have the same distribution across all three alloys.

`p = kruskalwallis(strength,alloy,'off')`

p = 0.0018

The returned value of `p`

indicates that the test rejects the null hypothesis at the 1% significance level.

## Input Arguments

`x`

— Sample data

vector | matrix

Sample data for the hypothesis test, specified as a vector or
an *m*-by-*n* matrix. If `x`

is
an *m*-by-*n* matrix, each of the *n* columns
represents an independent sample containing *m* mutually
independent observations.

**Data Types: **`single`

| `double`

`group`

— Grouping variable

numeric vector | logical vector | character array | string array | cell array of character vectors

Grouping variable, specified as a numeric or logical vector, a character or string array, or a cell array of character vectors.

If

`x`

is a vector, then each element in`group`

identifies the group to which the corresponding element in`x`

belongs, and`group`

must be a vector of the same length as`x`

. If a row of`group`

contains an empty value, that row and the corresponding observation in`x`

are disregarded.`NaN`

values in either`x`

or`group`

are similarly ignored.If

`x`

is a matrix, then each column in`x`

represents a different group, and you can use`group`

to specify labels for these columns. The number of elements in`group`

and the number of columns in`x`

must be equal.

The labels contained in `group`

also annotate
the box plot.

**Example: **`{'red','blue','green','blue','red','blue','green','green','red'}`

**Data Types: **`single`

| `double`

| `logical`

| `char`

| `string`

| `cell`

`displayopt`

— Display option

`'on'`

(default) | `'off'`

Display option, specified as `'on'`

or `'off'`

.
If `displayopt`

is `'on'`

, `kruskalwallis`

displays
the following figures:

An ANOVA table containing the sums of squares, degrees of freedom, and other quantities calculated based on the ranks of the data in

`x`

.A box plot of the data in each column of the data matrix

`x`

. The box plots are based on the actual data values, rather than on the ranks.

If `displayopt`

is `'off'`

, `kruskalwallis`

does
not display these figures.

If you specify a value for `displayopt`

, you
must also specify a value for `group`

. If you do
not have a grouping variable, specify `group`

as `[]`

.

**Example: **`'off'`

## Output Arguments

`p`

— *p*-value

scalar value in the range [0,1]

*p*-value of the test, returned as a scalar value in the range [0,1].
`p`

is the probability of observing a test statistic that is as
extreme as, or more extreme than, the observed value under the null hypothesis. A small
value of `p`

indicates that the null hypothesis might not be
valid.

`tbl`

— ANOVA table

cell array

ANOVA table of test results, returned as a cell array. `tbl`

includes
the sums of squares, degrees of freedom, and other quantities calculated
based on the ranks of the data in `x`

, as well
as column and row labels.

`stats`

— Test data

structure

Test data, returned as a structure. You can perform follow-up multiple comparison tests on
pairs of sample medians by using `multcompare`

, with
`stats`

as the input value.

## More About

### Kruskal-Wallis Test

The Kruskal-Wallis test is a nonparametric version of classical one-way
ANOVA, and an extension of the Wilcoxon rank sum test to more than two groups. The
Kruskal-Wallis test is valid for data that has two or more groups. It compares the
medians of the groups of data in `x`

to determine if the samples
come from the same population (or, equivalently, from different populations with the
same distribution).

The Kruskal-Wallis test uses ranks of the data, rather than
numeric values, to compute the test statistics. It finds ranks by
ordering the data from smallest to largest across all groups, and
taking the numeric index of this ordering. The rank for a tied observation
is equal to the average rank of all observations tied with it. The *F*-statistic
used in classical one-way ANOVA is replaced by a chi-square statistic,
and the *p*-value measures the significance of
the chi-square statistic.

The Kruskal-Wallis test assumes that all samples come from populations having the same continuous distribution, apart from possibly different locations due to group effects, and that all observations are mutually independent. By contrast, classical one-way ANOVA replaces the first assumption with the stronger assumption that the populations have normal distributions.

## Version History

**Introduced before R2006a**

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