ansaribradley

Ansari-Bradley test

Syntax

h = ansaribradley(x,y)

h = ansaribradley(x,y,Name,Value)

[h,p] = ansaribradley(___)

[h,p,stats] = ansaribradley(___)

Description

h = ansaribradley(x,y) returns a test decision for the null hypothesis that the data in vectors x and y comes from the same distribution, using the Ansari-Bradley test. The alternative hypothesis is that the data in x and y comes from distributions with the same median and shape but different dispersions (e.g., variances). The result h is 1 if the test rejects the null hypothesis at the 5% significance level, or 0 otherwise.

h = ansaribradley(x,y,Name,Value) returns a test decision for the Ansari-Bradley test with additional options specified by one or more name-value pair arguments. For example, you can change the significance level, conduct a one-sided test, or use a normal approximation to calculate the value of the test statistic.

[h,p] = ansaribradley(___) also returns the p-value, p, of the test, using any of the input arguments in the previous syntaxes.

[h,p,stats] = ansaribradley(___) also returns the structure stats containing information about the test statistic.

Examples

Ansari-Bradley Test for Equal Variances

Load the sample data. Create data vectors of miles per gallon (MPG) measurements for the model years 1982 and 1976.

load carsmall
x = MPG(Model_Year==82);
y = MPG(Model_Year==76);

Test the null hypothesis that the miles per gallon measured in cars from 1982 and 1976 have equal variances.

[h,p,stats] = ansaribradley(x,y)

h = 
0

p = 
0.8426

stats = struct with fields:
        W: 526.9000
    Wstar: 0.1986

The returned value of h = 0 indicates that ansaribradley does not reject the null hypothesis at the default 5% significance level.

Ansari-Bradley One-Sided Hypothesis Test

Load the sample data. Create data vectors of miles per gallon (MPG) measurements for the model years 1982 and 1976.

load carsmall
x = MPG(Model_Year==82);
y = MPG(Model_Year==76);

Test the null hypothesis that the miles per gallon measured in cars from 1982 and 1976 have equal variances, against the alternative hypothesis that the variance of cars from 1982 is greater than that of cars from 1976.

[h,p,stats] = ansaribradley(x,y,'Tail','right')

h = 
0

p = 
0.5787

stats = struct with fields:
        W: 526.9000
    Wstar: 0.1986

The returned value of h = 0 indicates that ansaribradley does not reject the null hypothesis that the variance in miles per gallon is the same for the two model years, when the alternative is that the variance of cars from 1982 is greater than that of cars from 1976.

Input Arguments

x — Sample data

vector | matrix | multidimensional array

Sample data, specified as a vector, matrix, or multidimensional array.

• If x and y are specified as vectors, they do not need to be the same length.

• If x and y are specified as matrices, they must have the same number of columns. ansaribradley performs separate tests along each column and returns a vector of results.

• If x and y are specified as multidimensional arrays, ansaribradley works along the first nonsingleton dimension. x and y must have the same size along all remaining dimensions.

Data Types: single | double

y — Sample data

vector | matrix | multidimensional array

Sample data, specified as a vector, matrix, or multidimensional array.

• If x and y are specified as vectors, they do not need to be the same length.

• If x and y are specified as matrices, they must have the same number of columns. ansaribradley performs separate tests along each column and returns a vector of results.

• If x and y are specified as multidimensional arrays, ansaribradley works along the first nonsingleton dimension. x and y must have the same size along all remaining dimensions.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'Tail','right','Alpha',0.01 specifies a right-tailed hypothesis test at the 1% significance level.

Alpha — Significance level

0.05 (default) | scalar value in the range (0,1)

Significance level of the hypothesis test, specified as the comma-separated pair consisting of 'Alpha' and a scalar value in the range (0,1).

Example: 'Alpha',0.01

Data Types: single | double

Dim — Dimension

first nonsingleton dimension (default) | positive integer value

Dimension of the input matrix along which to test the means, specified as the comma-separated pair consisting of 'Dim' and a positive integer value. For example, specifying 'Dim',1 tests the column means, while 'Dim',2 tests the row means.

Example: 'Dim',2

Data Types: single | double

Tail — Type of alternative hypothesis

'both' (default) | 'left' | 'right'

Type of alternative hypothesis to evaluate, specified as the comma-separated pair consisting of 'Tail' and one of the following.

'both'	Test the alternative hypothesis that the dispersion parameters of x and y are not equal.
'right'	Test the alternative hypothesis that the dispersion parameter of x is greater than that of y.
'left'	Test the alternative hypothesis that the dispersion parameter of x is less than that of y.

Example: 'Tail','right'

Method — Computation method

'exact' | 'approximate'

Computation method for the test statistic, specified as the comma-separated pair consisting of 'Method' and one of the following.

'exact'	Compute p using an exact calculation of the distribution of the test statistic W. This is the default if n, the total number of rows in x and y, is 25 or less. Note that n is computed before any NaN values (representing missing data) are removed.
'approximate'	Compute p using a normal approximation for the statistic W*. This is the default if n, the total number of rows in x and y, is greater than 25.

Example: 'Method','exact'

Output Arguments

h — Hypothesis test result

1 | 0

Hypothesis test result, returned as 1 or 0.

• A value of 1 indicates the rejection of the null hypothesis at the Alpha significance level.

• A value of 0 indicates a failure to reject the null hypothesis at the Alpha significance level.

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.

stats — Test statistics

structure

Test statistics for the Ansari-Bradley test, returned as a structure containing:

• W — Value of the test statistic, which is the sum of the Ansari-Bradley ranks for the x sample.

• Wstar — Approximate normal statistic W*.

More About

Ansari-Bradley Test

The Ansari-Bradley test is a nonparametric alternative to the two-sample F-test of equal variances. It does not require the assumption that x and y come from normal distributions. The dispersion of a distribution is generally measured by its variance or standard deviation, but the Ansari-Bradley test can be used with samples from distributions that do not have finite variances.

This test requires that the samples have equal medians. Under that assumption, and if the distributions of the samples are continuous and identical, the test is independent of the distributions. If the samples do not have the same medians, the results can be misleading. In that case, Ansari and Bradley recommend subtracting the median, but then the distribution of the resulting test under the null hypothesis is no longer independent of the common distribution of x and y. If you want to perform the tests with medians subtracted, you should subtract the medians from x and y before calling ansaribradley.

Multidimensional Array

A multidimensional array has more than two dimensions. For example, if x is a 1-by-3-by-4 array, then x is a three-dimensional array.

First Nonsingleton Dimension

The first nonsingleton dimension is the first dimension of an array whose size is not equal to 1. For example, if x is a 1-by-2-by-3-by-4 array, then the second dimension is the first nonsingleton dimension of x.

Version History

Introduced before R2006a