# @stdlib/stats-bartlett-test

0.2.1 • Public • Published

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# bartlettTest

Compute Bartlett’s test for equal variances.

Bartlett's test is used to test the null hypothesis that the variances of k groups are equal against the alternative that at least two of them are different.

For `k` groups each with `n_i` observations, the test statistic is

where `N` is the total number of observations, `S_i` are the biased group-level variances and `S^2` is a (biased) pooled estimate for the variance. Under the null hypothesis, the test statistic follows a chi-square distribution with `df = k - 1` degrees of freedom.

## Installation

`npm install @stdlib/stats-bartlett-test`

## Usage

`var bartlettTest = require( '@stdlib/stats-bartlett-test' );`

#### bartlettTest( a[,b,...,k][, opts] )

For input arrays `a`, `b`, ... holding numeric observations, this function calculates Bartlett’s test, which tests the null hypothesis that the variances in all `k` groups are the same.

```// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
/* returns
{
'rejected': false,
'alpha': 0.05,
'df': 2,
'pValue': ~0.573,
'statistic': ~1.112,
...
}
*/```

The function accepts the following `options`:

• alpha: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
• groups: an `array` of group indicators. If set, the function assumes that only a single numeric array is provided holding all observations.

By default, the test is carried out at a significance level of `0.05`. To choose a custom significance level, set the `alpha` option.

```var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z, {
'alpha': 0.01
});
/* returns
{
'rejected': false,
'alpha': 0.01,
'df': 2,
'pValue': ~0.573,
'statistic': ~1.112,
...
}
*/```

The function provides an alternate interface by supplying an array of group indicators to the `groups` option. In this case, it is assumed that only a single numeric array holding all observations is provided to the function.

```var arr = [
2.9, 3.0, 2.5, 2.6, 3.2,
3.8, 2.7, 4.0, 2.4,
2.8, 3.4, 3.7, 2.2, 2.0
];
var groups = [
'a', 'a', 'a', 'a', 'a',
'b', 'b', 'b', 'b',
'c', 'c', 'c', 'c', 'c'
];
var out = bartlettTest( arr, {
'groups': groups
});```

The returned object comes with a `.print()` method which when invoked will print a formatted output of the results of the hypothesis test. `print` accepts a `digits` option that controls the number of decimal digits displayed for the outputs and a `decision` option, which when set to `false` will hide the test decision.

```var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
console.log( out.print() );
/* =>
Bartlett's test of equal variances

Null hypothesis: The variances in all groups are the same.

pValue: 0.5735
statistic: 1.1122
df: 2

Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/```

## Examples

```var bartlettTest = require( '@stdlib/stats-bartlett-test' );

// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
/* returns
{
'rejected': false,
'alpha': 0.05,
'df': 2,
'pValue': ~0.573,
'statistic': ~1.112,
...
}
*/

var table = out.print();
/* returns
Bartlett's test of equal variances

Null hypothesis: The variances in all groups are the same.

pValue: 0.5735
statistic: 1.1122
df: 2

Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/```

## Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

## Package Sidebar

### Install

`npm i @stdlib/stats-bartlett-test`

### Repository

github.com/stdlib-js/stats-bartlett-test

stdlib.io

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