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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

$$\chi^2 = \frac{\displaystyle N\ln(S^2) - \sum_{i=0}^{k-1} n_i \ln(S_i^2)}{\displaystyle 1 + \frac{1}{3(k-1)}\left(\sum_{i=0}^{k-1} \frac{1}{n_i} - \frac{1}{N}\right)}$$

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

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

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
*/

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See Also

• @stdlib/stats-vartest: two-sample F-test for equal variances.
• @stdlib/stats-levene-test: Levene's test for equal variances.

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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.

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License

See LICENSE.

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