# 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.

## 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
*/