Two-sample F-test

Two-sample F-test for equal variances.

Usage

var vartest = require( '@stdlib/stats/vartest' );

vartest( x, y[, opts] )

By default, the function performs a two-sample F-test for the null hypothesis that the data in arrays or typed arrays x and y is independently drawn from normal distributions with equal variances.

var x = [ 610, 610, 550, 590, 565, 570 ];
var y = [ 560, 550, 580, 550, 560, 590, 550, 590 ];

var out = vartest( x, y );
/* returns
    {
        'rejected': false,
        'pValue': ~0.399,
        'statistic': ~1.976,
        'ci': [ ~0.374, ~13.542 ],
        // ...
    }
*/

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.

console.log( out.print() );
/* e.g., =>
    F test for comparing two variances

    Alternative hypothesis: True ratio in variances is not equal to 1

        pValue: 0.3992
        statistic: 1.976
        variance of x: 617.5 (df of x: 5)
        variance of y: 312.5 (df of y: 7)
        95% confidence interval: [0.3739,13.5417]

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

The function accepts the following options:

  • alpha: number in the interval [0,1] giving the significance level of the hypothesis test. Default: 0.05.
  • alternative: Either two-sided, less or greater. Indicates whether the alternative hypothesis is that the true ratio of variances is greater than one (greater), smaller than one (less), or that the variances are the same (two-sided). Default: two-sided.
  • ratio: positive number denoting the ratio of the two population variances under the null hypothesis. Default: 1.

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

var x = [ 610, 610, 550, 590, 565, 570, 500, 650, 500, 650 ];
var y = [ 560, 550, 580, 550, 560, 590, 550, 590 ];

var out = vartest( x, y, {
    'alpha': 0.01
});
var table = out.print();
/* e.g., returns
    F test for comparing two variances

    Alternative hypothesis: True ratio in variances is not equal to 1

        pValue: 0.0081
        statistic: 9.1458
        variance of x: 2858.0556 (df of x: 9)
        variance of y: 312.5 (df of y: 7)
        90% confidence interval: [2.4875,30.1147]

    Test Decision: Reject null in favor of alternative at 1% significance level

    Exited with status 0
*/

By default, a two-sided test is performed. To perform either of the one-sided tests, set the alternative option to less or greater.

var x = [ 610, 610, 550, 590, 565, 570, 500, 650, 500, 650 ];
var y = [ 560, 550, 580, 550, 560, 590, 550, 590 ];

var out = vartest( x, y, {
    'alternative': 'less'
});
var table = out.print();
/* e.g., returns
    Alternative hypothesis: True ratio in variances is less than 1

        pValue: 0.996
        statistic: 9.1458
        variance of x: 2858.0556 (df of x: 9)
        variance of y: 312.5 (df of y: 7)
        95% confidence interval: [0,30.1147]

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

    Exited with status 0
*/

out = vartest( x, y, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    Alternative hypothesis: True ratio in variances is greater than 1

        pValue: 0.004
        statistic: 9.1458
        variance of x: 2858.0556 (df of x: 9)
        variance of y: 312.5 (df of y: 7)
        95% confidence interval: [2.4875,Infinity]

    Test Decision: Reject null in favor of alternative at 5% significance level

    Exited with status 0
*/

To test whether the ratio in the population variances is equal to some other value than 1, set the ratio option.

var x = [ 610, 610, 550, 590, 565, 570, 500, 650, 500, 650 ];
var y = [ 560, 550, 580, 550, 560, 590, 550, 590 ];

var out = vartest( x, y, {
    'ratio': 10.0
});
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.879,
        'statistic': ~-0.915,
        'ci': [ ~1.896, ~38.385 ],
        // ...
    }
*/

var table = out.print();
/* e.g., returns
    F test for comparing two variances

    Alternative hypothesis: True ratio in variances is not equal to 10

        pValue: 0.8794
        statistic: 0.9146
        variance of x: 2858.0556 (df of x: 9)
        variance of y: 312.5 (df of y: 7)
        95% confidence interval: [1.8962,38.3853]

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

Examples

var rnorm = require( '@stdlib/random/base/normal' );
var vartest = require( '@stdlib/stats/vartest' );

var table;
var out;
var x;
var y;
var i;

x = new Array( 60 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = rnorm( 2.0, 1.0 );
}
y = new Array( 40 );
for ( i = 0; i < y.length; i++ ) {
    y[ i ] = rnorm( 1.0, 2.0 );
}

// Test whether the variances of `x` and `y` are the same:
out = vartest( x, y );
table = out.print();
/* e.g., returns
    F test for comparing two variances

    Alternative hypothesis: True ratio in variances is not equal to 1

        pValue: 0
        statistic: 0.1717
        variance of x: 0.6406 (df of x: 60)
        variance of y: 3.7306 (df of y: 40)
        95% confidence interval: [0.0953,0.2995]

    Test Decision: Reject null in favor of alternative at 5% significance level
*/

// Test whether the variance of `x` is one fourth of the variance of `y`:
out = vartest( x, y, {
    'ratio': 0.25
});
table = out.print();
/* e.g., returns
    F test for comparing two variances

    Alternative hypothesis: True ratio in variances is not equal to 0.25

        pValue: 0.1847
        statistic: 0.6869
        variance of x: 0.6406 (df of x: 60)
        variance of y: 3.7306 (df of y: 40)
        95% confidence interval: [0.0953,0.2995]

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