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# Two-sample F-test
> Two-sample F-test for equal variances.
<section class="usage">
## Usage
```javascript
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][mdn-array] or [typed arrays][mdn-typed-array] `x` and `y` is independently drawn from normal distributions with _equal_ variances.
```javascript
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.
<!-- run-disable -->
```javascript
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.
```javascript
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`.
```javascript
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.
```javascript
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
*/
```
</section>
<!-- /.usage -->
<section class="examples">
## Examples
<!-- eslint no-undef: "error" -->
```javascript
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
*/
```
</section>
<!-- /.examples -->
<section class="links">
[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays
</section>
<!-- /.links -->