# Student's t-Test
> Two-sample Student's t-Test.
## Usage
```javascript
var ttest2 = require( '@stdlib/stats/ttest2' );
```
#### ttest2( x, y\[, opts] )
By default, the function performs a two-sample t-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_ means.
```javascript
// Student's sleep data:
var x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];
var out = ttest2( x, y );
/* e.g., returns
{
'rejected': false,
'pValue': ~0.079,
'statistic': ~-1.861,
'ci': [ ~-3.365, ~0.205 ],
// ...
}
*/
```
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.
```javascript
console.log( out.print() );
/* e.g., =>
Welch two-sample t-test
Alternative hypothesis: True difference in means is not equal to 0
pValue: 0.0794
statistic: -1.8608
95% confidence interval: [-3.3655,0.2055]
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 `x` has a larger mean than `y` (`greater`), `x` has a smaller mean than `y` (`less`) or the means are the same (`two-sided`). Default: `two-sided`.
- **difference**: `number` denoting the difference in means under the null hypothesis. Default: `0`.
- **variance**: `string` indicating if the test should be conducted under the assumption that the unknown variances of the normal distributions are `equal` or `unequal`. Default: `unequal`.
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 = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];
var out = ttest2( x, y, {
'alpha': 0.1
});
var table = out.print();
/* e.g., returns
Welch two-sample t-test
Alternative hypothesis: True difference in means is not equal to 0
pValue: 0.0794
statistic: -1.8608
90% confidence interval: [-3.0534,-0.1066]
Test Decision: Reject null in favor of alternative at 10% significance level
*/
```
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
// Student's sleep data:
var x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];
var out = ttest2( x, y, {
'alternative': 'less'
});
var table = out.print();
/* e.g., returns
Welch two-sample t-test
Alternative hypothesis: True difference in means is less than 0
pValue: 0.0397
statistic: -1.8608
df: 17.7765
95% confidence interval: [-Infinity,-0.1066]
Test Decision: Reject null in favor of alternative at 5% significance level
*/
out = ttest2( x, y, {
'alternative': 'greater'
});
table = out.print();
/* e.g., returns
Welch two-sample t-test
Alternative hypothesis: True difference in means is greater than 0
pValue: 0.9603
statistic: -1.8608
df: 17.7765
95% confidence interval: [-3.0534,Infinity]
Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
```
As a default choice, the `ttest2` function carries out the Welch test (using the Satterthwaite approximation for the degrees of freedom), which does not have the requirement that the variances of the underlying distributions are equal. If the equal variances assumption seems warranted, set the `variance` option to `equal`.
```javascript
var x = [ 2, 3, 1, 4 ];
var y = [ 1, 2, 3, 1, 2, 5, 3, 4 ];
var out = ttest2( x, y, {
'variance': 'equal'
});
var table = out.print();
/* e.g., returns
Two-sample t-test
Alternative hypothesis: True difference in means is not equal to 0
pValue: 0.8848
statistic: -0.1486
df: 10
95% confidence interval: [-1.9996,1.7496]
Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
```
To test whether the difference in the population means is equal to some other value than `0`, set the `difference` option.
```javascript
var normal = require( '@stdlib/random/base/normal' ).factory;
var rnorm = normal({
'seed': 372
});
var x = [];
var i;
for ( i = 0; i < x.length; i++ ) {
x.push( rnorm( 2.0, 3.0 ) );
}
var y = [];
for ( i = 0; i < x.length; i++ ) {
y.push( rnorm( 1.0, 3.0 ) );
}
var out = ttest2( x, y, {
'difference': 1.0,
'variance': 'equal'
});
/* e.g., returns
{
'rejected': false,
'pValue': ~0.642,
'statistic': ~-0.466,
'ci': [ ~-0.0455, ~1.646 ],
// ...
}
*/
var table = out.print();
/* e.g., returns
Two-sample t-test
Alternative hypothesis: True difference in means is not equal to 1
pValue: 0.6419
statistic: -0.4657
df: 198
95% confidence interval: [-0.0455,1.646]
Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
```
## Examples
```javascript
var incrspace = require( '@stdlib/array/base/incrspace' );
var ttest2 = require( '@stdlib/stats/ttest2' );
var a = incrspace( 1, 11, 1 );
var b = incrspace( 7, 21, 1 );
var out = ttest2( a, b );
var table = out.print();
/* e.g., returns
Welch two-sample t-test
Alternative hypothesis: True difference in means is not equal to 0
pValue: 0
statistic: -5.4349
95% confidence interval: [-11.0528,-4.9472]
Test Decision: Reject null in favor of alternative at 5% significance level
*/
```
[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
[@stdlib/stats/ttest]: https://github.com/stdlib-js/stats/tree/main/ttest