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https://github.com/stdlib-js/stats-ttest2

Two-sample Student's t-Test.
https://github.com/stdlib-js/stats-ttest2

hypothesis javascript math mathematics mean node node-js nodejs normal statistics stats stdlib summary test two-sample welch

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Two-sample Student's t-Test.

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README 

        



  

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# Student's t-Test



[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] 



> Two-sample Student's t-Test.



## Installation



```bash

npm install @stdlib/stats-ttest2

```



Alternatively,



-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).



The [branches.md][branches-url] 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



```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

*/

```



* * *



## See Also



-   [`@stdlib/stats-ttest`][@stdlib/stats/ttest]: one-sample and paired Student's t-Test.



* * *



## Notice



This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib].



#### Community



[![Chat][chat-image]][chat-url]



---



## License



See [LICENSE][stdlib-license].



## Copyright



Copyright © 2016-2024. The Stdlib [Authors][stdlib-authors].



[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-ttest2.svg

[npm-url]: https://npmjs.org/package/@stdlib/stats-ttest2



[test-image]: https://github.com/stdlib-js/stats-ttest2/actions/workflows/test.yml/badge.svg?branch=main

[test-url]: https://github.com/stdlib-js/stats-ttest2/actions/workflows/test.yml?query=branch:main



[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-ttest2/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/stats-ttest2?branch=main



[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im



[stdlib]: https://github.com/stdlib-js/stdlib



[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors



[umd]: https://github.com/umdjs/umd

[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules



[deno-url]: https://github.com/stdlib-js/stats-ttest2/tree/deno

[deno-readme]: https://github.com/stdlib-js/stats-ttest2/blob/deno/README.md

[umd-url]: https://github.com/stdlib-js/stats-ttest2/tree/umd

[umd-readme]: https://github.com/stdlib-js/stats-ttest2/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/stats-ttest2/tree/esm

[esm-readme]: https://github.com/stdlib-js/stats-ttest2/blob/esm/README.md

[branches-url]: https://github.com/stdlib-js/stats-ttest2/blob/main/branches.md



[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-ttest2/main/LICENSE



[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-ttest