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# kruskalTest

> Compute the Kruskal-Wallis test for equal medians.

<section class="intro">

The Kruskal-Wallis rank sum test evaluates for multiple samples the null hypothesis that their medians are identical. The Kruskal-Wallis test is a nonparametric test which does not require the data to be normally distributed.

To carry out the test, the rank sums `S_h` of the individual groups are calculated. The test statistic is then calculated as

<!-- <equation class="equation" label="eq:kruskal_test_statistic" align="center" raw="H = \frac{\tfrac{12}{N(N+1)}\sum_h\tfrac{S_h^2}{n_h}-3(N+1)}{1-\tfrac{1}{(N^3-N)} \sum t_{r(i)}^3 - t_{r(i)}}" alt="Equation for the Kruskal-Wallis test statistic."> -->

<div class="equation" align="center" data-raw-text="H = \frac{\tfrac{12}{N(N+1)}\sum_h\tfrac{S_h^2}{n_h}-3(N+1)}{1-\tfrac{1}{(N^3-N)} \sum t_{r(i)}^3 - t_{r(i)}}" data-equation="eq:kruskal_test_statistic">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/1d8cc6e20147732234a178b3b719239a5778842e/lib/node_modules/@stdlib/stats/kruskal-test/docs/img/equation_kruskal_test_statistic.svg" alt="Equation for the Kruskal-Wallis test statistic.">
    <br>
</div>

<!-- </equation> -->

where `N` denotes the total number of observations and `t_{r(i)}` are the number of tied observations with rank _i_.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var kruskalTest = require( '@stdlib/stats/kruskal-test' );
```

#### kruskalTest( a\[,b,...,k]\[, opts] )

For input arrays `a`, `b`, ... holding numeric observations, this function calculates the Kruskal-Wallis rank sums test, which tests the null hypothesis that the medians in all `k` groups are the same. 

```javascript
// 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 = kruskalTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/
```

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.

```javascript
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 = kruskalTest( x, y, z, {
    'alpha': 0.01
});
/* returns
    {
        'rejected': false,
        'alpha': 0.01,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/
```

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.

<!-- eslint-disable array-element-newline -->

```javascript
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 = kruskalTest( 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.

```javascript
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 = kruskalTest( x, y, z );
console.log( out.print() );
/* =>
    Kruskal-Wallis Test

    Null hypothesis: the medians of all groups are the same

        pValue: 0.68
        statistic: 0.7714    df: 2

    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 kruskalTest = require( '@stdlib/stats/kruskal-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 = kruskalTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.68,
        'statistic': ~0.771,
        ...
    }
*/

var table = out.print();
/* returns
    Kruskal-Wallis Test

    Null hypothesis: the medians of all groups are the same

        pValue: 0.68
        statistic: 0.7714    df: 2

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

</section>

<!-- /.examples -->

<section class="references">

</section>

<!-- /.references -->

<section class="links">

</section>

<!-- /.links -->
