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# One Way ANOVA

> Perform a one-way analysis of variance.

<section class="usage">

## Usage

```javascript
var anova1 = require( '@stdlib/stats/anova1' );
```

#### anova1( x, factor\[, opts] )

For an [array][mdn-array] or [typed array][mdn-typed-array] of numeric values `x` and an [array][mdn-array] of classifications `factor`, a one-way analysis of variance is performed. The hypotheses are given as follows:

<!-- <equation class="equation" label="eq:hypotheses" align="center" raw="\begin{align*} H_{0}:& \; \mu_{1} = \mu_{2} = \dots = \mu_{k} \\ H_{a}:& \; \text{at least one} \; \mu_{i} \; \text{not equal to the others} \end{align*}" alt="Hypotheses of ANOVA"> -->

<div class="equation" align="center" data-raw-text="\begin{align*} H_{0}:&amp; \; \mu_{1} = \mu_{2} = \dots = \mu_{k} \\ H_{a}:&amp; \; \text{at least one} \; \mu_{i} \; \text{not equal to the others} \end{align*}" data-equation="eq:hypotheses">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/anova1/docs/img/equation_hypotheses.svg" alt="Hypotheses of ANOVA">
    <br>
</div>

<!-- </equation> -->

The function returns an object containing the treatment and error squared errors, degrees of freedom, mean squared errors, and both the p-value and F score.

```javascript
var out;
var x;
var y;

x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ];
y = [ 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control' ];

out = anova1( x, y );
/* returns
    {
        'treatment': { 'df': 11, 'ss': 15, 'ms': 5 },
        'error': { 'df': 8, 'ss': 128, 'ms': 16 },
        'statistic': 0.3125,
        'pValue': 0.81607947904798,
        'means':
          { 'Treatment A': { 'mean': 5, 'sampleSize': 3, 'SD': 4 },
            'Treatment B': { 'mean': 6, 'sampleSize': 3, 'SD': 4 },
            'Treatment C': { 'mean': 7, 'sampleSize': 3, 'SD': 4 },
            'Control': { 'mean': 8, 'sampleSize': 3, 'SD': 4 } },
        'method': 'One-Way ANOVA'
    }
*/
```

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 out;
var x;
var y;

x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ];
y = [ 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control' ];

out = anova1( x, y );
console.log( out.print() );
/* =>
    One-Way ANOVA

    Null Hypothesis: All Means Equal
    Alternate Hypothesis: At Least one Mean not Equal

                  df   SS     MS    F Score   P Value
    Treatment     3    15     5     0.3125    0.8161
    Errors        8    128    16

    Fail to Reject Null: 0.8161 >= 0.05
*/
```

The function accepts the following `options`:

-   **alpha**: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
-   **decision**: a `boolean` value indicating if function is to return a decision of either _rejection of the null hypothesis_ or _failure to reject the null hypothesis_. Default: `false`

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 = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ];
var y = [ 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control', 'Treatment A', 'Treatment B', 'Treatment C', 'Control' ];

var out = anova1( x, y );
var table = out.print();
/* e.g., returns
    One-Way ANOVA

    Null Hypothesis: All Means Equal
    Alternate Hypothesis: At Least one Mean not Equal

                  df   SS     MS    F Score   P Value
    Treatment     3    15     5     0.3125    0.8161
    Errors        8    128    16

    Fail to Reject Null: 0.8161 >= 0.05
*/

out = anova1( x, y, {
    'alpha': 0.9
});
table = out.print();
/* e.g., returns
    One-Way ANOVA

    Null Hypothesis: All Means Equal
    Alternate Hypothesis: At Least one Mean not Equal

                  df   SS     MS    F Score   P Value
    Treatment     3    15     5     0.3125    0.8161
    Errors        8    128    16

    Reject Null: 0.8161 <= 0.9
*/
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   The calculation for the p value is based on an F distribution as indicated on [this website][penn-state]. 

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var anova1 = require( '@stdlib/stats/anova1' );

var x = [ 3, 4, 5, 6, 2, 5, 10, 12, 8, 10 ];
var f = [ 'control', 'treatA', 'treatB', 'control', 'treatA', 'treatB', 'control', 'treatA', 'treatB', 'control' ];

var out = anova1( x, f, {
    'decision': true
});

console.log( out.print() );

out = anova1( x, f, {
    'alpha': 0.9
});

console.log( out.print() );
```

</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

[penn-state]: https://onlinecourses.science.psu.edu/stat502/node/141

</section>

<!-- /.links -->
