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# Chi-square goodness-of-fit test

> Perform a chi-square goodness-of-fit test.

<section class="usage">

## Usage

```javascript
var chi2gof = require( '@stdlib/stats/chi2gof' );
```

#### chi2gof( x, y\[, ...args]\[, opts] )

Computes a chi-square goodness-of-fit test for the **null hypothesis** that the values of `x` come from the discrete probability distribution specified by `y`.

```javascript
// Observed counts:
var x = [ 30, 20, 23, 27 ];

// Expected counts:
var y = [ 25, 25, 25, 25 ];

var res = chi2gof( x, y );
var o = res.toJSON();
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.5087,
        'df': 3,
        'statistic': ~2.32,
        ...
    }
*/
```

The second argument can either be an array-like object (or 1-dimensional [`ndarray`][@stdlib/ndarray/array]) of expected frequencies, an array-like object (or 1-dimensional [`ndarray`][@stdlib/ndarray/array]) of population probabilities summing to one, or a discrete probability distribution name to test against.

```javascript
// Observed counts:
var x = [ 89, 37, 30, 28, 2 ];

// Expected probabilities:
var y = [ 0.40, 0.20, 0.20, 0.15, 0.05 ];

var res = chi2gof( x, y );
var o = res.toJSON();
/* returns
    {
        'rejected': true,
        'alpha': 0.05,
        'pValue': ~0.0187,
        'df': 3,
        'statistic': ~9.9901,
        ...
    }
*/
```

When specifying a discrete probability distribution name, distribution parameters **must** be provided as additional arguments.

```javascript
var Int32Array = require( '@stdlib/array/int32' );
var discreteUniform = require( '@stdlib/random/base/discrete-uniform' );

var res;
var x;
var v;
var i;

// Simulate expected counts...
x = new Int32Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    v = discreteUniform( 0, 99 );
    x[ v ] += 1;
}

res = chi2gof( x, 'discrete-uniform', 0, 99 );
// returns {...}
```

The function accepts the following `options`:

-   **alpha**: significance level of the hypothesis test. Must be on the interval `[0,1]`. Default: `0.05`.
-   **ddof**: "delta degrees of freedom" adjustment. Must be a nonnegative integer. Default: `0`.
-   **simulate**: `boolean` indicating whether to calculate p-values by Monte Carlo simulation. Default: `false`.
-   **iterations**: number of Monte Carlo iterations. Default: `500`.

By default, the test is performed at a significance level of `0.05`. To adjust the significance level, set the `alpha` option.

```javascript
var x = [ 89, 37, 30, 28, 2 ];
var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ];

var res = chi2gof( x, p );

var table = res.toString();
/* e.g., returns

    Chi-square goodness-of-fit test

    Null hypothesis: population probabilities are equal to those in p

        pValue: 0.0186
        statistic: 9.9901
        degrees of freedom: 3

    Test Decision: Reject null in favor of alternative at 5% significance level

*/

res = chi2gof( x, p, {
    'alpha': 0.01
});

table = res.toString();
/* e.g., returns

    Chi-square goodness-of-fit test

    Null hypothesis: population probabilities are equal to those in p

        pValue: 0.0186
        statistic: 9.9901
        degrees of freedom: 3

    Test Decision: Fail to reject null in favor of alternative at 1% significance level

*/
```

By default, the p-value is computed using a chi-square distribution with `k-1` degrees of freedom, where `k` is the length of `x`. If provided distribution arguments are estimated (e.g., via maximum likelihood estimation), the degrees of freedom **should** be corrected. Set the `ddof` option to use `k-1-n` degrees of freedom, where `n` is the degrees of freedom adjustment.

```javascript
var x = [ 89, 37, 30, 28, 2 ];
var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ];

var res = chi2gof( x, p, {
    'ddof': 1
});

var o = res.toJSON();
// returns { 'pValue': ~0.0186, 'statistic': ~9.9901, 'df': 3, ... }
```

Instead of relying on chi-square approximation to calculate the p-value, one can use Monte Carlo simulation. When the `simulate` option is `true`, the simulation is performed by re-sampling from the discrete probability distribution specified by `y`.

```javascript
var x = [ 89, 37, 30, 28, 2 ];
var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ];

var res = chi2gof( x, p, {
    'simulate': true,
    'iterations': 1000 // explicitly set the number of Monte Carlo simulations
});
// returns {...}
```

The function returns a results `object` having the following properties:

-   **alpha**: significance level.
-   **rejected**: `boolean` indicating the test decision.
-   **pValue**: test p-value.
-   **statistic**: test statistic.
-   **df**: degrees of freedom.
-   **method**: test name.
-   **toString**: serializes results as formatted test output.
-   **toJSON**: serializes results as a JSON object.

To print formatted test output, invoke the `toString` method. The method accepts the following options:

-   **digits**: number of displayed decimal digits. Default: `4`.
-   **decision**: `boolean` indicating whether to show the test decision. Default: `true`.

```javascript
var x = [ 89, 37, 30, 28, 2 ];
var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ];

var res = chi2gof( x, p );

var table = res.toString({
    'decision': false
});
/* e.g., returns

    Chi-square goodness-of-fit test

    Null hypothesis: population probabilities are equal to those in p

        pValue: 0.0186
        statistic: 9.9901
        degrees of freedom: 3

*/
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   The chi-square approximation may be incorrect if the observed or expected frequencies in each category are too small. Common practice is to require frequencies **greater than** five.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var poisson = require( '@stdlib/random/base/poisson' );
var Int32Array = require( '@stdlib/array/int32' );
var chi2gof = require( '@stdlib/stats/chi2gof' );

var N = 400;
var lambda = 3.0;
var rpois = poisson.factory( lambda );

// Draw samples from a Poisson distribution:
var x = [];
var i;
for ( i = 0; i < N; i++ ) {
    x.push( rpois() );
}

// Generate a frequency table:
var freqs = new Int32Array( N );
for ( i = 0; i < N; i++ ) {
    freqs[ x[ i ] ] += 1;
}

// Assess whether the simulated values come from a Poisson distribution:
var out = chi2gof( freqs, 'poisson', lambda );
// returns {...}

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

</section>

<!-- /.examples -->

<section class="links">

[@stdlib/ndarray/array]: https://www.npmjs.com/package/@stdlib/ndarray-array

</section>

<!-- /.links -->