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# incrgrubbs

> [Grubbs' test][grubbs-test] for outliers.

<section class="intro">

[Grubbs' test][grubbs-test] (also known as the **maximum normalized residual test** or **extreme studentized deviate test**) is a statistical test used to detect outliers in a univariate dataset assumed to come from a normally distributed population. [Grubbs' test][grubbs-test] is defined for the hypothesis:

-   **H_0**: the dataset does **not** contain outliers.
-   **H_1**: the dataset contains **exactly** one outlier.

The [Grubbs' test][grubbs-test] statistic for a two-sided alternative hypothesis is defined as

<!-- <equation class="equation" label="eq:grubbs_test_statistic" align="center" raw="G = \frac{\max_{i=0,\ldots,N-1} |Y_i - \bar{Y}|}{s}" alt="Grubbs' test statistic."> -->

<div class="equation" align="center" data-raw-text="G = \frac{\max_{i=0,\ldots,N-1} |Y_i - \bar{Y}|}{s}" data-equation="eq:grubbs_test_statistic">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/d6af7da0801d2116a9507668d13ef7bf607fd275/lib/node_modules/@stdlib/stats/incr/grubbs/docs/img/equation_grubbs_test_statistic.svg" alt="Grubbs' test statistic.">
    <br>
</div>

<!-- </equation> -->

where `s` is the sample standard deviation. The [Grubbs test][grubbs-test] statistic is thus the largest absolute deviation from the sample mean in units of the sample standard deviation.

The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that the minimum value is an outlier is defined as

<!-- <equation class="equation" label="eq:grubbs_test_statistic_min" align="center" raw="G = \frac{\bar{Y} - Y_{\textrm{min}}}{s}" alt="Grubbs' test statistic for testing whether the minimum value is an outlier."> -->

<div class="equation" align="center" data-raw-text="G = \frac{\bar{Y} - Y_{\textrm{min}}}{s}" data-equation="eq:grubbs_test_statistic_min">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/d6af7da0801d2116a9507668d13ef7bf607fd275/lib/node_modules/@stdlib/stats/incr/grubbs/docs/img/equation_grubbs_test_statistic_min.svg" alt="Grubbs' test statistic for testing whether the minimum value is an outlier.">
    <br>
</div>

<!-- </equation> -->

The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that the maximum value is an outlier is defined as

<!-- <equation class="equation" label="eq:grubbs_test_statistic_max" align="center" raw="G = \frac{Y_{\textrm{max}} - \bar{Y}}{s}" alt="Grubbs' test statistic for testing whether the maximum value is an outlier."> -->

<div class="equation" align="center" data-raw-text="G = \frac{Y_{\textrm{max}} - \bar{Y}}{s}" data-equation="eq:grubbs_test_statistic_max">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/d6af7da0801d2116a9507668d13ef7bf607fd275/lib/node_modules/@stdlib/stats/incr/grubbs/docs/img/equation_grubbs_test_statistic_max.svg" alt="Grubbs' test statistic for testing whether the maximum value is an outlier.">
    <br>
</div>

<!-- </equation> -->

For a two-sided test, the hypothesis that a dataset does **not** contain an outlier is rejected at significance level α if

<!-- <equation class="equation" label="eq:grubbs_test_two_sided" align="center" raw="G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/(2N),N-2}}{N - 2 + t^2_{\alpha/(2N),N-2}}}" alt="Two-sided Grubbs' test."> -->

<div class="equation" align="center" data-raw-text="G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/(2N),N-2}}{N - 2 + t^2_{\alpha/(2N),N-2}}}" data-equation="eq:grubbs_test_two_sided">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/d6af7da0801d2116a9507668d13ef7bf607fd275/lib/node_modules/@stdlib/stats/incr/grubbs/docs/img/equation_grubbs_test_two_sided.svg" alt="Two-sided Grubbs' test.">
    <br>
</div>

<!-- </equation> -->

where `t` denotes the upper critical value of the _t_-distribution with `N-2` degrees of freedom and a significance level of `α/(2N)`.

For a one-sided test, the hypothesis that a dataset does **not** contain an outlier is rejected at significance level α if

<!-- <equation class="equation" label="eq:grubbs_test_one_sided" align="center" raw="G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/N,N-2}}{N - 2 + t^2_{\alpha/N,N-2}}}" alt="One-sided Grubbs' test."> -->

<div class="equation" align="center" data-raw-text="G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/N,N-2}}{N - 2 + t^2_{\alpha/N,N-2}}}" data-equation="eq:grubbs_test_one_sided">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/d6af7da0801d2116a9507668d13ef7bf607fd275/lib/node_modules/@stdlib/stats/incr/grubbs/docs/img/equation_grubbs_test_one_sided.svg" alt="One-sided Grubbs' test.">
    <br>
</div>

<!-- </equation> -->

where `t` denotes the upper critical value of the _t_-distribution with `N-2` degrees of freedom and a significance level of `α/N`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var incrgrubbs = require( '@stdlib/stats/incr/grubbs' );
```

#### incrgrubbs( \[options] )

Returns an accumulator `function` which incrementally performs [Grubbs' test][grubbs-test] for outliers.

```javascript
var accumulator = incrgrubbs();
```

The function accepts the following `options`:

-   **alpha**: significance level. Default: `0.05`.

-   **alternative**: alternative hypothesis. The option may be one of the following values:

    -   `'two-sided'`: test whether the minimum or maximum value is an outlier.
    -   `'min'`: test whether the minimum value is an outlier.
    -   `'max'`: test whether the maximum value is an outlier.

    Default: `'two-sided'`.

-   **init**: number of data points the accumulator should use to compute initial statistics **before** testing for an outlier. Until the accumulator is provided the number of data points specified by this option, the accumulator returns `null`. Default: `100`.

#### accumulator( \[x] )

If provided an input value `x`, the accumulator function returns updated test results. If not provided an input value `x`, the accumulator function returns the current test results.

```javascript
var rnorm = require( '@stdlib/random/base/normal' );

var opts = {
    'init': 0
};
var accumulator = incrgrubbs( opts );

var results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null

results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null

results = accumulator( rnorm( 10.0, 5.0 ) );
// returns <Object>

results = accumulator();
// returns <Object>
```

The accumulator function returns an `object` having the following fields:

-   **rejected**: boolean indicating whether the null hypothesis should be rejected.
-   **alpha**: significance level.
-   **criticalValue**: critical value.
-   **statistic**: test statistic.
-   **df**: degrees of freedom.
-   **mean**: sample mean.
-   **sd**: corrected sample standard deviation.
-   **min**: minimum value.
-   **max**: maximum value.
-   **alt**: alternative hypothesis.
-   **method**: method name.
-   **print**: method for pretty-printing test output.

The `print` method accepts the following options:

-   **digits**: number of digits after the decimal point. Default: `4`.
-   **decision**: `boolean` indicating whether to print the test decision. Default: `true`.

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   [Grubbs' test][grubbs-test] **assumes** that data is normally distributed. Accordingly, one should first **verify** that the data can be _reasonably_ approximated by a normal distribution before applying the [Grubbs' test][grubbs-test].
-   The accumulator must be provided **at least** three data points before performing [Grubbs' test][grubbs-test]. Until at least three data points are provided, the accumulator returns `null`.
-   Input values are **not** type checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the test statistic is `NaN` for **all** future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var incrgrubbs = require( '@stdlib/stats/incr/grubbs' );

var data;
var opts;
var acc;
var i;

// Define a data set (8 mass spectrometer measurements of a uranium isotope; see Tietjen and Moore. 1972. "Some Grubbs-Type Statistics for the Detection of Several Outliers".)
data = [ 199.31, 199.53, 200.19, 200.82, 201.92, 201.95, 202.18, 245.57 ];

// Create a new accumulator:
opts = {
    'init': data.length,
    'alternative': 'two-sided'
};
acc = incrgrubbs( opts );

// Update the accumulator:
for ( i = 0; i < data.length; i++ ) {
    acc( data[ i ] );
}

// Print the test results:
console.log( acc().print() );
/* e.g., =>
Grubbs' Test

Alternative hypothesis: The maximum value (245.57) is an outlier

    criticalValue: 2.1266
    statistic: 2.4688
    df: 6

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

*/
```

</section>

<!-- /.examples -->

<section class="references">

* * *

## References

-   Grubbs, Frank E. 1950. "Sample Criteria for Testing Outlying Observations." _The Annals of Mathematical Statistics_ 21 (1). The Institute of Mathematical Statistics: 27–58. doi:[10.1214/aoms/1177729885][@grubbs:1950a].
-   Grubbs, Frank E. 1969. "Procedures for Detecting Outlying Observations in Samples." _Technometrics_ 11 (1). Taylor & Francis: 1–21. doi:[10.1080/00401706.1969.10490657][@grubbs:1969a].    

</section>

<!-- /.references -->

<section class="links">

[grubbs-test]: https://en.wikipedia.org/wiki/Grubbs%27_test_for_outliers

[@grubbs:1950a]: https://doi.org/10.1214/aoms/1177729885

[@grubbs:1969a]: https://doi.org/10.1080/00401706.1969.10490657

</section>

<!-- /.links -->
