# incrgrubbs

Grubbs' test for outliers.

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 is defined for the hypothesis:

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

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

where `s`

is the sample standard deviation. The Grubbs test statistic is thus the largest absolute deviation from the sample mean in units of the sample standard deviation.

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

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

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

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

where `t`

denotes the upper critical value of the *t*-distribution with `N-2`

degrees of freedom and a significance level of `α/N`

.

## Usage

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

#### incrgrubbs( [options] )

Returns an accumulator `function`

which incrementally performs Grubbs' test for outliers.

```
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.

```
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`

.

## Notes

- 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. - The accumulator must be provided
**at least**three data points before performing 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.

## Examples

```
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
*/
```

## 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, Frank E. 1969. "Procedures for Detecting Outlying Observations in Samples."
*Technometrics*11 (1). Taylor & Francis: 1–21. doi:10.1080/00401706.1969.10490657.