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

upper G equals StartFraction max Underscript i equals 0 comma ellipsis comma upper N minus 1 Endscripts StartAbsoluteValue upper Y Subscript i Baseline minus upper Y overbar EndAbsoluteValue Over s EndFraction

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

upper G equals StartFraction upper Y overbar minus upper Y Subscript min Baseline Over s EndFraction

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

upper G equals StartFraction upper Y Subscript max Baseline minus upper Y overbar Over s EndFraction

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

upper G greater-than StartFraction upper N minus 1 Over StartRoot upper N EndRoot EndFraction StartRoot StartFraction t Subscript alpha slash left-parenthesis 2 upper N right-parenthesis comma upper N minus 2 Superscript 2 Baseline Over upper N minus 2 plus t Subscript alpha slash left-parenthesis 2 upper N right-parenthesis comma upper N minus 2 Superscript 2 Baseline EndFraction EndRoot

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

upper G greater-than StartFraction upper N minus 1 Over StartRoot upper N EndRoot EndFraction StartRoot StartFraction t Subscript alpha slash upper N comma upper N minus 2 Superscript 2 Baseline Over upper N minus 2 plus t Subscript alpha slash upper N comma upper N minus 2 Superscript 2 Baseline EndFraction EndRoot

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.