ShapiroWilkTest Class

Shapiro-Wilk test for normality.

Inheritance Hierarchy

SystemObject
Accord.Statistics.TestingHypothesisTestShapiroWilkDistribution
Accord.Statistics.TestingShapiroWilkTest

Namespace: Accord.Statistics.Testing
Assembly: Accord.Statistics (in Accord.Statistics.dll) Version: 3.8.0

Syntax

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[SerializableAttribute]
public class ShapiroWilkTest : HypothesisTest<ShapiroWilkDistribution>, 
	IHypothesisTest<ShapiroWilkDistribution>, IHypothesisTest

<SerializableAttribute>
Public Class ShapiroWilkTest
	Inherits HypothesisTest(Of ShapiroWilkDistribution)
	Implements IHypothesisTest(Of ShapiroWilkDistribution), IHypothesisTest

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The ShapiroWilkTest type exposes the following members.

Constructors

	Name	Description
	ShapiroWilkTest	Creates a new Shapiro-Wilk test.

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Properties

	Name	Description
	CriticalValue	Gets the critical value for the current significance level. (Inherited from HypothesisTestTDistribution.)
	PValue	Gets the P-value associated with this test. (Inherited from HypothesisTestTDistribution.)
	Significant	Gets whether the null hypothesis should be rejected. (Inherited from HypothesisTestTDistribution.)
	Size	Gets the significance level for the test. Default value is 0.05 (5%). (Inherited from HypothesisTestTDistribution.)
	Statistic	Gets the test statistic. (Inherited from HypothesisTestTDistribution.)
	StatisticDistribution	Gets the distribution associated with the test statistic. (Inherited from HypothesisTestTDistribution.)
	Tail	Gets the test type. (Inherited from HypothesisTestTDistribution.)

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as the default hash function. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	OnSizeChanged	Called whenever the test significance level changes. (Inherited from HypothesisTestTDistribution.)
	PValueToStatistic	Converts a given p-value to a test statistic. (Overrides HypothesisTestTDistributionPValueToStatistic(Double).)
	StatisticToPValue	Converts a given test statistic to a p-value. (Overrides HypothesisTestTDistributionStatisticToPValue(Double).)
	ToString	Converts the numeric P-Value of this test to its equivalent string representation. (Inherited from HypothesisTestTDistribution.)
	ToString(String, IFormatProvider)	Converts the numeric P-Value of this test to its equivalent string representation. (Inherited from HypothesisTestTDistribution.)

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Extension Methods

	Name	Description
	HasMethod	Checks whether an object implements a method with the given name. (Defined by ExtensionMethods.)
	IsEqual	Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices. (Defined by Matrix.)
	To(Type)	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)
	ToT	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)

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Remarks

The Shapiro–Wilk test is a test of normality in frequentist statistics. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk. The The Shapiro–Wilk test tests the null hypothesis that a sample came from a normally distributed population.

The null-hypothesis of this test is that the population is normally distributed. Thus, if the p-value is less than the chosen alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not from a normally distributed population; in other words, the data are not normal. On the contrary, if the p-value is greater than the chosen alpha level, then the null hypothesis that the data came from a normally distributed population cannot be rejected (e.g., for an alpha level of 0.05, a data set with a p-value of 0.02 rejects the null hypothesis that the data are from a normally distributed population). However, since the test is biased by sample size, the test may be statistically significant from a normal distribution in any large samples. Thus a Q–Q plot is required for verification in addition to the test.

References:

Wikipedia, The Free Encyclopedia. Shapiro-Wilk test. Available on: http://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test

Examples

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// Let's say we would like to determine whether a set
// of observations come from a normal distribution:
double[] samples =
{
    0.11, 7.87, 4.61, 10.14, 7.95, 3.14, 0.46, 4.43, 
    0.21, 4.75, 0.71, 1.52, 3.24, 0.93, 0.42, 4.97,
    9.53, 4.55, 0.47, 6.66 
};

// For this, we can use the Shapiro-Wilk test. This test tests the null hypothesis 
// that samples come from a Normal distribution, vs. the alternative hypothesis that 
// the samples do not come from such distribution. In other words, should this test
// come out significant, it means our samples do not come from a Normal distribution.

// Create a new Shapiro-Wilk test:
var sw = new ShapiroWilkTest(samples);

double W = sw.Statistic; // should be 0.90050
double p = sw.PValue;    // should be 0.04209
bool significant = sw.Significant; // should be true

// The test is significant, therefore we should reject the null 
// hypothesis that the samples come from a Normal distribution.

Reference

Accord.Statistics.Testing Namespace