ShapiroWilkTest

ShapiroWilkTest[data]

tests whether data is normally distributed using the ShapiroWilk test.

ShapiroWilkTest[data,"property"]

returns the value of "property".

Details and Options

  • ShapiroWilkTest performs the ShapiroWilk goodness-of-fit test with null hypothesis that data was drawn from a NormalDistribution and alternative hypothesis that it was not.
  • By default, a probability value or -value is returned.
  • A small -value suggests that it is unlikely that the data is normally distributed.
  • The data can be univariate {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
  • The ShapiroWilk test effectively compares the order statistics of data to the theoretical order statistics of a NormalDistribution.
  • ShapiroWilkTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • ShapiroWilkTest[data,dist,"property"] can be used to directly give the value of "property".
  • Properties related to the reporting of test results include:
  • "PValue"-value
    "PValueTable"formatted version of "PValue"
    "ShortTestConclusion"a short description of the conclusion of a test
    "TestConclusion"a description of the conclusion of a test
    "TestData"test statistic and -value
    "TestDataTable"formatted version of "TestData"
    "TestStatistic"test statistic
    "TestStatisticTable"formatted "TestStatistic"
  • The following properties are independent of which test is being performed.
  • Properties related to the data distribution include:
  • "FittedDistribution"fitted distribution of data
    "FittedDistributionParameters"distribution parameters of data
  • The following options can be given:
  • MethodAutomaticthe method to use for computing -values
    SignificanceLevel0.05cutoff for diagnostics and reporting
  • For a test for goodness of fit, a cutoff is chosen such that is rejected only if . The value of used for the "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. By default, is set to 0.05.
  • With the setting Method->"MonteCarlo", datasets of the same length as the input are generated under using the fitted distribution. The EmpiricalDistribution from ShapiroWilkTest[si,"TestStatistic"] is then used to estimate the -value.

Examples

open allclose all

Basic Examples  (2)

Perform a ShapiroWilk test for normality:

Perform a test for multivariate normality:

The full test table:

The test statistic and -value:

Scope  (6)

Testing  (3)

Perform a ShapiroWilk test for normality:

The -value for the normal data is large compared to the -value for the non-normal data:

Test for multivariate normality:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Reporting  (3)

Tabulate the results of the ShapiroWilk test:

The full test table:

A -value table:

The test statistic:

Retrieve the entries from a ShapiroWilk test table for custom reporting:

Report test conclusions using "ShortTestConclusion" and "TestConclusion":

The conclusion may differ at a different significance level:

Options  (3)

Method  (3)

Use Monte Carlo-based methods or a computation formula:

Set the number of samples to use for Monte Carlo-based methods:

The Monte Carlo estimate converges to the true -value with increasing samples:

Set the random seed used in Monte Carlo-based methods:

The seed affects the state of the generator and has some effect on the resulting -value:

Applications  (2)

A power curve for the ShapiroWilk test:

Visualize the approximate power curve:

Estimate the power of the ShapiroWilk test when the underlying distribution is a CauchyDistribution[0,1], the test size is 0.05, and the sample size is 12:

The boiling point of water was measured at varying altitudes in the Alps. The barometric pressure was recorded for each boiling point. Determine if a linear model is appropriate for use in predicting boiling points given pressure:

A plot of the model and the data:

For the model to be appropriate, the residuals should be normally distributed:

A QuantilePlot confirms that the linear model is not appropriate for this data:

Properties & Relations  (3)

ShapiroWilkTest compares the order statistics of the data to their expectations under :

Expected values of the order statistics and an estimate of their covariance matrix:

These are used to compute weights:

The statistic using the estimated covariance matrix is slightly different from the reported value:

For tests of multivariate normality, a transformation to univariate data is made:

The data has been transformed to approximate univariate normal data:

Perform the test on the transformed data:

The result agrees with a test of the original data:

The ShapiroWilk test works with the values only when the input is a TimeSeries:

Possible Issues  (1)

The ShapiroWilk test requires sample sizes be less than 5000 for -values to be valid:

Neat Examples  (1)

Compute the statistic when the null hypothesis is true:

The test statistic given a particular alternative:

Compare the distributions of the test statistics:

Introduced in 2010
 (8.0)