This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# MardiaCombinedTest

 MardiaCombinedTest[data] tests whether data follows a MultinormalDistribution using the Mardia combined test. MardiaCombinedTest returns the value of .
• By default a probability value or -value is returned.
• A small -value suggests that it is unlikely that the data came from dist.
• The data can be univariate or multivariate .
• Properties related to the reporting of test results include:
 "DegreesOfFreedom" the degrees of freedom used in a test "PValue" -value "PValueTable" formatted version of "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 "TestStatistic" test statistic "TestStatisticTable" formatted
• 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:
 Method Automatic the method to use for computing -values SignificanceLevel 0.05 cutoff 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 and properties is controlled by the SignificanceLevel option. By default is set to .
Perform a test for multivariate normality:
Extract the test statistic from the Mardia combined test:
Obtain a formatted test table:
Perform a test for multivariate normality:
 Out[2]=
 Out[3]=

Extract the test statistic from the Mardia combined test:
 Out[2]=

Obtain a formatted test table:
 Out[2]=
 Scope   (5)
Perform a Mardia test for multivariate normality:
The -value for the normal data is large compared to the -value for the non-normal data:
Create a HypothesisTestData object for repeated property extraction:
The properties available for extraction:
Tabulate the results of the Mardia combined test:
The full test table:
A -value table:
The test statistic:
Retrieve the entries from a Mardia combined test table for custom reporting:
Report test conclusions using and :
The conclusion may differ at a different significance level:
 Options   (4)
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:
Set the significance level used for and :
By default is used:
 Applications   (2)
A power curve for the Mardia combined test:
Visualize the approximate power curve:
Estimate the power of the Mardia combined test when the underlying distribution is a MultivariateTDistribution, the test size is 0.05, and the sample size is 12:
Five morphological measures were recorded for two separate varieties of a crab species. A researcher hopes to simultaneously compare all the measures across the species using multivariate analysis of variance, which requires that the data is multivariate normal:
Use MardiaCombinedTest to determine if both sets of data are multivariate normal:
Univariate density estimates of the five measures for each species:
The deviation from normality appears to be in the skewness:
The multivariate test statistic:
Under the test statistic asymptotically follows a ChiSquareDistribution:
For univariate data the test is equivalent to the JarqueBeraALMTest:
If the covariance matrix of the data is not positive definite the test will fail:
The number of data points must be greater than the dimension of the data:
The distribution of the Mardia combined test statistic:
New in 8