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MardiaCombinedTest
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MardiaCombinedTest

✖

tests whether data follows a MultinormalDistribution using the Mardia combined test.

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MardiaCombinedTest[data,"property"]

returns the value of "property".

Details and Options

  • MardiaCombinedTest performs a goodness-of-fit test with null hypothesis that data was drawn from a MultinormalDistribution 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 Mardia combined test effectively pools the results from MardiaSkewnessTest and MardiaKurtosisTest.
  • MardiaCombinedTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • MardiaCombinedTest[data,dist,"property"] can be used to directly give the value of "property".
  • Properties related to the reporting of test results include:
  • "DegreesOfFreedom"the degrees of freedom used in a test
    "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:
  • Method Automaticthe method to use for computing -values
    SignificanceLevel 0.05cutoff 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 MardiaCombinedTest[si,"TestStatistic"] is then used to estimate the -value.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Perform a test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-ijo5
In[2]:=2
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https://wolfram.com/xid/01la8cwjkiw46mp3q-hnfx0t
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-dfzlwy
Out[3]=3

Extract the test statistic from the Mardia combined test:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-m3chh3
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-cm7ucc
Out[2]=2

Obtain a formatted test table:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-p7hfx
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-cr9vsu
Out[2]=2

Scope  (5)Survey of the scope of standard use cases

Testing  (2)

Perform a Mardia test for multivariate normality:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-lqxbgq

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

In[3]:=3
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https://wolfram.com/xid/01la8cwjkiw46mp3q-mks25
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-kjd2s
Out[4]=4

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
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https://wolfram.com/xid/01la8cwjkiw46mp3q-cdnr2n
In[2]:=2
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https://wolfram.com/xid/01la8cwjkiw46mp3q-bolz27
Out[2]=2

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-e40fsc
Out[3]=3

Reporting  (3)

Tabulate the results of the Mardia combined test:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-cqxopz
In[2]:=2
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https://wolfram.com/xid/01la8cwjkiw46mp3q-be6kk1

The full test table:

In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-ef4et7
Out[3]=3

A -value table:

In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bfqugt
Out[4]=4

The test statistic:

In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-oi9r56
Out[5]=5

Retrieve the entries from a Mardia combined test table for custom reporting:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-s3qsd
In[2]:=2
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https://wolfram.com/xid/01la8cwjkiw46mp3q-ga3bij
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/01la8cwjkiw46mp3q-gg2n22
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/01la8cwjkiw46mp3q-65k71
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/01la8cwjkiw46mp3q-bkir6y
In[2]:=2
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https://wolfram.com/xid/01la8cwjkiw46mp3q-cd96sm
In[3]:=3
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https://wolfram.com/xid/01la8cwjkiw46mp3q-ggy9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-el9mb

The conclusion may differ at a different significance level:

In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-c53cri
In[6]:=6
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-byyexa
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bc67bi

Options  (4)Common values & functionality for each option

Method  (3)

Use Monte Carlo-based methods or a computation formula:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-b56tvj
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-eyrfe
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-evuhgg
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-xg6xc
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-499mh

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

In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-eli8sg
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-ba5c5u
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/01la8cwjkiw46mp3q-ccet45
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-ip8pt1

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

In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-go8plt
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-pfg0ok
Out[4]=4

SignificanceLevel  (1)

Set the significance level used for "TestConclusion" and "ShortTestConclusion":

In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-kj9pa
In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-qafg
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bu4h57
Out[6]=6

By default, 0.05 is used:

In[7]:=7
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-b50sx2

Applications  (2)Sample problems that can be solved with this function

A power curve for the Mardia combined test:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-f9ry9j
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bhp5v
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-fyqopk

Visualize the approximate power curve:

In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-eel2vq
Out[4]=4

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:

In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-z6f7c
Out[5]=5

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:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-k2w6px
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-9eqgxb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-dwl2bl

Use MardiaCombinedTest to determine if both sets of data are multivariate normal:

In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-pheluk
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bkgw16
Out[5]=5

Univariate density estimates of the five measures for each species:

In[6]:=6
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-eg3o0g
In[7]:=7
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bxncu5
Out[7]=7

The deviation from normality appears to be in the skewness:

In[8]:=8
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-qzq80f
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-eyjyx
Out[9]=9

Properties & Relations  (3)Properties of the function, and connections to other functions

The multivariate test statistic:

In[8]:=8
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-et46gq
In[9]:=9
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-mvgdy
In[10]:=10
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-e2j9t7
Out[10]=10
In[11]:=11
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-dpmzkv
Out[11]=11

Under , the test statistic asymptotically follows a ChiSquareDistribution:

In[12]:=12
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-jxo3sh
In[13]:=13
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-q4iyo4
In[14]:=14
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-fl2g5b
Out[14]=14

For univariate data, the test is equivalent to the JarqueBeraALMTest:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-gih5gu
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-bvdksn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-mmzpw
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-of2jx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-57bf57
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-vvd88
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

If the covariance matrix of the data is not positive definite, the test will fail:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-e2fnxq
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-dmqkla
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-k118o6
Out[3]=3

The number of data points must be greater than the dimension of the data:

In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-0q09i
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-0curas
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-wlr4s0
Out[7]=7

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-2qqg3c
In[2]:=2
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-ywy3ty

The test statistic given a particular alternative:

In[3]:=3
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-612aqt
In[4]:=4
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-c5cy2n

Compare the distributions of the test statistics:

In[5]:=5
✖
https://wolfram.com/xid/01la8cwjkiw46mp3q-87eb6q
Out[5]=5
Wolfram Research (2010), MardiaCombinedTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.
✖
Wolfram Research (2010), MardiaCombinedTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.

Text

Wolfram Research (2010), MardiaCombinedTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.

✖
Wolfram Research (2010), MardiaCombinedTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.

CMS

Wolfram Language. 2010. "MardiaCombinedTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.

✖
Wolfram Language. 2010. "MardiaCombinedTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MardiaCombinedTest.html.

APA

Wolfram Language. (2010). MardiaCombinedTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MardiaCombinedTest.html

✖
Wolfram Language. (2010). MardiaCombinedTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MardiaCombinedTest.html

BibTeX

@misc{reference.wolfram_2025_mardiacombinedtest, author="Wolfram Research", title="{MardiaCombinedTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MardiaCombinedTest.html}", note=[Accessed: 30-April-2025 ]}

✖
@misc{reference.wolfram_2025_mardiacombinedtest, author="Wolfram Research", title="{MardiaCombinedTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MardiaCombinedTest.html}", note=[Accessed: 30-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mardiacombinedtest, organization={Wolfram Research}, title={MardiaCombinedTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MardiaCombinedTest.html}, note=[Accessed: 30-April-2025 ]}

✖
@online{reference.wolfram_2025_mardiacombinedtest, organization={Wolfram Research}, title={MardiaCombinedTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MardiaCombinedTest.html}, note=[Accessed: 30-April-2025 ]}