PearsonChiSquareTest

PearsonChiSquareTest[data]

tests whether data is normally distributed using the Pearson test.

PearsonChiSquareTest[data,dist]

tests whether data is distributed according to dist using the Pearson test.

PearsonChiSquareTest[data,dist,"property"]

returns the value of "property".

Details and Options

  • PearsonChiSquareTest performs the Pearson goodness-of-fit test with null hypothesis that data was drawn from a population with distribution dist, 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 came from dist.
  • The dist can be any symbolic distribution with numeric and symbolic parameters or a dataset.
  • The data can be univariate {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
  • The Pearson test effectively compares a histogram of data to a theoretical histogram based on dist. The bins are chosen to have equal probability in dist. »
  • For univariate data, the test statistic is given by , where and are the observed and expected counts for the ^(th) histogram bin, respectively.
  • For multivariate tests, the sum of the univariate marginal -values is used and is assumed to follow a UniformSumDistribution under .
  • PearsonChiSquareTest[data,dist,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • PearsonChiSquareTest[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:
  • 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 PearsonChiSquareTest[si,dist,"TestStatistic"] is then used to estimate the -value.

Examples

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Basic Examples  (4)

Perform the Pearson test for normality:

Test the fit of some data to a particular distribution:

Compare the distributions of two datasets:

Extract the test statistic from the Pearson test:

Scope  (9)

Testing  (6)

Perform a Pearson test for normality:

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

Test the goodness of fit to a particular distribution:

Compare the distributions of two datasets:

The two datasets do not have the same distribution:

Test for multivariate normality:

Test for goodness of fit to any multivariate distribution:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Reporting  (3)

Tabulate the results of the Pearson test:

The full test table:

A -value table:

The test statistic:

Retrieve the entries from a Pearson 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 Pearson test:

Visualize the approximate power curve:

Estimate the power of the Pearson test when the underlying distribution is UniformDistribution[{-4,4}], the test size is 0.05, and the sample size is 12:

The number of auto accidents was recorded for a city over the course of 30 days. The city council is planning on lowering speed limits in the city and wants a model of the accident rate as a baseline for later comparison:

Count data is often modeled well by PoissonDistribution:

Suppose the city collected data over another 30-day period after reducing the speed limit. Compare the distributions before and after the reduction:

The distributions are significantly different:

Properties & Relations  (10)

By default, univariate data is compared to NormalDistribution:

The parameters have been estimated from the data:

Multivariate data is compared to MultinormalDistribution by default:

The parameters of the test distribution are estimated from the data if not specified:

Specified parameters are not estimated:

Maximum likelihood estimates are used for unspecified parameters of the test distribution:

PearsonChiSquareTest effectively compares the observed and expected histograms:

The data is binned into approximately bins that are equiprobable under :

Under , each bin will contain an equal number of points:

Observed histograms for when is true and false, respectively:

The degrees of freedom are equal to the number of non-empty bins minus one:

One degree of freedom is removed for each parameter that is estimated from the data:

If the parameters are unknown, PearsonChiSquareTest corrects the degrees of freedom:

No correction is applied when the parameters are specified:

The fitted distribution is equivalent, but the degrees of freedom and -value are corrected:

The Pearson statistic asymptotically follows ChiSquareDistribution under :

Independent marginal densities are assumed in tests for multivariate goodness of fit:

The test statistic is identical when independence is assumed:

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

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)