SignedRankTest

SignedRankTest[data]

tests whether the median of data is zero.

SignedRankTest[{data1,data2}]

tests whether the median of data1-data2 is zero.

SignedRankTest[dspec,μ0]

tests a location measure against μ0.

SignedRankTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • SignedRankTest tests the null hypothesis against the alternative hypothesis :
  • data
    {data1,data2}
  • where μ is the population median for data and μ12 is the median of the paired differences of the two datasets .
  • By default, a probability value or -value is returned.
  • A small -value suggests that it is unlikely that is true.
  • The data in dspec can be univariate {x1,x2,} or multivariate {{x1,y1,},{x2,y2,},}.
  • If two samples are given, they must be of equal length.
  • The argument μ0 can be a real number or a real vector with length equal to the dimension of the data.
  • SignedRankTest assumes that the data is symmetric about the median in the univariate case and elliptically symmetric in the multivariate case. For this reason, SignedRankTest is also a test of means.
  • SignedRankTest[dspec,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • SignedRankTest[dspec,μ0,"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"list of -values
    "PValueTable"formatted table of -values
    "ShortTestConclusion"a short description of the conclusion of a test
    "TestConclusion"a description of the conclusion of a test
    "TestData"list of pairs of test statistics and -values
    "TestDataTable"formatted table of -values and test statistics
    "TestStatistic"list of test statistics
    "TestStatisticTable"formatted table of test statistics
  • The SignedRankTest is a more powerful alternative to the SignTest.
  • For univariate samples, SignedRankTest performs the Wilcoxon signed rank test for medians of paired samples. A correction for ties is applied for permutation-based -values. By default, the test statistic is corrected for continuity and an asymptotic result is returned.
  • For multivariate samples, SignedRankTest performs an affine invariant test for paired samples using standardized spatial signed ranks. The test statistic is assumed to follow a ChiSquareDistribution[dim] where dim is the dimension of the data.
  • The following options can be used:
  • AlternativeHypothesis"Unequal"the inequality for the alternative hypothesis
    MaxIterationsAutomaticmax iterations for multivariate median tests
    MethodAutomaticthe method to use for computing -values
    SignificanceLevel0.05cutoff for diagnostics and reporting
    VerifyTestAssumptionsAutomaticwhat assumptions to verify
  • For the SignedRankTest, 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. This value is also used in diagnostic tests of assumptions including a test for symmetry. By default, is set to 0.05.
  • Named settings for VerifyTestAssumptions in SignedRankTest include:
  • "Symmetry"verify that all data is symmetric

Examples

open allclose all

Basic Examples  (4)

Test whether the median of a population is zero:

Compare the median difference for paired data to a particular value:

Report the test results in a table:

Test whether the spatial median of a multivariate population is some value:

Compute the test statistic:

Create a HypothesisTestData object for repeated property extraction:

A list of available properties:

Extract a single property or a list of properties:

Scope  (13)

Testing  (10)

Test versus :

The -values are typically large when the median is close to μ0:

The -values are typically small when the location is far from μ0:

Using Automatic is equivalent to testing for a median of zero:

Test versus :

The -values are typically large when median is close to μ0:

The -values are typically small when the location is far from μ0:

Test whether the median vector of a multivariate population is the zero vector:

Alternatively, test against {0.1,0,-.5,0}:

Test versus :

The -values are generally small when the locations are not equal:

The -values are generally large when the locations are equal:

Test versus :

The order of the datasets affects the test results:

Test whether the median difference vector of two multivariate populations is the zero vector:

Alternatively, test against {1,0,-1,0}:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Extract some properties from a HypothesisTestData object:

The -value and test statistic:

Extract any number of properties simultaneously:

The -value and test statistic:

Reporting  (3)

Tabulate the test results:

Retrieve the entries from a test table for customized reporting:

Tabulate -values or test statistics:

The -value from the table:

The test statistic from the table:

Options  (12)

AlternativeHypothesis  (3)

A two-sided test is performed by default:

Test versus :

Perform a two-sided test or a one-sided alternative:

Test versus :

Test versus :

Test versus :

Perform tests with one-sided alternatives when μ0 is given:

Test versus :

Test versus :

MaxIterations  (2)

Set the maximum number of iterations to use for multivariate tests:

By default, 500 iterations are allowed:

Setting the maximum number of iterations may result in lack of convergence:

The -values are not equivalent:

Method  (4)

By default, -values are computed using asymptotic test statistic distributions:

Permutation methods can be used:

Set the number of permutations to use:

By default, random permutations are used:

Set the seed used for generating random permutations:

SignificanceLevel  (1)

The significance level is used for "TestConclusion" and "ShortTestConclusion":

VerifyTestAssumptions  (2)

Diagnostics can be controlled as a group using All or None:

Verify all assumptions:

Check no assumptions:

Diagnostics can be controlled independently:

Check for symmetry:

Set the symmetry assumption to True:

Applications  (1)

Twelve sets of identical twins were given psychological tests to measure aggressiveness. It is hypothesized that the first-born twin will tend to be more aggressive than the second-born:

There is insufficient evidence to reject that birth order has no effect on aggressiveness:

Properties & Relations  (8)

The SignedRankTest is generally more powerful than the SignTest:

The univariate Wilcoxon signed rank test statistic:

In the absence of ties, Ordering can be used to compute ranks:

The asymptotic two-sided -value:

For univariate data, the test statistic is asymptotically normal:

For multivariate data, the test statistic follows a ChiSquareDistribution under :

The degree of freedom is equal to the dimension of the data:

For multivariate data, the SignedRankTest effectively tests uniformity about a unit sphere:

A function for computing the spatial signed ranks of a matrix:

Deviations from μ0 yield clustering of spatial signed ranks and larger test statistics:

The test statistic is affine invariant for multivariate data:

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

The signed rank test works with all the values together when the input is a TemporalData:

Test all the values only:

Test the difference of the medians of the two paths:

Possible Issues  (1)

SignedRankTest requires that the data be symmetric about a common median:

Use SignTest if the data is not symmetric:

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)