SignTest

SignTest[data]

tests whether the median of data is zero.

SignTest[{data1,data2}]

tests whether the median of data1 data2 is zero.

SignTest[dspec,μ0]

tests a location measure against μ0.

SignTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • SignTest 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.
  • SignTest[dspec,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • SignTest[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
  • For univariate samples, SignTest performs the sign test for medians of paired samples. The test statistic is assumed to follow a BinomialDistribution[n,1/2] where n is the number of elements in dspec not equal to μ0.
  • For multivariate samples, SignTest performs an affine invariant test for paired samples using spatial signs. The test statistic is assumed to follow a ChiSquareDistribution[dim] where dim is the dimension of dspec.
  • 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
  • For the SignTest, 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.

Examples

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Basic Examples  (4)

Test whether the median of a population is zero:

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

Compute the test statistic:

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

Report the test results in a table:

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 the 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,-0.05,0}:

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  (9)

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  (3)

By default, -values are computed using the BinomialDistribution for univariate data:

Asymptotic methods can be used for univariate data:

For multivariate data, only the asymptotic result is available:

SignificanceLevel  (1)

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

Applications  (2)

A new sleeping aid was tested on eight patients. The number of minutes taken for each subject to fall asleep was recorded for a night taking the medication and for a night with a placebo:

The SignTest does not detect a difference in the sleep aid and placebo:

The datasets, while very small, do not fail a test for normality:

A more powerful PairedTTest shows a significant reduction in time to sleep with the sleep aid:

A group of 10 students with low assessments in mathematics and science was asked to participate in tutoring program. A test similar to the original assessment was administered after the program. The students' scores on the math and science portions of both assessments are as follows:

There is a significant improvement in scores overall:

The Bonferroni-corrected tests of the individual components suggest that math scores alone account for the detected improvement:

Properties & Relations  (6)

Conceptually, the SignTest counts the number of positive signs in a dataset:

For univariate data, the test statistic follows a BinomialDistribution, ignoring zeros:

The SignTest is generally less powerful than other hypothesis tests for location:

For multivariate data, spatial signs are used when computing the test statistic:

Spatial signs tend to cluster when the spatial median is nonzero:

The amount of clustering is quantified by the test statistic:

The test statistic follows a ChiSquareDistribution[p]:

The test statistic is affine invariant for multivariate data:

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

The sign 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:

Neat Examples  (2)

Compute the statistic when the null hypothesis is true:

The test statistic given a particular alternative:

Compare the distributions of the test statistics:

The distribution of spatial signs in three dimensions shows that larger deviations from a zero mean vector produce more highly clustered spatial signs and larger sign statistics:

Introduced in 2010
 (8.0)