--- title: "SignedRankTest" language: "en" type: "Symbol" summary: "SignedRankTest[data] tests whether the median of data is zero. SignedRankTest[{data1, data2}] tests whether the median of data1 - data2 is zero. SignedRankTest[dspec, \\[Mu] 0] tests a location measure against \\[Mu] 0. SignedRankTest[dspec, \\[Mu] 0, property] returns the value of property." keywords: - hypothesis testing - test of location - mean test - mean difference test - equal mean test - test of shift - Oja signed rank test - signed rank test - Wilcoxon signed rank test - wilcoxon mann whitney test - wilcoxon rank sum test - rank sum test - Wilcoxon-Mann-Whitney U test - nonparametric test - median difference - equal medians canonical_url: "https://reference.wolfram.com/language/ref/SignedRankTest.html" source: "Wolfram Language Documentation" related_guides: - title: "Hypothesis Tests" link: "https://reference.wolfram.com/language/guide/HypothesisTests.en.md" related_functions: - title: "HypothesisTestData" link: "https://reference.wolfram.com/language/ref/HypothesisTestData.en.md" - title: "LocationTest" link: "https://reference.wolfram.com/language/ref/LocationTest.en.md" - title: "LocationEquivalenceTest" link: "https://reference.wolfram.com/language/ref/LocationEquivalenceTest.en.md" - title: "VarianceTest" link: "https://reference.wolfram.com/language/ref/VarianceTest.en.md" - title: "VarianceEquivalenceTest" link: "https://reference.wolfram.com/language/ref/VarianceEquivalenceTest.en.md" - title: "DistributionFitTest" link: "https://reference.wolfram.com/language/ref/DistributionFitTest.en.md" - title: "MannWhitneyTest" link: "https://reference.wolfram.com/language/ref/MannWhitneyTest.en.md" - title: "PairedTTest" link: "https://reference.wolfram.com/language/ref/PairedTTest.en.md" - title: "PairedZTest" link: "https://reference.wolfram.com/language/ref/PairedZTest.en.md" - title: "SignTest" link: "https://reference.wolfram.com/language/ref/SignTest.en.md" - title: "TTest" link: "https://reference.wolfram.com/language/ref/TTest.en.md" - title: "ZTest" link: "https://reference.wolfram.com/language/ref/ZTest.en.md" --- # 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 $H_0$ against the alternative hypothesis $H_a$: | | | | | -------------- | ------------------------------------------------ | ---------------------------------------------------- | | | $H_0$ | $H_a$ | | data | $\mu =\mu _0$ | $\mu \neq \mu _0$ | | {data1, data2} | $\mu _{12}=\mu _0$ | $\mu _{12}\neq \mu _0$ | * where ``μ`` is the population median for ``data`` and ``μ12`` is the median of the paired differences of the two datasets $\text{\textit{data}}_1-\text{\textit{data}}_2$. * By default, a probability value or $p$-value is returned. * A small $p$-value suggests that it is unlikely that $H_0$ 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 $p$-values | | "PValueTable" | formatted table of $p$-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 $p$-values | | "TestDataTable" | formatted table of $p$-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 $p$-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 | | MaxIterations | Automatic | max iterations for multivariate median tests | | Method | Automatic | the method to use for computing $p$-values | | SignificanceLevel | 0.05 | cutoff for diagnostics and reporting | | VerifyTestAssumptions | Automatic | what assumptions to verify | * For the ``SignedRankTest``, a cutoff $\alpha$ is chosen such that $H_0$ is rejected only if $p<\alpha$. The value of $\alpha$ used for the ``"TestConclusion"`` and ``"ShortTestConclusion"`` properties is controlled by the ``SignificanceLevel`` option. This value $\alpha$ is also used in diagnostic tests of assumptions including a test for symmetry. By default, $\alpha$ is set to ``0.05``. * Named settings for ``VerifyTestAssumptions`` in ``SignedRankTest`` include: "Symmetry" verify that all data is symmetric --- ## Examples (40) ### Basic Examples (4) Test whether the median of a population is zero: ```wl In[1]:= data = RandomVariate[NormalDistribution[], 100]; In[2]:= DistributionChart[data] Out[2]= [image] In[3]:= SignedRankTest[data] Out[3]= 0.152126 ``` --- Compare the median difference for paired data to a particular value: ```wl In[1]:= data = RandomVariate[MultinormalDistribution[{2, 0}, {{1, .75}, {.75, 1}}], 1000]; In[2]:= BoxWhiskerChart[Transpose@data] Out[2]= [image] In[3]:= SignedRankTest[Transpose@data, 2] Out[3]= 0.995939 ``` Report the test results in a table: ```wl In[4]:= SignedRankTest[Transpose@data, 2, "TestDataTable"] Out[4]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 250297. | 0.995939 | ``` --- Test whether the spatial median of a multivariate population is some value: ```wl In[1]:= data = RandomVariate[MultinormalDistribution[{4, 5, 6}, IdentityMatrix[3]], 10^2]; In[2]:= SignedRankTest[data, {4, 5, 6}] Out[2]= 0.145411 ``` Compute the test statistic: ```wl In[3]:= SignedRankTest[data, {4, 5, 6}, "TestStatistic"] Out[3]= 5.38932 ``` --- Create a ``HypothesisTestData`` object for repeated property extraction: ```wl In[1]:= data = RandomVariate[StudentTDistribution[4], 100]; In[2]:= ℋ = SignedRankTest[data, Automatic, "HypothesisTestData"] Out[2]= HypothesisTestData[SignedRankTest, {CompressedData["«1170»"], "SampleToNull", 1, Rational[1, 20], {0, "Unequal", Automatic, Automatic}, Automatic, {}}, {"Normality" -> 0, "EqualVariance" -> 0, "Symmetry" -> 0}] ``` A list of available properties: ```wl In[3]:= ℋ["Properties"] Out[3]= {"DegreesOfFreedom", "HypothesisTestData", "Properties", "PValue", "PValueTable", "ShortTestConclusion", "SignedRank", "TestConclusion", "TestData", "TestDataTable", "TestEntries", "TestStatistic", "TestStatisticTable"} ``` Extract a single property or a list of properties: ```wl In[4]:= ℋ["TestData"] Out[4]= {2621., 0.742639} In[5]:= ℋ["TestDataTable", "PValue", "TestStatistic"] Out[5]= {| "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 2621. | 0.742639 |, 0.742639, 2621.} ``` ### Scope (13) #### Testing (10) Test $H_0:$$\mu =0$ versus $H_a:$$\mu \neq 0$ : ```wl In[1]:= data1 = RandomVariate[NormalDistribution[0, 1], 500]; data2 = RandomVariate[NormalDistribution[3, 1], 500]; ``` The $p$-values are typically large when the median is close to ``μ0`` : ```wl In[2]:= SignedRankTest[data1] Out[2]= 0.0327488 ``` The $p$-values are typically small when the location is far from ``μ0`` : ```wl In[3]:= SignedRankTest[data2] Out[3]= 1.2685250602309246`*^-83 ``` --- Using ``Automatic`` is equivalent to testing for a median of zero: ```wl In[1]:= data = RandomVariate[NormalDistribution[0, 1], 500]; In[2]:= SignedRankTest[data, 0] Out[2]= 0.752217 In[3]:= SignedRankTest[data, Automatic] Out[3]= 0.752217 ``` --- Test $H_0:$$\mu =3$ versus $H_a:$$\mu \neq 3$ : ```wl In[1]:= data1 = RandomVariate[NormalDistribution[3, 1], 500]; data2 = RandomVariate[NormalDistribution[0, 1], 500]; ``` The $p$-values are typically large when median is close to ``μ0`` : ```wl In[2]:= SignedRankTest[data1, 3] Out[2]= 0.405548 ``` The $p$-values are typically small when the location is far from ``μ0`` : ```wl In[3]:= SignedRankTest[data2, 3] Out[3]= 1.2685250602309246`*^-83 ``` --- Test whether the median vector of a multivariate population is the zero vector: ```wl In[1]:= data = RandomVariate[MultinormalDistribution[{.1, 0, -.5, 0}, IdentityMatrix[4]], 10^2]; In[2]:= SignedRankTest[data] Out[2]= 4.954575224600739`*^-7 ``` Alternatively, test against ``{0.1, 0, -.5, 0}`` : ```wl In[3]:= SignedRankTest[data, {0.1, 0, -.5, 0}] Out[3]= 0.102291 ``` --- Test $H_0$$:\mu _1-\mu _2=0$ versus $H_a$$:\mu _1-\mu _2\neq 0$ : ```wl In[1]:= data1 = RandomVariate[NormalDistribution[0, 1], 100]; data2 = RandomVariate[NormalDistribution[1, 1], 100]; data3 = RandomVariate[NormalDistribution[0, 2], 100]; ``` The $p$-values are generally small when the locations are not equal: ```wl In[2]:= SignedRankTest[{data1, data2}, 0] Out[2]= 5.733770393107215`*^-10 ``` The $p$-values are generally large when the locations are equal: ```wl In[3]:= SignedRankTest[{data1, data3}, 0] Out[3]= 0.314555 ``` --- Test $H_0$$:\mu _1-\mu _2=3$ versus $H_a$$:\mu _1-\mu _2\neq 3$ : ```wl In[1]:= data1 = RandomVariate[NormalDistribution[3, 1], 100]; data2 = RandomVariate[NormalDistribution[0, 1], 100]; ``` The order of the datasets affects the test results: ```wl In[2]:= SignedRankTest[{data1, data2}, 3] Out[2]= 0.444255 In[3]:= SignedRankTest[{data2, data1}, 3] Out[3]= 3.955911608899571`*^-18 ``` --- Test whether the median difference vector of two multivariate populations is the zero vector: ```wl In[1]:= data1 = RandomVariate[MultinormalDistribution[{.5, 0, -.5, 0}, IdentityMatrix[4]], 10^2]; In[2]:= data2 = RandomVariate[MultinormalDistribution[{-.5, 0, .5, 0}, IdentityMatrix[4]], 10^2]; In[3]:= SignedRankTest[{data1, data2}] Out[3]= 5.7562468325158316`*^-11 ``` Alternatively, test against ``{1, 0, -1, 0}`` : ```wl In[4]:= SignedRankTest[{data1, data2}, {1, 0, -1, 0}] Out[4]= 0.808417 ``` --- Create a ``HypothesisTestData`` object for repeated property extraction: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 10^4}]; In[2]:= ℋ = SignedRankTest[data, 0, "HypothesisTestData"]; ``` The properties available for extraction: ```wl In[3]:= ℋ["Properties"] Out[3]= {"DegreesOfFreedom", "HypothesisTestData", "Properties", "PValue", "PValueTable", "ShortTestConclusion", "SignedRank", "TestConclusion", "TestData", "TestDataTable", "TestEntries", "TestStatistic", "TestStatisticTable"} ``` --- Extract some properties from a ``HypothesisTestData`` object: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 10^4}]; In[2]:= ℋ = SignedRankTest[data, 0, "HypothesisTestData"]; ``` The $p$-value and test statistic: ```wl In[3]:= ℋ["PValue"] Out[3]= 0.0722162 In[4]:= ℋ["TestStatistic"] Out[4]= 2.4483494*^7 ``` --- Extract any number of properties simultaneously: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 10^4}]; In[2]:= ℋ = SignedRankTest[data, 0, "HypothesisTestData"]; ``` The $p$-value and test statistic: ```wl In[3]:= ℋ["PValue", "TestStatistic"] Out[3]= {0.865568, 2.5051374*^7} ``` #### Reporting (3) Tabulate the test results: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 20}]; In[2]:= ℋ = SignedRankTest[data, 0, "HypothesisTestData"]; In[3]:= ℋ["TestDataTable"] Out[3]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 148. | 0.112595 | ``` --- Retrieve the entries from a test table for customized reporting: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {1000, 25}]; In[2]:= res = Table[SignedRankTest[i, 0, "TestData", VerifyTestAssumptions -> None], {i, data}]; In[3]:= ListPlot[res, PlotStyle -> PointSize[Medium]] Out[3]= [image] ``` --- Tabulate $p$-values or test statistics: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 10^2}]; In[2]:= ℋ = SignedRankTest[data, 0, "HypothesisTestData"]; In[3]:= ℋ["PValueTable"] Out[3]= | "" | "P‐Value" | | :------------ | :-------- | | "Signed‐Rank" | 0.763526 | ``` The $p$-value from the table: ```wl In[4]:= ℋ["PValue"] Out[4]= 0.763526 In[5]:= ℋ["TestStatisticTable"] Out[5]= | "" | "Statistic" | | :------------ | :---------- | | "Signed‐Rank" | 2613. | ``` The test statistic from the table: ```wl In[6]:= ℋ["TestStatistic"] Out[6]= 2613. ``` ### Options (12) #### AlternativeHypothesis (3) A two-sided test is performed by default: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 100}]; ``` Test $H_0$$: \mu _1-\mu _2=0$ versus $H_a$$: \mu _1-\mu _2\neq 0$ : ```wl In[2]:= SignedRankTest[data, 0, AlternativeHypothesis -> "Unequal"] Out[2]= 0.375958 In[3]:= SignedRankTest[data, 0, AlternativeHypothesis -> Automatic] Out[3]= 0.375958 ``` --- Perform a two-sided test or a one-sided alternative: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 100}]; ``` Test $H_0:\mu =0$ versus $H_a:\mu \neq 0$ : ```wl In[2]:= SignedRankTest[data, 0, AlternativeHypothesis -> "Unequal"] Out[2]= 0.147267 ``` Test $H_0:\mu \geq 0$ versus $H_a:\mu <0$ : ```wl In[3]:= SignedRankTest[data, 0, AlternativeHypothesis -> "Less"] Out[3]= 0.926845 ``` Test $H_0:\mu \leq 0$ versus $H_a:\mu >0$ : ```wl In[4]:= SignedRankTest[data, 0, AlternativeHypothesis -> "Greater"] Out[4]= 0.0736333 ``` --- Perform tests with one-sided alternatives when ``μ0`` is given: ```wl In[1]:= data1 = RandomVariate[NormalDistribution[2.9, 1], 1000]; data2 = RandomVariate[NormalDistribution[0, 1], 1000]; In[2]:= Mean[data1 - data2] Out[2]= 2.88212 ``` Test $H_0: \mu \geq 3$ versus $H_a: \mu <3$ : ```wl In[3]:= SignedRankTest[{data1, data2}, 3, AlternativeHypothesis -> "Less"] Out[3]= 0.00445705 ``` Test $H_0: \mu \geq 2.9$ versus $H_a: \mu <2.9$ : ```wl In[4]:= SignedRankTest[{data1, data2}, 2.9, AlternativeHypothesis -> "Less"] Out[4]= 0.33818 ``` #### MaxIterations (2) Set the maximum number of iterations to use for multivariate tests: ```wl In[1]:= data1 = RandomVariate[BinormalDistribution[.5], 25]; data2 = RandomVariate[BinormalDistribution[.5], 25]; ``` By default, 500 iterations are allowed: ```wl In[2]:= SignedRankTest[{data1, data2}, Automatic, MaxIterations -> 500] Out[2]= 0.367252 In[3]:= SignedRankTest[{data1, data2}, Automatic, MaxIterations -> Automatic] Out[3]= 0.367252 ``` --- Setting the maximum number of iterations may result in lack of convergence: ```wl In[1]:= BlockRandom[SeedRandom[2]; data1 = RandomVariate[BinormalDistribution[.5], 25]; data2 = RandomVariate[BinormalDistribution[.5], 25];] ``` The $p$-values are not equivalent: ```wl In[2]:= SignedRankTest[{data1, data2}, Automatic, MaxIterations -> 1] ``` SignedRankTest::cvmit: Failed to converge to the requested accuracy or precision within 1 iterations. ```wl Out[2]= 0.854572 In[3]:= SignedRankTest[{data1, data2}, Automatic, MaxIterations -> Automatic] Out[3]= 0.862316 ``` #### Method (4) By default, $p$-values are computed using asymptotic test statistic distributions: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 100}]; In[2]:= SignedRankTest[data, 0, Method -> Automatic] Out[2]= 0.136995 In[3]:= SignedRankTest[data, 0, Method -> "Asymptotic"] Out[3]= 0.136995 ``` --- Permutation methods can be used: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 35}]; In[2]:= SignedRankTest[data, 0, Method -> "Asymptotic"] Out[2]= 0.58321 In[3]:= SignedRankTest[data, 0, Method -> "Permutation"] Out[3]= 0.5808 ``` --- Set the number of permutations to use: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 35}]; In[2]:= SignedRankTest[data, 0, Method -> {"Permutation", "MonteCarloSamples" -> 10^3}] Out[2]= 0.17 In[3]:= SignedRankTest[data, 0, Method -> {"Permutation", "MonteCarloSamples" -> 10^5}] Out[3]= 0.18 ``` By default, $10^4$ random permutations are used: ```wl In[4]:= SignedRankTest[data, 0, Method -> {"Permutation", "MonteCarloSamples" -> 10^4}] Out[4]= 0.18 ``` --- Set the seed used for generating random permutations: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {2, 35}]; In[2]:= SignedRankTest[data, 0, Method -> {"Permutation", "RandomSeed" -> 0}] Out[2]= 0.1174 In[3]:= SignedRankTest[data, 0, Method -> {"Permutation", "RandomSeed" -> 9}] Out[3]= 0.1256 ``` #### SignificanceLevel (1) The significance level is used for ``"TestConclusion"`` and ``"ShortTestConclusion"`` : ```wl In[1]:= BlockRandom[SeedRandom[1];data = RandomVariate[NormalDistribution[0, 1], {2, 100}]]; In[2]:= ℋ1 = SignedRankTest[data, .2, "HypothesisTestData", SignificanceLevel -> .05]; In[3]:= ℋ2 = SignedRankTest[data, .2, "HypothesisTestData", SignificanceLevel -> .001]; In[4]:= ℋ1["TestConclusion"]//TraditionalForm Out[4]//TraditionalForm= $$\text{The null hypothesis that }\text{the }\text{median}\text{ difference is }0.2 \text{is rejected at the }5.\text{ percent level }\text{based on the }\text{Signed-Rank}\text{ test.}$$ In[5]:= ℋ2["TestConclusion"]//TraditionalForm Out[5]//TraditionalForm= $$\text{The null hypothesis that }\text{the }\text{median}\text{ difference is }0.2 \text{is not rejected at the }0.1\text{ percent level }\text{based on the }\text{Signed-Rank}\text{ test.}$$ In[6]:= ℋ1["ShortTestConclusion"] Out[6]= "Reject" In[7]:= ℋ2["ShortTestConclusion"] Out[7]= "Do not reject" ``` #### VerifyTestAssumptions (2) Diagnostics can be controlled as a group using ``All`` or ``None`` : ```wl In[1]:= data1 = RandomVariate[ExponentialDistribution[1], 100]; data2 = RandomVariate[ExponentialDistribution[1], 100]; ``` Verify all assumptions: ```wl In[2]:= SignedRankTest[{data1, data2}, 0, VerifyTestAssumptions -> All] ``` SignedRankTest::ntsymmd: The p-value in {1.05156\*10^-8}, resulting from a test for symmetry, is below 0.05\`. The tests in {SignedRank} require that the data is symmetric about a common median. ```wl Out[2]= 0.571663 ``` Check no assumptions: ```wl In[3]:= SignedRankTest[{data1, data2}, 0, VerifyTestAssumptions -> None] Out[3]= 0.571663 ``` --- Diagnostics can be controlled independently: ```wl In[1]:= data1 = RandomVariate[ExponentialDistribution[1], 1000]; data2 = RandomVariate[ExponentialDistribution[1], 1000]; ``` Check for symmetry: ```wl In[2]:= SignedRankTest[{data1, data2}, 0, VerifyTestAssumptions -> "Symmetry"] ``` SignedRankTest::ntsymmd: The p-value in {1.1619\*10^-116}, resulting from a test for symmetry, is below 0.05\`. The tests in {SignedRank} require that the data is symmetric about a common median. ```wl Out[2]= 0.0494125 ``` Set the symmetry assumption to ``True`` : ```wl In[3]:= SignedRankTest[{data1, data2}, 0, VerifyTestAssumptions -> "Symmetry" -> True] Out[3]= 0.0494125 ``` ### 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: ```wl In[1]:= first = {86, 71, 77, 68, 91, 72, 77, 91, 70, 71, 88, 87}; second = {88, 77, 76, 64, 96, 72, 65, 90, 65, 80, 81, 72}; In[2]:= BarChart[first - second] Out[2]= [image] ``` There is insufficient evidence to reject $H_0$ that birth order has no effect on aggressiveness: ```wl In[3]:= SignedRankTest[{first, second}, 0, "TestDataTable", AlternativeHypothesis -> "Greater"] Out[3]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 41.5 | 0.238235 | ``` ### Properties & Relations (8) The ``SignedRankTest`` is generally more powerful than the ``SignTest`` : ```wl In[1]:= data = RandomVariate[NormalDistribution[1, 1], {250, 15}]; In[2]:= Subscript[p, SR] = Table[SignedRankTest[i, VerifyTestAssumptions -> None], {i, data}]; Subscript[p, SGN] = Table[SignTest[i], {i, data}]; In[3]:= Probability[x ≤ .05, x\[Distributed]Subscript[p, SR]]//N Out[3]= 0.924 In[4]:= Probability[x ≤ .05, x\[Distributed]Subscript[p, SGN]]//N Out[4]= 0.8 ``` --- The univariate Wilcoxon signed rank test statistic: ```wl In[1]:= data = RandomVariate[NormalDistribution[], 100]; In[2]:= Subscript[μ, 0] = 0; ``` In the absence of ties, ``Ordering`` can be used to compute ranks: ```wl In[3]:= WSR[data_, μ0_] := With[{dat = data - μ0}, Total[Pick[Sign[dat] * Ordering[Ordering[Abs@dat]], Sign[dat], 1]]] In[4]:= WSR[data, Subscript[μ, 0]] Out[4]= 2858 In[5]:= SignedRankTest[data, Subscript[μ, 0], "TestDataTable"] Out[5]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 2858. | 0.252938 | ``` The asymptotic two-sided $p$-value: ```wl In[6]:= WSRP[data_] := With[{ranks = Sign[data] Ordering[Ordering[Abs[data]]], d = NormalDistribution[]}, 2. Min[CDF[d, (Total[ranks] + 1/Sqrt[ranks.ranks])], SurvivalFunction[d, (Total[ranks] - 1/Sqrt[ranks.ranks])]]] In[7]:= WSRP[data - Subscript[μ, 0]] Out[7]= 0.252938 ``` --- For univariate data, the test statistic is asymptotically normal: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {1000, 100}]; In[2]:= T = Table[Quiet@SignedRankTest[i, Automatic, "TestStatistic"], {i, data}]; In[3]:= DistributionFitTest[T, NormalDistribution[μ, σ]] Out[3]= 0.325971 ``` --- For multivariate data, the test statistic follows a ``ChiSquareDistribution`` under $H_0$ : ```wl In[1]:= data = RandomVariate[BinormalDistribution[.5], {250, 15}]; In[2]:= T = Table[SignedRankTest[i, Automatic, "TestStatistic"], {i, data}]; ``` The degree of freedom is equal to the dimension of the data: ```wl In[3]:= df = Dimensions[data[[1]]][[2]] Out[3]= 2 In[4]:= DistributionFitTest[T, ChiSquareDistribution[df]] Out[4]= 0.276203 ``` --- For multivariate data, the ``SignedRankTest`` effectively tests uniformity about a unit sphere: ```wl In[1]:= μ = {{0, 0}, {.5, .5}, {1, 1}, {10, 10}}; In[2]:= data = Table[RandomVariate[ProductDistribution[NormalDistribution[i[[1]], 1], NormalDistribution[i[[2]], 1]], 100], {i, μ}]; ``` A function for computing the spatial signed ranks of a matrix: ```wl In[3]:= SSR[data_] := Block[{n, p, item, dmt, dpt, dm, dp, out = {}}, {n, p} = Dimensions[data];Do[dmt = dpt = Table[0., {p}];Do[dm = i - j;dp = i + j;dmt = dmt + (If[#1 == 0, dm, (dm/#1)]&)[Sqrt[dm.dm]];dpt = dpt + (If[#1 == 0, dp, (dp/#1)]&)[Sqrt[dp.dp]];, {j, data}];item = (dmt + dpt/2 n);out = Join[{item}, out], {i, data}];out] ``` Deviations from ``μ0`` yield clustering of spatial signed ranks and larger test statistics: ```wl In[4]:= Table[Show[ListPlot[SSR[i], AspectRatio -> 1, PlotStyle -> PointSize[Medium], PlotLabel -> Row[{"SR: ", SignedRankTest[i, Automatic, "TestStatistic"]}], PlotRange -> {{-1, 1}, {-1, 1}}], Graphics[Circle[]]], {i, data}] Out[4]= {[image], [image], [image], [image]} ``` --- The test statistic is affine invariant for multivariate data: ```wl In[1]:= data = RandomVariate[MultinormalDistribution[{0, 0}, IdentityMatrix[2]], 100]; In[2]:= \[ScriptCapitalT] = AffineTransform[{{{1, 2}, {3, 4}}, {1, 2}}]; In[3]:= SignedRankTest[data, {0, 0}, "TestDataTable"] Out[3]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 0.0432064 | 0.978628 | In[4]:= SignedRankTest[\[ScriptCapitalT][data], \[ScriptCapitalT][{0, 0}], "TestDataTable"] Out[4]= | "" | "Statistic" | "P‐Value" | | :------------ | :---------- | :-------- | | "Signed‐Rank" | 0.0432064 | 0.978628 | ``` --- The signed rank test works with the values only when the input is a ``TimeSeries`` : ```wl In[1]:= ts = TemporalData[TimeSeries, {CompressedData["«1188»"], {{0, 100, 1}}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 10.1]; In[2]:= SignedRankTest[ts] Out[2]= 0.783778 In[3]:= SignedRankTest[ts["Values"]] Out[3]= 0.783778 ``` --- The signed rank test works with all the values together when the input is a ``TemporalData`` : ```wl In[1]:= td = TemporalData[Automatic, {CompressedData["«2294»"], {{0, 100, 1}}, 2, {"Continuous", 2}, {"Discrete", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 10.1]; In[2]:= SignedRankTest[td] Out[2]= 0.476672 ``` Test all the values only: ```wl In[3]:= data = td["ValueList"]//Flatten; SignedRankTest[data] Out[3]= 0.476672 ``` Test the difference of the medians of the two paths: ```wl In[4]:= {data1, data2} = td["ValueList"]; In[5]:= SignedRankTest[{data1, data2}] Out[5]= 0.466404 ``` ### Possible Issues (1) ``SignedRankTest`` requires that the data be symmetric about a common median: ```wl In[1]:= BlockRandom[SeedRandom[3];data1 = RandomVariate[NormalDistribution[], 100]; data2 = RandomVariate[ExponentialDistribution[2], 100]]; In[2]:= SignedRankTest[{data1, data2}, VerifyTestAssumptions -> "Symmetry"] ``` SignedRankTest::ntsymmd: The p-value in {0.0097608}, resulting from a test for symmetry, is below 0.05\`. The tests in {SignedRank} require that the data is symmetric about a common median. ```wl Out[2]= 0.0000158295 ``` Use ``SignTest`` if the data is not symmetric: ```wl In[3]:= SignTest[{data1, data2}] Out[3]= 0.00178993 ``` ### Neat Examples (1) Compute the statistic when the null hypothesis $H_0$ is true: ```wl In[1]:= data1 = RandomVariate[NormalDistribution[], {2500, 100}]; data2 = RandomVariate[NormalDistribution[], {2500, 100}]; In[2]:= T1 = MapThread[SignedRankTest[{#1, #2}, 0, "TestStatistic"]&, {data1, data2}]; ``` The test statistic given a particular alternative: ```wl In[3]:= T2 = MapThread[SignedRankTest[{#1, #2}, 1, "TestStatistic"]&, {data1, data2}]; ``` Compare the distributions of the test statistics: In[4]:= SmoothHistogram[{T1,T2},Filling->Axis,PlotLegends->{"Subscript[H, 0] is True","Subscript[H, 0] is False"}] ```wl Out[4]= [image] ``` ## See Also * [`HypothesisTestData`](https://reference.wolfram.com/language/ref/HypothesisTestData.en.md) * [`LocationTest`](https://reference.wolfram.com/language/ref/LocationTest.en.md) * [`LocationEquivalenceTest`](https://reference.wolfram.com/language/ref/LocationEquivalenceTest.en.md) * [`VarianceTest`](https://reference.wolfram.com/language/ref/VarianceTest.en.md) * [`VarianceEquivalenceTest`](https://reference.wolfram.com/language/ref/VarianceEquivalenceTest.en.md) * [`DistributionFitTest`](https://reference.wolfram.com/language/ref/DistributionFitTest.en.md) * [`MannWhitneyTest`](https://reference.wolfram.com/language/ref/MannWhitneyTest.en.md) * [`PairedTTest`](https://reference.wolfram.com/language/ref/PairedTTest.en.md) * [`PairedZTest`](https://reference.wolfram.com/language/ref/PairedZTest.en.md) * [`SignTest`](https://reference.wolfram.com/language/ref/SignTest.en.md) * [`TTest`](https://reference.wolfram.com/language/ref/TTest.en.md) * [`ZTest`](https://reference.wolfram.com/language/ref/ZTest.en.md) ## Related Guides * [Hypothesis Tests](https://reference.wolfram.com/language/guide/HypothesisTests.en.md) ## History * [Introduced in 2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80.en.md)