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VarianceTest

VarianceTest[data]

检验 data 的方差是否为1.

VarianceTest[{data₁,data₂}]

检验 data₁ 和 data₂ 的方差是否相等.

VarianceTest[dspec,σ₀²]

对 σ₀² 检验一个离散量数.

VarianceTest[dspec,σ₀²,"property"]

返回 "property" 的值.

范例

打开所有单元关闭所有单元

基本范例 (3)

检验两个总体方差的相等性：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 100}];

Wolfram Language code: VarianceTest[data]

创建一个 HypothesisTestData 对象，用以进行进一步属性提取：

Wolfram Language code: ℋ = VarianceTest[data, Automatic, "HypothesisTestData"]

完全检验表格：

Wolfram Language code: ℋ["TestDataTable", All]

将一个总体的方差和一个特定值比较：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[1, 2], 10^3];

Wolfram Language code: VarianceTest[data, 4, "TestDataTable"]

Wolfram Language code: Variance[data]

检验两个总体方差的比率和一个特定值的关系：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[0, 1], 100];
data2 = RandomVariate[NormalDistribution[0, 2.5], 250];

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, "TestDataTable"]

Wolfram Language code: Variance[data1] / Variance[data2]

对备择假设进行检验：

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, AlternativeHypothesis -> "Less"]

范围 (15)

检验 (11)

检验总体的方差是否为 1：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[], 500];
data2 = RandomVariate[NormalDistribution[1, 1 / 3], 500];

在下，值通常较大：

Wolfram Language code: VarianceTest[data1]

当为假时，值通常较小：

Wolfram Language code: VarianceTest[data2]

将一个总体的方差与一个特定值进行比较：

Wolfram Language code:

SeedRandom[2];data1 = RandomVariate[NormalDistribution[0, 2], 10^4];
data2 = RandomVariate[NormalDistribution[0, 2.1], 10^4];

Wolfram Language code: VarianceTest[data1, 2^2]

Wolfram Language code: VarianceTest[data2, 2^2]

比较量数据的方差和一个特殊值：

Wolfram Language code:

SeedRandom["K"];
tempData = RandomVariate[NormalDistribution[Quantity[273, "Kelvins"], Quantity[5, "Kelvins"]], 26]

Wolfram Language code: VarianceTest[tempData, (Quantity[4.8, "Kelvins"])^2]

比较两个总体的方差：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[], 500];
data2 = RandomVariate[StudentTDistribution[5], 500];
data3 = RandomVariate[NormalDistribution[], 350];

当方差相等时，值通常较大：

Wolfram Language code: VarianceTest[{data1, data3}]

当方差不相等时，值通常较小：

Wolfram Language code: VarianceTest[{data1, data2}]

检验两个总体的方差的比率是否为一个特定的值：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[0, 2], 500];
data2 = RandomVariate[NormalDistribution[0, 1], 500];

Wolfram Language code: Subscript[σ, 0] = 4;

以下形式是等价的：

Wolfram Language code: VarianceTest[{data2, data1}, 1 / Subscript[σ, 0]]

Wolfram Language code: VarianceTest[{data1, data2}, Subscript[σ, 0]]

当确定时，应该考虑数据集的顺序：

Wolfram Language code: VarianceTest[{data2, data1}, Subscript[σ, 0]]

使用 Automatic 应用最强大并且适合的检验方法：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 10^4}];

Wolfram Language code: VarianceTest[data, 1, Automatic]

属性 "AutomaticTest" 可以用来确定选择哪个检验方法：

Wolfram Language code: VarianceTest[data, 1, "AutomaticTest"]

执行一个特定的检验，以对等方差进行检验：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 10^3}];

Wolfram Language code: VarianceTest[data, 1, "BrownForsythe"]

可以同时执行任意数目的检验：

Wolfram Language code: VarianceTest[data, 1, {"PValue", {"Levene", "BrownForsythe"}}]

同时执行适用于数据的所有检验方法：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 250}];

Wolfram Language code: VarianceTest[data, 1, All]

使用属性 "AllTests" 来识别使用的检验方法：

Wolfram Language code: VarianceTest[data, 1, "AllTests"]

创建一个 HypothesisTestData 对象，以进行重复属性提取：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 10^4}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

可以提取的属性：

Wolfram Language code: ℋ["Properties"]

从一个 HypothesisTestData 对象提取某些属性：

Wolfram Language code: SeedRandom[2];data = RandomVariate[NormalDistribution[], {2, 10^4}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

来自一个 Levene 检验的值和检验统计量：

Wolfram Language code: ℋ["PValue", "Levene"]

Wolfram Language code: ℋ["TestStatistic", "Levene"]

同时提取任意数目的属性：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 100}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

来自一个 Brown–Forsythe检验的值和检验统计量：

Wolfram Language code: ℋ[{{"PValue", "BrownForsythe"}, {"TestStatistic", "BrownForsythe"}}]

报告 (4)

将来自多个检验的结果制作成表格：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[], {2, 10^3}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

Wolfram Language code: ℋ["TestDataTable"]

所有合适的检验结果组成的完全表：

Wolfram Language code: ℋ["TestDataTable", All]

选定的检验结果组成的表格：

Wolfram Language code: ℋ["TestDataTable", {"Conover", "BrownForsythe"}]

从检验表格获取项目，以生成定制的报告：

Wolfram Language code: data = BlockRandom[SeedRandom[1];RandomVariate[NormalDistribution[], {2, 10^3}]];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

Wolfram Language code: res = ℋ["TestData", All];

Wolfram Language code: ℋ["AllTests"]

值大于 0.05，所以在 0.05 水平下，没有足够的证据拒绝正态性：

Wolfram Language code:

Show[BarChart[res[[All, 2]], ChartLabels -> Placed[ℋ["AllTests"], Center], BarOrigin -> Left], Graphics[{Dashed, InfiniteLine[{{.05, 0}, {.05, 1}}]}]]

对于一个检验或者一组检验，将值制作成表格：

Wolfram Language code: SeedRandom[2];data = RandomVariate[NormalDistribution[], {2, 10^4}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

Wolfram Language code: ℋ["PValueTable", "Levene"]

来自表格的值：

Wolfram Language code: ℋ["PValue", "Levene"]

来自所有适当的检验的值组成的表格：

Wolfram Language code: ℋ["PValueTable", All]

来自部分检验方法的值组成的表格：

Wolfram Language code: ℋ["PValueTable", {"Levene", "Conover", "BrownForsythe"}]

从一个检验或者一组检验报告检验统计量：

Wolfram Language code: SeedRandom[1];data = RandomVariate[CauchyDistribution[1, 2], {2, 10^3}];

Wolfram Language code: ℋ = VarianceTest[data, 1, "HypothesisTestData"];

Wolfram Language code: ℋ["TestStatisticTable"]

来自表格的检验统计量：

Wolfram Language code: ℋ["TestStatistic"]

来自所有适当的检验的检验统计量组成的表格：

Wolfram Language code: ℋ["TestStatisticTable", All]

选项 (10)

AlternativeHypothesis (3)

默认情况下，执行一个双侧检验：

Wolfram Language code:

SeedRandom[1];
data = RandomVariate[NormalDistribution[], 100];

检验与：

Wolfram Language code: VarianceTest[data, 1, AlternativeHypothesis -> "Unequal"]

Wolfram Language code: VarianceTest[data, 1, AlternativeHypothesis -> Automatic]

执行一个双侧检验或者单侧检验：

Wolfram Language code: SeedRandom[1];data = RandomVariate[NormalDistribution[0, 1.25], 100];

检验与：

Wolfram Language code: VarianceTest[data, Automatic, AlternativeHypothesis -> "Unequal"]

检验与：

Wolfram Language code: VarianceTest[data, Automatic, AlternativeHypothesis -> "Less"]

检验与：

Wolfram Language code: VarianceTest[data, Automatic, AlternativeHypothesis -> "Greater"]

当给定空值时，执行带有单侧备择假设的检验：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[0, .9], 100];
data2 = RandomVariate[NormalDistribution[0, 1], 100];

Wolfram Language code: Variance[data1] / Variance[data2]

检验与：

Wolfram Language code: VarianceTest[{data1, data2}, 1, AlternativeHypothesis -> "Less"]

检验与：

Wolfram Language code: VarianceTest[{data1, data2}, 1.5, AlternativeHypothesis -> "Less"]

SignificanceLevel (3)

对诊断检验，设置显著性水平：

Wolfram Language code: SeedRandom[1];data = BlockRandom[SeedRandom[5];RandomVariate[StudentTDistribution[3], 50]];

Wolfram Language code: VarianceTest[data, 1, "FisherRatio", SignificanceLevel -> .005]

默认情况下，使用 0.05：

Wolfram Language code: VarianceTest[data, 1, "FisherRatio", SignificanceLevel -> Automatic]

设置显著性水平可能改变自动选择的检验方法：

Wolfram Language code: data = BlockRandom[SeedRandom[1];RandomVariate[StudentTDistribution[3], {2, 50}]];

Wolfram Language code: VarianceTest[data, 1, "AutomaticTest", SignificanceLevel -> .0001]

默认情况下会选择一个基于秩的检验方法：

Wolfram Language code: VarianceTest[data, 1, "AutomaticTest", SignificanceLevel -> Automatic]

显著性水平也用于 "TestConclusion" 和 "ShortTestConclusion":

Wolfram Language code: data = BlockRandom[SeedRandom[1];RandomVariate[NormalDistribution[0, 1], 100]];

Wolfram Language code: ℋ1 = VarianceTest[data, 1.5, "HypothesisTestData", SignificanceLevel -> .05];

Wolfram Language code: ℋ2 = VarianceTest[data, 1.5, "HypothesisTestData", SignificanceLevel -> .01];

Wolfram Language code: ℋ1["TestConclusion"]//TraditionalForm

Wolfram Language code: ℋ2["TestConclusion"]//TraditionalForm

Wolfram Language code: ℋ1["ShortTestConclusion"]

Wolfram Language code: ℋ2["ShortTestConclusion"]

VerifyTestAssumptions (4)

可以使用 All 或 None 将诊断作为一个组来控制：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[StudentTDistribution[3], 1000];
data2 = RandomVariate[NormalDistribution[0, Sqrt[3]], 1000];

验证所有假定：

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, "Levene", VerifyTestAssumptions -> All]

不检查假定：

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, "Levene", VerifyTestAssumptions -> None]

可以独立控制诊断：

Wolfram Language code:

SeedRandom[2];data1 = RandomVariate[StudentTDistribution[3], 1000];
data2 = RandomVariate[NormalDistribution[0, Sqrt[3]], 1000];

假定正态性，但是检查对称性：

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, "Levene", VerifyTestAssumptions -> "Symmetry"]

只检查正态性：

Wolfram Language code: VarianceTest[{data1, data2}, 1 / 4, "Levene", VerifyTestAssumptions -> "Normality"]

检验假设值可以明确设置：

Wolfram Language code: data = RandomVariate[CauchyDistribution[0, 1], 100];

Wolfram Language code: VarianceTest[data, 1, "TestDataTable"]

Wolfram Language code: VarianceTest[data, 1, "TestDataTable", VerifyTestAssumptions -> "Normality" -> True]

对于模拟而言，避免诊断检验往往是有用的：

Wolfram Language code: data = RandomVariate[NormalDistribution[], {1000, 100}];

Wolfram Language code: AbsoluteTiming[T = Quiet@VarianceTest[#, Automatic, {"TestStatistic", "FisherRatio"}]& /@ data;]

由于检验的假定在设计上被抑制，因此可以节省大量时间：

Wolfram Language code:

AbsoluteTiming[T2 = Quiet@VarianceTest[#, Automatic, {"TestStatistic", "FisherRatio"}, VerifyTestAssumptions -> None]& /@ data;]

结果是完全相同的：

Wolfram Language code: SmoothHistogram[{T, T2}]

应用 (2)

检验某些总体的方差是否相同：

Wolfram Language code:

SeedRandom[1];data1 = RandomVariate[NormalDistribution[0, 1], 100];
data2 = RandomVariate[NormalDistribution[0, 1], 100];
data3 = RandomVariate[NormalDistribution[0, 3], 100];

Wolfram Language code: BoxWhiskerChart[{data1, data2, data3}, "Notched", ChartStyle -> "SolarColors"]

前两个总体的方差相似：

Wolfram Language code: VarianceTest[{data1, data2}, Automatic, "TestDataTable"]

第三个总体与第一个总体的方差不同：

Wolfram Language code: VarianceTest[{data1, data3}, Automatic, "TestDataTable"]

两个班共 25 个小学生用同一把尺子测量一个长度为 2.15 厘米的物体：

Wolfram Language code:

group1 = {2.28, 2.13, 2.21, 2., 2.06, 2.09, 2.12, 2.27, 2.17, 2.09, 2.02, 2.28, 2.28, 2.03, 2.11, 2.05, 2.06, 2.28, 2.1, 2.35, 2.07, 2.22, 2.19, 2.33, 2.02};

Wolfram Language code:

group2 = {2.03, 2.12, 1.92, 1.96, 2.05, 2.05, 2.02, 2.12, 2.02, 1.95, 2., 2.03, 1.93, 1.96, 2.08, 1.92, 2.12, 2.01, 1.84, 2.06, 2., 1.95, 1.90, 2., 1.94};

Wolfram Language code: {Variance[group1], Variance[group2]}

第二组有较高的精度：

Wolfram Language code: VarianceTest[{group1, group2}, 1, "TestDataTable"]

利用平方差比较两组的准确度：

Wolfram Language code: SqrErr1 = (group1 - 2.15)^2;SqrErr2 = (group2 - 2.15)^2;

Wolfram Language code: {Mean[SqrErr1], Mean[SqrErr2]}

第一组明显更准确：

Wolfram Language code: LocationTest[{SqrErr1, SqrErr2}, 0, "TestDataTable"]

属性和关系 (7)

值表明假阳性（第一类错误）的期望比率：

Wolfram Language code: data = RandomVariate[NormalDistribution[], {1000, 10}];

Wolfram Language code: pvals = Table[VarianceTest[i, 1, "FisherRatio", VerifyTestAssumptions -> None], {i, data}];

把检验样本的大小设置为 0.05 时，错误否定的概率为大概为 5% ：

Wolfram Language code: Probability[x ≤ 0.05, xpvals]//N

各检验的功效指的是当为假时，拒绝它的概率：

Wolfram Language code: n = 100;

Wolfram Language code:

data1 = RandomVariate[NormalDistribution[0, .75], {1000, n}];
data2 = RandomVariate[NormalDistribution[0, 1], {1000, n}];

Wolfram Language code: tests = VarianceTest[{data1[[1]], data2[[1]]}, 1, "AllTests"]

Wolfram Language code: pvs = MapThread[VarianceTest[{#1, #2}, 1, tests, VerifyTestAssumptions -> None]&, {data1, data2}];

Wolfram Language code: α = {.001, .005, .01, .025, .05, .1};

检验功效有六种不同的水平. Siegel–Tukey 检验的功效最低：

Wolfram Language code:

Grid[Join[{Join[{""}, α]}, Table[Join[{tests[[i]]}, (Probability[x ≤ #1, xpvs[[All, i]]]&) /@ α], {i, Length[tests]}]], Alignment -> Left, Background -> {None, {StandardGray, {StandardBlue, StandardPink}}}, Dividers -> {{False, True}, {False, True}}]//N

检验功效与样本大小成正比：

Wolfram Language code: n = 10;

Wolfram Language code:

data1 = RandomVariate[NormalDistribution[0, .75], {1000, n}];
data2 = RandomVariate[NormalDistribution[0, 1], {1000, n}];

Wolfram Language code: tests = VarianceTest[{data1[[1]], data2[[1]]}, 1, "AllTests"]

Wolfram Language code: pvs = MapThread[VarianceTest[{#1, #2}, 1, tests, VerifyTestAssumptions -> None]&, {data1, data2}];

Wolfram Language code: α = {.001, .005, .01, .025, .05, .1};

与前面的例子相比，该检验的功效较低：

Wolfram Language code:

Grid[Join[{Join[{""}, α]}, Table[Join[{tests[[i]]}, (Probability[x ≤ #1, xpvs[[All, i]]]&) /@ α], {i, Length[tests]}]], Alignment -> Left, Background -> {None, {StandardGray, {StandardPurple, StandardBlue}}}, Dividers -> {{False, True}, {False, True}}]//N

双侧值是两个单侧值中较小的值的两倍：

Wolfram Language code:

data1 = RandomVariate[NormalDistribution[0, 1], 100];
data2 = RandomVariate[NormalDistribution[0, .5], 100];

Wolfram Language code: twoSided = VarianceTest[{data1, data2}, 1, All, AlternativeHypothesis -> "Unequal"];

Wolfram Language code:

minOneSided = Min /@ Transpose[{VarianceTest[{data1, data2}, 1, All, AlternativeHypothesis -> "Less"], VarianceTest[{data1, data2}, 1, All, AlternativeHypothesis -> "Greater"]}];

Wolfram Language code: twoSided - 2 minOneSided

对于单个样本，Brown–Forsythe 和 Levene 检验等价于 Fisher Ratio 检验：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];

Wolfram Language code: VarianceTest[data, Automatic, {"TestDataTable", {"FisherRatio", "BrownForsythe", "Levene"}}]

当输入是一个 TimeSeries 时，方差检验只作用于数值：

Wolfram Language code:

ts = TemporalData[TimeSeries, {{{1.224578634529677, 0.47929635789978015, 0.6572781300178168, 
    0.21496048742669355, 0.7299608014554928, -0.2495111111278263, -1.3286551762002712, 
    0.552725018274874, 0.19272112205837066, 1.1809144012420882, -1.1671 ... 40938613662046, 1.052394590214582, 0.9345044123980388, 0.38537803109557855, 
    -0.48660931166089394, -0.71203560340161}}, {{0, 100, 1}}, 1, {"Continuous", 1}, 
  {"Discrete", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 10.1];

Wolfram Language code: VarianceTest[ts, Automatic, {"TestDataTable", All}]

Wolfram Language code: VarianceTest[ts["Values"], Automatic, {"TestDataTable", All}]

Wolfram Language code: %% == %

当输入为 TemporalData 时，方差检验同时作用于所有数值：

Wolfram Language code:

td = TemporalData[Automatic, {{{-0.25275046867718637, -0.7175779198306353, -1.9370139837317764, 
    0.006665621735740701, -0.3730807122324292, 0.6740106823161018, 0.8562214990564344, 
    0.955785083955732, 1.7020898014886303, 1.8523009430646802, 0.244 ... 3951759101545, -1.1611722313627828, 1.1602446901533021, 
    1.1052173095128992, 1.1089143920161917, -0.13837156328402}}, {{0, 100, 1}}, 2, 
  {"Continuous", 2}, {"Discrete", 1}, 1, {ValueDimensions -> 1, ResamplingMethod -> None}}, False, 
 10.1];

Wolfram Language code: VarianceTest[td]

只检验所有数值：

Wolfram Language code:

data = td["ValueList"]//Flatten;
VarianceTest[data]

检验两条路径的方差是否相等：

Wolfram Language code: {data1, data2} = td["ValueList"];

Wolfram Language code: VarianceTest[{data1, data2}]

可能存在的问题 (2)

对于单个样本，Conover 和 Siegel–Tukey 检验未定义：

Wolfram Language code: data = RandomVariate[NormalDistribution[], 100];

Wolfram Language code: VarianceTest[data, Automatic, {"TestDataTable", {"Conover", "SiegelTukey"}}]

某些检验假定数据服从正态分布：

Wolfram Language code: data = RandomVariate[LaplaceDistribution[1, 2], {2, 100}];

Wolfram Language code: VarianceTest[data, Automatic, {"TestDataTable", {"BrownForsythe", "FisherRatio", "Levene"}}]

Conover 检验和 Siegel–Tukey 检验不假定正态性：

Wolfram Language code: VarianceTest[data, Automatic, {"TestDataTable", {"Conover", "SiegelTukey"}}]

巧妙范例 (1)

当零假设为真时，计算统计量：

Wolfram Language code: data = RandomVariate[NormalDistribution[], {250, 100}];

Wolfram Language code: T1 = VarianceTest[#, 1, "TestStatistic", VerifyTestAssumptions -> None]& /@ data;

给定一个特定的备择假设的检验统计量：

Wolfram Language code: T2 = VarianceTest[#, 2, "TestStatistic", VerifyTestAssumptions -> None]& /@ data;

比较检验统计量的分布：

Wolfram Language code:

SmoothHistogram[{T1, T2}, Filling -> Axis, PlotLegends -> {"SubscriptBox[H, 0] is True", "SubscriptBox[H, 0] is False"}]

Top

	"AllTests"	所有可用检验的列表
	"AutomaticTest"	如果使用 Automatic，选择的检验方法
	"DegreesOfFreedom"	检验中所用的自由度
	"PValue"	值列表
	"PValueTable"	值组成的格式化表格
	"ShortTestConclusion"	检验结论的简短描述
	"TestConclusion"	检验结论的描述
	"TestData"	检验统计量和值的成对列表
	"TestDataTable"	值和检验统计量组成的格式化表格
	"TestStatistic"	检验统计量组成的列表
	"TestStatisticTable"	检验统计量组成的格式化表格

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VarianceTest

更多信息和选项

范例

基本范例 (3)

范围 (15)

检验 (11)

报告 (4)

选项 (10)

AlternativeHypothesis (3)

SignificanceLevel (3)

VerifyTestAssumptions (4)

应用 (2)

属性和关系 (7)

可能存在的问题 (2)

巧妙范例 (1)

文本

CMS

APA

BibTeX

BibLaTeX


	data
	{data₁,data₂}

"BrownForsythe"	稳健	稳健 Levene 检验
"Conover"	对称性	基于 data 的平方秩
"FisherRatio"	正态性	基于
"Levene"	稳健、对称性	比较单个和群体方差
"SiegelTukey"	对称性	基于合并 data 的秩

	"Normality"	验证所有数据都服从正态分布
	"Symmetry"	验证是否关于公共中位数对称

VarianceTest

更多信息和选项

范例

基本范例 (3)

范围 (15)

检验 (11)

报告 (4)

选项 (10)

AlternativeHypothesis (3)

SignificanceLevel (3)

VerifyTestAssumptions (4)

应用 (2)

属性和关系 (7)

可能存在的问题 (2)

巧妙范例 (1)

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