克鲁斯卡尔-沃利斯检验(Kruskal-Wallis test)亦称“K-W检验”、“H检验”等。用以检验两个以上样本是否来自 同一个概率分布的一种非参数方法。被检验的几个样本必须是独立的或不相关的。与此检验对等的参数检验是单因素方差分析，但与之不同的是，K-W检验不假设样本来自正态分布。它的原假设是各样本服从的概率分布具有相同的中位数，原假设被拒绝意味着至少一个样本的概率分布的中位数不同于其他样本。此检验并未识别出这些差异发生在哪些样本之间以及差异的大小1。

基本介绍

克鲁斯卡尔-沃利斯检验是一种秩检验，是威尔科克逊检验的推广， 用于多个连续型独立样本的比较。方差分析(ANOVA)程序关注的是，几个总体的均值是否相等。数据是间隔测量尺度或比率测量尺度的数据。另外还要假定这些总体服从正态概率分布，并且有相等的标准差。如果数据是顺序测量尺度的和(或)总体不服从正态分布会怎样呢?W.H.克鲁斯卡尔(Kruskal)和W.A.沃利斯(Wallis)于1952年提出了仅仅要求顺序(排序)测量尺度数据的非参数检验。不需要对总体分布形态做任何假定。该检验被称为克鲁斯卡尔-沃利斯单因素秩方差分析(Kruskal-Wallis one-way analysis of variance by ranks)。

为了利用克鲁斯卡尔-沃利斯检验，从总体中抽取的样本必须是独立的。例如，从三个组(经理、员工、管理人员)中抽取样本，并且进行访谈。一组人员(如经理)的回答决不能影响其他两组的回答。

为了计算克鲁斯卡尔-沃利斯检验统计量：①合并所有的样本；②将合并后的样本值从低到高排序；③将排序后的值用秩代替，从最小值1开始。

要应用方差分析技术，我们假定: (1) 总体都服从正态分布； (2) 这些总体有相等的标准差；(3) 样本是独立抽取的。如果这些假定都满足，我们可以利用F分布作为检验统计量。如果这些假定不能被满足，我们应用不依赖于分布的克鲁斯卡尔-沃利斯检验2。

检验步骤

假设有m个相互独立的简单随机样本(X1，…，Xni) (i=1，…， m)2。

检验步骤：

1)将各样本全部 个观测值按递增顺序排成一 列；

2)以Ri(i=1，…，m)表示第 i个样本的ni个观测值X1，…，Xni在此排列中的秩的和；

3) 计算统计量

假如各样本有r个相同数据，设t1(i=1，…，r)是各样本的第i个公共观测值在全部N观测值中出现 的次数，则计算如下修正统计量

(当N充分大时H及H′近似服从分布，自由度v=m-1)；

4)对于 给定的显著性水平α和自由度v= m-1，由附表2查出分布上侧分位数 当 (或 )时，认为m个样本不全来自同 一总体(无齐一性)，否则可以利用 概率积分表(附表1)计算检验的拟合优度 (参见“拟合优度检验”)。

附表1 x²概率积分的p值p=P{x²≥c} (υ——自由度)
v p c12345678
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

22

24

26

28

30

.3173

.1573

.0833

.0455

.0254

.0143

.0082

.0047

.0027

.0016

.0009

.0005

.0003

.0002

.0001

.0001

.0000

.6065

.3679

.2231

.1353

.0821

.0498

.0302

.0183

.0111

.0067

.0041

. 0025

.0015

.0009

.0006

，0003

.0002

.0001

.O001

.0000

.0000

.0000

.0000

.0000

.0000

.8013

.5724

.3916

.2615

.1718

.1116

.0719

.0460

.0293

.0186

.0117

.0074

.0046

.0029

.0018

.0011

.0007

.0004

.0003

.0002

.0001

.0000

.0000

.0000

.0000

.9098

.7358

.5578

.4060

.2873

.1992

.1359

.0916

.0611

.0404

.0266

.0174

.0113

.0073

.0047

.0030

.0019

.0012

.0008

.0005

.0002

.0001

.0000

.0000

.0000

.9626

.8492

.7000

.5494

.4159

.3062

.2206

.1562

.1091

.0752

.0514

.0348

.0234

.0156

.0104

.0068

.0045

.0029

.0019

.0013

.0005

.0002

.0001

.0000

.0000

.9856

.9197

.8089

.6767

.5438

.4232

.3209

.2381

.1736

.1247

.0884

.0620

.0430

.0296

.0203

.0138

.0093

.0062

.0042

.0028

.0012

.0005

.0002

.0001

.0000

.9948

.9598

.8850

.7798

.6600

.5398

.4289

.3326

.2527

.1886

.1386

.1006

.0721

.0512

.0360

.0251

.0174

.0120

.0082

.0056

.0025

.0011

.0005

.0002

.0001

.9983

.9810

.9344

.8571

.7576

.6472

.5366

.4335

.3423

.2650

.2017

.1512

.1119

.0818

.0592

.0424

.0301

.0212

.0149

.0103

.0049

.0023

.0011

.0005

.0002

(续)


　　
v p c9101112131415
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

22

24

26

28

30

.9994

.9915

.9643

.9114

.8343

.7399

.6371

.5342

.4373

.3505

.2757

.2133

.1626

.1223

.0909

.0669

.0487

.0352

.0252

.0179

.0089

.0043

.0020

.0010

.0004

.9998

.9963

.9814

.9474

.8912

.8153

.7254

.6288

.5321

.4405

.3575

.2851

.2237

.1730

.1321

.0996

.0744

.0550

.0403

.0293

.0151

.0076

.0037

.0018

.0009

.9999

.9985

.9907

.9699

.9312

.8734

.7991

.7133

.6219

.5304

.4433

.3636

.2933

.2330

.1825

.1411

.1079

.0816

.0611

.0453

.0244

.0127

.0065

.0032

.0016

1.0000

.9994

.9955

.9834

.9580

.9161

.8576

.7851

.7029

.6160

.5289

.4457

.3690

.3007

.2414

.1912

.1496

.1157

.0885

.0671

.0375

.0203

.0107

.0055

.0028

1.0000

.9998

.9979

.9912

.9752

.9462

.9022

.8436

.7729

.6939

.6108

.5276

.4478

.3738

.3074

.2491

.1993

.1575

.1231

.0952

.0554

.0311

.0170

.0091

.0047

1.0000

.9999

.9991

.9955

.9858

.9665

.9347

.8893

.8311

.7622

.6860

.6063

.5265

.4497

.3782

.3134

.2562

.2068

.1649

.1301

.0786

.0458

.0259

.0142

.0076

1.0000

1.0000

.9996

.9977

.9921

.9798

.9577

.9238

.8775

.8197

.7526

.6790

.6023

.5255

. 4514

.3821

.3189

.2627

.2137

.1719

.1078

.0651

.0380

.0216

.0119

当υ≥45时，使用近似公式：

其中υp，是N(0，1) 的双侧分位数2。

附表2 X²分布上侧分位数X²n，υ(1≤υ≤45)
α v0.9950.9900.9750.950.900.700.50
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

4×10-5

.010

.072

.207

.412

.676

.989

1.443

1.537

2.651

2.306

3.470

3.565

4.570

4.106

5.241

5.796

6.562

6.448

7.434

2×10-4

.020

.115

.297

.554

.872

1.239

1.646

2.088

2.558

3.053

3.571

4.107

4.660

5.229

5.812

6.408

7.015

7.633

8.260

.001

.051

.216

.484

.831

1.237

1.690

2.180

2.700

3.247

3.816

4.404

5.009

5.629

6.262

6.908

7.564

8.231

8.907

9.591

.004

.103

.352

.711

1.145

1.635

2.167

2.733

3.325

3.940

4.575

5.226

5.892

6.571

7.261

7.962

8.672

9.390

10.117

10.851

.016

.211

.584

1.064

1.610

2.204

2.833

3.490

4.168

4.865

5.578

6.304

7.042

7.790

8.547

9.312

10.085

10.865

11.651

12.443

.148

.713

1.424

2.195

3.000

3.828

4.671

5.527

6.393

7.267

8.148

9.034

9.926

10.821

11.721

12.624

13.531

14.440

15.352

16.266

.455

1.386

2.366

3.357

4.351

5.348

6.346

7.344

8.343

9.342

10.341

11.340

12.340

13.339

14.339

15.338

16.338

17.338

18.338

19.337

(续)

α v0.300.100.050.0250.010.0050.001
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1.074

2.408

3.665

4.878

6.064

7.231

8.383

9.524

10.656

11.781

12.899

14.011

15.119

16.222

17.322

18.418

19.511

20.601

21.689

22.775

2.706

4.605

6.251

7.779

9.236

10.645

12.017

13.362

14.684

15.987

17.275

18.549

19.812

21.064

22.307

23.542

24.769

25.989

27.204

28.412

3.841

5.991

7.815

9.488

11.070

12.592

14.067

15.507

16.919

18.307

19.675

21.026

22.362

23.685

24.996

26.296

27.587

28.869

30.144

31.410

5.024

7.378

9.348

11.143

12.832

14.449

16.013

17.535

19.023

20.483

21.920

23.336

24.736

26.119

27.488

28.845

30.191

31.526

32.852

34.170

6.635

9.210

11.345

13.277

15.086

16.912

18.475

20.090

21.666

23.209

24.725

26.217

27.688

29.141

30.578

32.000

33.409

34.805

36.191

37.566

7.879

10.597

12.838

14.860

16.750

18.548

20.278

21.955

23.589

25.188

26.757

28.300

29.819

31.319

32.801

34.267

35.718

37.156

38.582

39.997

10.828

13.816

16.266

18.467

20.515

22.458

24.322

26.125

27.877

29.588

31.264

32.909

34.528

36.123

37.697

39.252

40.790

42.312

43.820

45.315

(续)

α v0.9950.9900.9750.950.900.700.50
21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

8.430

8.346

9.062

9.688

10.025

11.061

11.808

12.461

13.121

13.787

14.458

15.134

15.815

16.01

17.192

17.887

18.586

19.289

19.996

20.707

21.421

22.138

22.859

23.584

24.311

8.897

9.542

10.196

10.856

11.524

12.198

12.879

13.565

14.256

14.953

15.655

16.362

17.073

17.789

18.509

19.233

19.960

20.691

21.426

22.164

22.906

23.650

24.398

25.148

25.901

10.283

10.982

11.688

12.401

13.120

13.844

14.573

15.308

16.047

16.791

17.539

18.291

19.047

19.806

20.569

21.336

22.106

22.878

23.654

24.433

24.215

25.999

26.785

27.575

28.366

11.591

12.338

13.091

13.848

14.611

15.379

16.151

16.928

17.708

18.493

19.281

20.072

20.867

21.664

22.465

23.269

24.075

24.884

25.695

26.509

27.326

28.144

Z8.965

29.787

30.612

13.240

14.041

14.848

15.659

16.473

17.292

18.114

18.939

19.768

20.599

21.434

22.271

23.110

23.952

24.797

25.643

26.492

27.343

28.196

29.051

29.907

30.765

31.625

32.487

33.350

17.182

18.101

19.021

19.943

20.867

21.792

22.719

23. 647

24.577

25.508

26.440

27.373

28.307

29.242

30.178

31.115

32.053

32.992

33.932

34.872

35.813

36.755

37.698

38.641

39.585

20.337

21.337

22.337

23.337

24.337

25.336

26.336

27.336

28.336

29.336

30.336

31.336

32.336

33.336

34.336

35.336

36.336

37.335

38.335

39.335

40.335

41.335

42.335

43.335

44.335

(续)

α v0.300.100.050.0250.010.0050.001
21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

23.858

24.939

26.018

27.096

28.172

29.246

30.319

31.391

32.461

33.530

34.598

35.665

36.731

37.795

38.859

39.922

40.984

42.045

43.105

44.165

45.224

46.282

47.339

48.396

49.452

29.615

30.813

32.007

33.196

34.382

35.563

36.741

37.916

39.087

40.256

41.422

42.585

43.745

44.903

46.059

47.212

48.363

49.513

50.660

51.805

55.949

54.090

55.230

56.369

57.505

32.671

33.924

35.172

36.415

37.652

38.885

40.113

41.337

42.557

43.773

44.985

46.194

47.400

48.602

49.802

50.998

52.192

53.384

54.572

55.758

56.942

58.124

59.304

60.481

61.656

35.479

36.781

38.076

39.364

40.646

41.923

43.194

44.461

45.722

46.979

48.232

49.480

50.725

51.966

53.203

54.437

55.668

56.895

58.120

59.342

60.561

61.777

62.990

64.201

65.410

38.932

40.289

41.638

42.980

44.314

45.642

46.963

48.278

49.588

50.892

52.191

53.486

54.776

56.061

57.342

58.619

59.892

61.162

62.428

63.691

64.950

66.206

67.459

68.709

69.957

41.401

42.796

44.181

45.558

46.928

48.290

49.645

50.993

52.336

53.672

55.003

56.328

57.648

58.964

60.275

61.581

62.882

64.181

65.476

66.766

68.053

69.336

70. 616

71.893

73.166

46.797

48.268

49.728

51.179

52.618

54.052

55.476

56.892

58.301

59.703

61.098

62.487

63.870

65.247

66.619

67.985

69.346

70.703

72.055

73.402

74. 745

76.084

77.419

78.749

80.077

本词条内容贡献者为:

任毅如 - 副教授 - 湖南大学

克鲁斯卡尔-沃利斯检验

图文简介

克鲁斯卡尔-沃利斯检验(Kruskal-Wallis test)亦称“K-W检验”、“H检验”等。用以检验两个以上样本是否来自 同一个概率分布的一种非参数方法。被检验的几个样本必须是独立的或不相关的。与此检验对等的参数检验是单因素方差分析，但与之不同的是，K-W检验不假设样本来自正态分布。它的原假设是各样本服从的概率分布具有相同的中位数，原假设被拒绝意味着至少一个样本的概率分布的中位数不同于其他样本。此检验并未识别出这些差异发生在哪些样本之间以及差异的大小。