克魯斯卡爾-沃利斯檢驗

克魯斯卡爾-沃利斯檢驗(Kruskal-Wallis test)亦稱“K-W檢驗”、“H檢驗”等。用以檢驗兩個以上樣本是否來自 同一個機率分布的一種非參數方法。被檢驗的幾個樣本必須是獨立的或不相關的。與此檢驗對等的參數檢驗是單因素方差分析，但與之不同的是，K-W檢驗不假設樣本來自常態分配。它的原假設是各樣本服從的機率分布具有相同的中位數，原假設被拒絕意味著至少一個樣本的機率分布的中位數不同於其他樣本。此檢驗並未識別出這些差異發生在哪些樣本之間以及差異的大小。

基本介紹

• 中文名：克魯斯卡爾-沃利斯檢驗
• 外文名：Kruskal-Wallis test
• 別稱：K-W檢驗、H檢驗
• 所屬學科：數學
• 所屬問題：統計學（試驗方法）

基本介紹

克魯斯卡爾-沃利斯檢驗是一種秩檢驗，是威爾科克遜檢驗的推廣， 用於多個連續型獨立樣本的比較。方差分析(ANOVA)程式關注的是，幾個總體的均值是否相等。數據是間隔測量尺度或比率測量尺度的數據。另外還要假定這些總體服從正態機率分布，並且有相等的標準差。如果數據是順序測量尺度的和(或)總體不服從常態分配會怎樣呢?W.H.克魯斯卡爾(Kruskal)和W.A.沃利斯(Wallis)於1952年提出了僅僅要求順序(排序)測量尺度數據的非參數檢驗。不需要對總體分布形態做任何假定。該檢驗被稱為克魯斯卡爾-沃利斯單因素秩方差分析(Kruskal-Wallis one-way analysis of variance by ranks)。

為了利用克魯斯卡爾-沃利斯檢驗，從總體中抽取的樣本必須是獨立的。例如，從三個組(經理、員工、管理人員)中抽取樣本，並且進行訪談。一組人員(如經理)的回答決不能影響其他兩組的回答。

為了計算克魯斯卡爾-沃利斯檢驗統計量：①合併所有的樣本；②將合併後的樣本值從低到高排序；③將排序後的值用秩代替，從最小值1開始。

要套用方差分析技術，我們假定: (1) 總體都服從常態分配； (2) 這些總體有相等的標準差；(3) 樣本是獨立抽取的。如果這些假定都滿足，我們可以利用F分布作為檢驗統計量。如果這些假定不能被滿足，我們套用不依賴於分布的克魯斯卡爾-沃利斯檢驗。

檢驗步驟

假設有m個相互獨立的簡單隨機樣本(X₁，…，X_ni) (i=1，…， m)。

檢驗步驟：

1)將各樣本全部 個觀測值按遞增順序排成一 列；

2)以R_i(i=1，…，m)表示第 i個樣本的n_i個觀測值X₁，…，X_ni在此排列中的秩的和；

3) 計算統計量

假如各樣本有r個相同數據，設t₁(i=1，…，r)是各樣本的第i個公共觀測值在全部N觀測值中出現 的次數，則計算如下修正統計量

(當N充分大時H及H′近似服從分布，自由度v=m-1)；

4)對於 給定的顯著性水平α和自由度v= m-1，由附表2查出分布上側分位數 當 (或 )時，認為m個樣本不全來自同 一總體(無齊一性)，否則可以利用 機率積分表(附表1)計算檢驗的擬合優度 (參見“擬合優度檢驗”)。

附表1 x2機率積分的p值p=P{x2≥c} (υ——自由度)

v p c	1	2	3	4	5	6	7	8
12; 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

(續)

v p c	9	10	11	12	13	14	15
12; 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	1.00001.0000; .9996; .9977; .9921; .9798; .9577; .9238; .8775; .8197; .7526; .6790; .6023; .5255; . 4514; .3821; .3189; .2627; .2137; .1719; .1078; .0651; .0380; .0216; .0119

當υ≥45時，使用近似公式：

其中υ_p，是N(0，1) 的雙側分位數。

圖1

附表2 X2分布上側分位數X2n，υ(1≤υ≤45)

α v	0.995	0.990	0.975	0.95	0.90	0.70	0.50
12; 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	.4551.386; 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

(續)

α v	0.30	0.10	0.05	0.025	0.01	0.005	0.001
12; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20	1.0742.408; 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	2.7064.605; 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	3.8415.991; 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	5.0247.378; 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	6.6359.210; 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	7.87910.597; 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	10.82813.816; 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

(續)

α v	0.995	0.990	0.975	0.95	0.90	0.70	0.50
2122; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45	8.4308.346; 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	8.8979.542; 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	10.28310.982; 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	11.59112.338; 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	13.24014.041; 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	17.18218.101; 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	20.33721.337; 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

(續)

α v	0.30	0.10	0.05	0.025	0.01	0.005	0.001
2122; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45	23.85824.939; 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	29.61530.813; 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	32.67133.924; 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	35.47936.781; 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	38.93240.289; 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	41.40142.796; 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	46.79748.268; 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