克魯斯卡爾-沃利斯檢驗

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
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

(續)

α v	0.995	0.990	0.975	0.95	0.90	0.70	0.50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	4×10 .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 .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

α v	0.30	0.10	0.05	0.025	0.01	0.005	0.001
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

α v	0.995	0.990	0.975	0.95	0.90	0.70	0.50
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

α v	0.30	0.10	0.05	0.025	0.01	0.005	0.001
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

克魯斯卡爾-沃利斯檢驗

基本介紹

基本介紹

檢驗步驟

相關詞條

熱門詞條

v p c	9	10	11	12	13	14	15
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