假设检验之卡方检验

假设检验简介

假设检验(Hypothesis Testing),或者叫做显著性检验(Significance Testing)是数理统计学中根据一定假设条件由样本推断总体的一种方法。其基本原理是先对总体的特征作出某种假设,然后通过抽样研究的统计推理,对此假设应该被拒绝还是接受作出推断。既然以假设为前提,那么在进行检验前需要提出相应的假设:

  • H0:原假设或零假设(null hypothesis),即需要去验证的假设;一般首先认定原假设是正确的,然后根据显著性水平选择是接受还是拒绝原假设。
  • H1:备择假设(alternative hypothesis),一般是原假设的否命题;当原假设被拒绝时,默认接受备择假设。

如原假设是假设总体均值$\mu = 0$,则备择假设为总体均值$\mu \neq 0$,检验的过程就是计算相应的统计量和显著性概率,来验证原假设应该被接受还是拒绝。

卡方检验简介

卡方检验是一种用途很广的计数资料的假设检验方法。它属于非参数检验的范畴,主要是比较两个及两个以上样本率( 构成比)以及两个分类变量的关联性分析。其根本思想就是在于比较理论频数和实际频数的吻合程度或拟合优度问题。

卡方检验是以$\chi^2$分布为基础的一种常用假设检验方法,它的无效假设H0是:观察频数与期望频数没有差别。

该检验的基本思想是:首先假设H0成立,基于此前提计算出$\chi^2$值,它表示观察值与理论值之间的偏离程度。根据$\chi^2$分布及自由度可以确定在H0假设成立的情况下获得当前统计量及更极端情况的概率P。如果P值很小,说明观察值与理论值偏离程度太大,应当拒绝无效假设,表示比较资料之间有显著差异;否则就不能拒绝无效假设,尚不能认为样本所代表的实际情况和理论假设有差别。

$\chi^2$值表示观察值与理论值之问的偏离程度。计算这种偏离程度的基本思路如下。

  • 设A代表某个类别的观察频数,E代表基于H0计算出的期望频数,A与E之差称为残差。
  • 残差可以表示某一个类别观察值和理论值的偏离程度,但如果将残差简单相加以表示各类别观察频数与期望频数的差别,则有一定的不足之处。因为残差有正有负,相加后会彼此抵消,总和仍然为0,为此可以将残差平方后求和。
  • 残差大小是一个相对的概念,相对于期望频数为10时,期望频数为20的残差非常大,但相对于期望频数为1 000时20的残差就很小了。考虑到这一点,人们又将残差平方除以期望频数再求和,以估计观察频数与期望频数的差别。

进行上述操作之后,就得到了常用的$\chi^2$统计量,由于它最初是由英国统计学家Karl Pearson在1900年首次提出的,因此也称之为Pearson $\chi^2$,其计算公式为:

$$\chi^2 = \sum \frac{(A-E)^2}{E}=\sum_{i=1}^{k}\frac{(A_i-E_i)^2}{E_i}=\sum_{i=1}^{k}\frac{(A_i-np_i)^2}{np_i}$$

其中,$A_i$为i水平的观察频数,$E_i$为i水平的期望频数,n为总频数,$p_i$为i水平的期望频率。i水平的期望频数$E_i$等于总频数$n \times i$水平的期望概率$p_i$,k为单元格数。当n比较大时,$\chi^2$统计量近似服从k-1(计算$E_i$时用到的参数个数)个自由度的卡方分布。

由卡方的计算公式可知:

  • 观察频数与期望频数完全一致时,$\chi^2$值为0;
  • 观察频数与期望频数越接近,两者之间的差异越小,$\chi^2$值越小;
  • 观察频数与期望频数差别越大,两者之间的差异越大,$\chi^2$值越大

换言之,大的$\chi^2$值表明观察频数远离期望频数,即表明远离假设。小的$\chi^2$值表明观察频数接近期望频数,接近假设。至于$\chi^2$在每个具体研究中究竟要大到什么程度才能拒绝H0,则要借助于卡方分布求出所对应的P值来确定。

1-1000自由度的卡方临界值查询表:

 显著性水平(a)
自由度0.9950.9750.20.10.050.0250.020.010.0050.0020.001
13.93E-050.0009821.6422.7063.8415.0245.4126.6357.8799.5510.828
20.010.05063.2194.6055.9917.3787.8249.2110.59712.42913.816
30.07170.2164.6426.2517.8159.3489.83711.34512.83814.79616.266
40.2070.4845.9897.7799.48811.14311.66813.27714.8616.92418.467
50.4120.8317.2899.23611.0712.83313.38815.08616.7518.90720.515
60.6761.2378.55810.64512.59214.44915.03316.81218.54820.79122.458
70.9891.699.80312.01714.06716.01316.62218.47520.27822.60124.322
81.3442.1811.0313.36215.50717.53518.16820.0921.95524.35226.124
91.7352.712.24214.68416.91919.02319.67921.66623.58926.05627.877
102.1563.24713.44215.98718.30720.48321.16123.20925.18827.72229.588
112.6033.81614.63117.27519.67521.9222.61824.72526.75729.35431.264
123.0744.40415.81218.54921.02623.33724.05426.21728.330.95732.909
133.5655.00916.98519.81222.36224.73625.47227.68829.81932.53534.528
144.0755.62918.15121.06423.68526.11926.87329.14131.31934.09136.123
154.6016.26219.31122.30724.99627.48828.25930.57832.80135.62837.697
165.1426.90820.46523.54226.29628.84529.6333234.26737.14639.252
175.6977.56421.61524.76927.58730.19130.99533.40935.71838.64840.79
186.2658.23122.7625.98928.86931.52632.34634.80537.15640.13642.312
196.8448.90723.927.20430.14432.85233.68736.19138.58241.6143.82
207.4349.59125.03828.41231.4134.1735.0237.56639.99743.07245.315
218.03410.28326.17129.61532.67135.47936.34338.93241.40144.52246.797
228.64310.98227.30130.81333.92436.78137.65940.28942.79645.96248.268
239.2611.68928.42932.00735.17238.07638.96841.63844.18147.39149.728
249.88612.40129.55333.19636.41539.36440.2742.9845.55948.81251.179
2510.5213.1230.67534.38237.65240.64641.56644.31446.92850.22352.62
2611.1613.84431.79535.56338.88541.92342.85645.64248.2951.62754.052
2711.80814.57332.91236.74140.11343.19544.1446.96349.64553.02355.476
2812.46115.30834.02737.91641.33744.46145.41948.27850.99354.41156.892
2913.12116.04735.13939.08742.55745.72246.69349.58852.33655.79258.301
3013.78716.79136.2540.25643.77346.97947.96250.89253.67257.16759.703
3114.45817.53937.35941.42244.98548.23249.22652.19155.00358.53661.098
3215.13418.29138.46642.58546.19449.4850.48753.48656.32859.89962.487
3315.81519.04739.57243.74547.450.72551.74354.77657.64861.25663.87
3416.50119.80640.67644.90348.60251.96652.99556.06158.96462.60865.247
3517.19220.56941.77846.05949.80253.20354.24457.34260.27563.95566.619
3617.88721.33642.87947.21250.99854.43755.48958.61961.58165.29667.985
3718.58622.10643.97848.36352.19255.66856.7359.89362.88366.63369.346
3819.28922.87845.07649.51353.38456.89657.96961.16264.18167.96670.703
3919.99623.65446.17350.6654.57258.1259.20462.42865.47669.29472.055
4020.70724.43347.26951.80555.75859.34260.43663.69166.76670.61873.402
4121.42125.21548.36352.94956.94260.56161.66564.9568.05371.93874.745
4222.13825.99949.45654.0958.12461.77762.89266.20669.33673.25476.084
4322.85926.78550.54855.2359.30462.9964.11667.45970.61674.56677.419
4423.58427.57551.63956.36960.48164.20165.33768.7171.89375.87478.75
4524.31128.36652.72957.50561.65665.4166.55569.95773.16677.17980.077
4625.04129.1653.81858.64162.8366.61767.77171.20174.43778.48181.4
4725.77529.95654.90659.77464.00167.82168.98572.44375.70479.7882.72
4826.51130.75555.99360.90765.17169.02370.19773.68376.96981.07584.037
4927.24931.55557.07962.03866.33970.22271.40674.91978.23182.36785.351
5027.99132.35758.16463.16767.50571.4272.61376.15479.4983.65786.661
5128.73533.16259.24864.29568.66972.61673.81877.38680.74784.94387.968
5229.48133.96860.33265.42269.83273.8175.02178.61682.00186.22789.272
5330.2334.77661.41466.54870.99375.00276.22379.84383.25387.50790.573
5430.98135.58662.49667.67372.15376.19277.42281.06984.50288.78691.872
5531.73536.39863.57768.79673.31177.3878.61982.29285.74990.06193.168
5632.4937.21264.65869.91974.46878.56779.81583.51386.99491.33594.461
5733.24838.02765.73771.0475.62479.75281.00984.73388.23692.60595.751
5834.00838.84466.81672.1676.77880.93682.20185.9589.47793.87497.039
5934.7739.66267.89473.27977.93182.11783.39187.16690.71595.1498.324
6035.53440.48268.97274.39779.08283.29884.5888.37991.95296.40499.607
6136.30141.30370.04975.51480.23284.47685.76789.59193.18697.665100.888
6237.06842.12671.12576.6381.38185.65486.95390.80294.41998.925102.166
6337.83842.9572.20177.74582.52986.8388.13792.0195.649100.182103.442
6438.6143.77673.27678.8683.67588.00489.3293.21796.878101.437104.716
6539.38344.60374.35179.97384.82189.17790.50194.42298.105102.691105.988
6640.15845.43175.42481.08585.96590.34991.68195.62699.33103.942107.258
6740.93546.26176.49882.19787.10891.51992.8696.828100.554105.192108.526
6841.71347.09277.57183.30888.2592.68994.03798.028101.776106.44109.791
6942.49447.92478.64384.41889.39193.85695.21399.228102.996107.685111.055
7043.27548.75879.71585.52790.53195.02396.388100.425104.215108.929112.317
7144.05849.59280.78686.63591.6796.18997.561101.621105.432110.172113.577
7244.84350.42881.85787.74392.80897.35398.733102.816106.648111.412114.835
7345.62951.26582.92788.8593.94598.51699.904104.01107.862112.651116.092
7446.41752.10383.99789.95695.08199.678101.074105.202109.074113.889117.346
7547.20652.94285.06691.06196.217100.839102.243106.393110.286115.125118.599
7647.99753.78286.13592.16697.351101.999103.41107.583111.495116.359119.85
7748.78854.62387.20393.2798.484103.158104.576108.771112.704117.591121.1
7849.58255.46688.27194.37499.617104.316105.742109.958113.911118.823122.348
7950.37656.30989.33895.476100.749105.473106.906111.144115.117120.052123.594
8051.17257.15390.40596.578101.879106.629108.069112.329116.321121.28124.839
8151.96957.99891.47297.68103.01107.783109.232113.512117.524122.507126.083
8252.76758.84592.53898.78104.139108.937110.393114.695118.726123.733127.324
8353.56759.69293.60499.88105.267110.09111.553115.876119.927124.957128.565
8454.36860.5494.669100.98106.395111.242112.712117.057121.126126.179129.804
8555.1761.38995.734102.079107.522112.393113.871118.236122.325127.401131.041
8655.97362.23996.799103.177108.648113.544115.028119.414123.522128.621132.277
8756.77763.08997.863104.275109.773114.693116.184120.591124.718129.84133.512
8857.58263.94198.927105.372110.898115.841117.34121.767125.913131.057134.745
8958.38964.79399.991106.469112.022116.989118.495122.942127.106132.273135.978
9059.19665.647101.054107.565113.145118.136119.648124.116128.299133.489137.208
9160.00566.501102.117108.661114.268119.282120.801125.289129.491134.702138.438
9260.81567.356103.179109.756115.39120.427121.954126.462130.681135.915139.666
9361.62568.211104.241110.85116.511121.571123.105127.633131.871137.127140.893
9462.43769.068105.303111.944117.632122.715124.255128.803133.059138.337142.119
9563.2569.925106.364113.038118.752123.858125.405129.973134.247139.546143.344
9664.06370.783107.425114.131119.871125126.554131.141135.433140.755144.567
9764.87871.642108.486115.223120.99126.141127.702132.309136.619141.962145.789
9865.69472.501109.547116.315122.108127.282128.849133.476137.803143.168147.01
9966.5173.361110.607117.407123.225128.422129.996134.642138.987144.373148.23
10067.32874.222111.667118.498124.342129.561131.142135.807140.169145.577149.449
10168.14675.083112.726119.589125.458130.7132.287136.971141.351146.78150.667
10268.96575.946113.786120.679126.574131.838133.431138.134142.532147.982151.884
10369.78576.809114.845121.769127.689132.975134.575139.297143.712149.183153.099
10470.60677.672115.903122.858128.804134.111135.718140.459144.891150.383154.314
10571.42878.536116.962123.947129.918135.247136.86141.62146.07151.582155.528
10672.25179.401118.02125.035131.031136.382138.002142.78147.247152.78156.74
10773.07580.267119.078126.123132.144137.517139.143143.94148.424153.977157.952
10873.89981.133120.135127.211133.257138.651140.283145.099149.599155.173159.162
10974.72482121.192128.298134.369139.784141.423146.257150.774156.369160.372
11075.5582.867122.25129.385135.48140.917142.562147.414151.948157.563161.581
11176.37783.735123.306130.472136.591142.049143.7148.571153.122158.757162.788
11277.20484.604124.363131.558137.701143.18144.838149.727154.294159.95163.995
11378.03385.473125.419132.643138.811144.311145.975150.882155.466161.141165.201
11478.86286.342126.475133.729139.921145.441147.111152.037156.637162.332166.406
11579.69287.213127.531134.813141.03146.571148.247153.191157.808163.523167.61
11680.52288.084128.587135.898142.138147.7149.383154.344158.977164.712168.813
11781.35388.955129.642136.982143.246148.829150.517155.496160.146165.9170.016
11882.18589.827130.697138.066144.354149.957151.652156.648161.314167.088171.217
11983.01890.7131.752139.149145.461151.084152.785157.8162.481168.275172.418
12083.85291.573132.806140.233146.567152.211153.918158.95163.648169.461173.617
12184.68692.446133.861141.315147.674153.338155.051160.1164.814170.647174.816
12285.5293.32134.915142.398148.779154.464156.183161.25165.98171.831176.014
12386.35694.195135.969143.48149.885155.589157.314162.398167.144173.015177.212
12487.19295.07137.022144.562150.989156.714158.445163.546168.308174.198178.408
12588.02995.946138.076145.643152.094157.839159.575164.694169.471175.38179.604
12688.86696.822139.129146.724153.198158.962160.705165.841170.634176.562180.799
12789.70497.698140.182147.805154.302160.086161.834166.987171.796177.743181.993
12890.54398.576141.235148.885155.405161.209162.963168.133172.957178.923183.186
12991.38299.453142.288149.965156.508162.331164.091169.278174.118180.103184.379
13092.222100.331143.34151.045157.61163.453165.219170.423175.278181.282185.571
13193.063101.21144.392152.125158.712164.575166.346171.567176.438182.46186.762
13293.904102.089145.444153.204159.814165.696167.473172.711177.597183.637187.953
13394.746102.968146.496154.283160.915166.816168.6173.854178.755184.814189.142
13495.588103.848147.548155.361162.016167.936169.725174.996179.913185.99190.331
13596.431104.729148.599156.44163.116169.056170.851176.138181.07187.165191.52
13697.275105.609149.651157.518164.216170.175171.976177.28182.226188.34192.707
13798.119106.491150.702158.595165.316171.294173.1178.421183.382189.514193.894
13898.964107.372151.753159.673166.415172.412174.224179.561184.538190.688195.08
13999.809108.254152.803160.75167.514173.53175.348180.701185.693191.861196.266
140100.655109.137153.854161.827168.613174.648176.471181.84186.847193.033197.451
141101.501110.02154.904162.904169.711175.765177.594182.979188.001194.205198.635
142102.348110.903155.954163.98170.809176.882178.716184.118189.154195.376199.819
143103.196111.787157.004165.056171.907177.998179.838185.256190.306196.546201.002
144104.044112.671158.054166.132173.004179.114180.959186.393191.458197.716202.184
145104.892113.556159.104167.207174.101180.229182.08187.53192.61198.885203.366
146105.741114.441160.153168.283175.198181.344183.2188.666193.761200.054204.547
147106.591115.326161.202169.358176.294182.459184.321189.802194.912201.222205.727
148107.441116.212162.251170.432177.39183.573185.44190.938196.062202.39206.907
149108.291117.098163.3171.507178.485184.687186.56192.073197.211203.557208.086
150109.142117.985164.349172.581179.581185.8187.678193.208198.36204.723209.265
151109.994118.871165.398173.655180.676186.914188.797194.342199.509205.889210.443
152110.846119.759166.446174.729181.77188.026189.915195.476200.657207.054211.62
153111.698120.646167.495175.803182.865189.139191.033196.609201.804208.219212.797
154112.551121.534168.543176.876183.959190.251192.15197.742202.951209.383213.973
155113.405122.423169.591177.949185.052191.362193.267198.874204.098210.547215.149
156114.259123.312170.639179.022186.146192.474194.384200.006205.244211.71216.324
157115.113124.201171.686180.094187.239193.584195.5201.138206.39212.873217.499
158115.968125.09172.734181.167188.332194.695196.616202.269207.535214.035218.673
159116.823125.98173.781182.239189.424195.805197.731203.4208.68215.197219.846
160117.679126.87174.828183.311190.516196.915198.846204.53209.824216.358221.019
161118.536127.761175.875184.382191.608198.025199.961205.66210.968217.518222.191
162119.392128.651176.922185.454192.7199.134201.076206.79212.111218.678223.363
163120.249129.543177.969186.525193.791200.243202.19207.919213.254219.838224.535
164121.107130.434179.016187.596194.883201.351203.303209.047214.396220.997225.705
165121.965131.326180.062188.667195.973202.459204.417210.176215.539222.156226.876
166122.823132.218181.109189.737197.064203.567205.53211.304216.68223.314228.045
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