单因素方差分析的计算步.pdf

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1、一、单因素方差分析的计算步骤假定实验或观察中只有一个因素（因子）A,且A有m个水平，分别记为,21mAAA在每一种水平下，做n次实验，在每一次试验后可得一实验值，记做ijx表示在第j 个水平下的第 i 个试验值mjni,2,1;,2,1。结果如下表3.1：表 3.1 单因素方差分析数据结构表观测值水平1A2AmA1 11x12xmx12 22x22xmx2n1nx2nxnmx为了考察因素A对实验结果是否有显著性影响，我们把因素A的m个水平mAAA,21看成是m个正态总体，而mjnixij,2,1;,2,1看成是取自第j总体的第i个样品，因此，可设mjniaNxjij,2,1;,2,1,2。可以

2、认为jjja,是因素A的第j个水平jA所引起的差异。因此检验因素A的各水平之间是否有显著的差异，就相当于检验：maaaH210:或者0:210mH具体的分析检验步骤是：（一）计算水平均值令jx表示第j种水平的样本均值，jniijjnxxj1式中，ijx是第j种水平下的第i个观察值，jn表示第j种水平的观察值次数（二）计算离差平方和在单因素方差分析中，离差平方和有三个，它们分别是总离差平方和，组内离差平方和以及组间平方和。首先，总离差平方和，用SST代表，则，2)(xxSSTij其中,nxxij它反映了离差平方和的总体情况。其次，组内离差平方和，用SSE表示，其计算公式为：jijijxxSSE2

3、其中jx反映的是水平内部或组内观察值的离散状况，即反映了随机因素带来的影响。最后，组间平方和，用SSA表示，SSA的计算公式为：22xxnxxSSAjjj用各组均值减去总均值的离差的平方，乘以各组观察值个数，然后加总，即得到SSA。可以看出，它所表现的是组间差异。其中既包括随机因素，也包括系统因素。根据证明，SSASSESST,之间存在着一定的联系，这种联系表现在：SSASSESST因为：22xxxxxxjjijijxxxxxxxxjjijjjij222在各组同为正态分布，等方差的条件下，等式右边最后一项为零，故有，222)()()(xxxxxxjjijij即SSASSESST文档编码:CF5

4、U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6

5、O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6

6、X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H

7、6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI

8、9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N

9、10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档

10、编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2（三）计算平均平方用离差平方和除以各自自由度即可得到平均平方。对SST来说，其自由度为1n，因为它只有一个约束条件，即0)(xxij。对SSA来说，其自由度是1m，这里m表示水平的个数

11、，SSA反映的是组间的差异，它也有一个约束条件，即要求：0)(xxnjj对SSE来说，其自由度为mn，因为对每一种水平而言，其观察值个数为jn，该水平下的自由度为1jn，总共有m个水平，因此拥有自由度的个数为mnnmj)1(。与离差平方和一样，SSESSASST,之间的自由度也存在着关系，即)()1(1mnmn这样对SSA，其平均平方MSA为：1mSSAMSA对于SSE，平均平方MSE为：mnSSEMSE（四）方差分析表由 F 分布知，F值的计算公式为：MSEMSAF组内方差组间方差为了将方差分析的主要过程表现的更加清楚，通常把有关计算结果列成方差分析表如下表 3.2：表 3.2 方差分析表方

12、差来源离 差 平 方 和（SS）自由度（df）平均平方（MS）F值组间组内总差异SSASSESST1mmn1nMSAMSEMSEMSA/文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X1

13、0E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P

14、5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z

15、9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10

16、H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码

17、:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3

18、Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2（五）作出统计判断对于给定的显著

19、性水平，由F分布表查出自由度为),1(mnm的临界值F，如果FF，则拒绝原假设，说明因素对指标起显著影响；如果FF，则接受原假设，说明因素的不同水平对试验结果影响不显著。文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:

20、CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y

21、8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8

22、HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10

23、E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5

24、 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9

25、F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2文档编码:CF5U3Y8I6O8 HM6X10E1H6P5 ZI9Z9F7N10H2