數(shù)理統(tǒng)計CH_方差分析(2)ppt課件

1、6.1 單向分組數(shù)據方差分析 方差分析原理 單向分組數(shù)據6.2 兩向分組數(shù)據方差分析 有重復兩向分組數(shù)據 無重復兩向分組數(shù)據本章內容本章內容6 方差分析6.2 兩向分組數(shù)據方差分析Two way analysis of variance6 方差分析6.2 兩向分組數(shù)據方差分析(1)案例和問題推進器(B)B1B2B3燃料(A)A158.252.656.241.265.360.8A249.142.854.150.551.648.4A360.158.370.973.239.240.7A475.871.558.251.048.741.4火箭射程試驗數(shù)據 選用4種燃料和3種推進器進行火箭射程試驗，每個試

2、驗處理重復2次。因素響應水平需要弄清的問題是：選用不同的燃料對火箭的射程是否有影響，影響多大？選用不同的推進器對火箭的射程是否有影響，影響多大？不同推進器選配不同燃料對火箭的射程是否有影響，影響多大？(1)案例和問題6.2 兩向分組數(shù)據方差分析B1B2BbA1x111 x11,n11x121 x12,n12x1b1 x1b,n1bA2x211 x21,n21x221 x22,n21x2b1 x2b,n2bAaxa11 xa1,na1xa21 xa2,na2xab1 xab,nab(3)數(shù)據模式AB兩向分組B單向分組A單向分組6.2 兩向分組數(shù)據方差分析各個處理(因素A與B的水平組合)分別獨立試

3、驗，第ij處理重復試驗nij次獲得nij個觀測，這nij個觀測視作第ij正態(tài)總體的一個樣本；全部觀測整個樣本由ab個獨立正態(tài)總體的ab個樣本組成，稱作兩向分組數(shù)據；按因素A的a個水平分組，則全部觀測可視作因素A單向分組數(shù)據，包括因素B所有b個水平的試驗結果；按因素B的b個水平分組，則全部觀測可視作因素B單向分組數(shù)據，包括因素A所有a個水平的試驗結果。(2)數(shù)據模式6.2 兩向分組數(shù)據方差分析2()(0,)1,2,1,2,1,2,ijkijijkijijkijijijkijkijXNiajbkn(3)統(tǒng)計模型The statistical model is:()()ijkijijkijijijk

4、ijijijijX效應模型響應模型6.2 兩向分組數(shù)據方差分析(3)統(tǒng)計模型11111111111定義總平均定義行平均 因素定義列平均 因素baabiijjijijjiijabijijijbiiijijjiajjijijijnnnnnnnnAnnBnn定義幾個總體的均值6.2 兩向分組數(shù)據方差分析(3)統(tǒng)計模型因素的總效應因素 的效應因素 的效應因素的互作效應ijijijijijijiiijjjijijijijijijABABAB定義因素的效應因素影響決定的響應值稱作效應6.2 兩向分組數(shù)據方差分析(4)均值的無偏估計1111111111111111111總平均行平均列平均ijijijijni

5、jijkijkijnababijkijijijkijnbbiijkiijijjkjiinaajijkjijijikijjxxnxxnnnxxnnnxxnnn 1111baiijjijjiabijijnnnnnn6.2 兩向分組數(shù)據方差分析(5)效應的無偏估計()誤差效應估計因素總效應估計因素 效應估計因素 總效應估計因素互作效應估計ijkijijkijijkijijijijiiiijjjjijijijijijijxxXABxxAxxBxxABxxxx 樣本均值差作為效應的估計6.2 兩向分組數(shù)據方差分析0111211112:,ababHH不全相等 問題歸結為檢驗ab個方差齊性正態(tài)總體的均值是否

6、相等0111211112:0:,ababHH不全為零總效應假設(6)統(tǒng)計假設總效應亦稱作模型效應總效應假設6.2 兩向分組數(shù)據方差分析01121112:,aaHH不全相等因素A效應假設02121212:,bbHH不全相等0313:ijijijijHH恒為零不全為零(6)統(tǒng)計假設因素B效應假設因素AB互作效應假設總效應分解成各個因素效應6.2 兩向分組數(shù)據方差分析01121112:0:,aaHH 不全為零02121212:0:,bbHH 不全為零031112131112:()()()0:() ,() ,()ababHH不全為零(6)統(tǒng)計假設因素A效應假設因素B效應假設因素AB互作效應假設總效應分

7、解成各個因素效應6.2 兩向分組數(shù)據方差分析211122111221111111ijijijijnabijkijknabijkijknnababijkijkijkijkSSTxxxnxxxn(7)計算校正總平方和SST表征數(shù)據的總變異021TTHSSTMSTdfdfnEMST6.2 兩向分組數(shù)據方差分析21112211111221111111ijijijijnabijkijijknababijkijijijkijnnababijkijkijkijkijSSExxxn xxxn(8)計算誤差平方和SSE2EESSEMSEdfdfnabE MSE誤差均方MSE6.2 兩向分組數(shù)據方差分析11211

8、12211112221111121222111111(1)(1)(1)(1)(1)(1)(1)ijabnabijkijijknnijkijkabkkabababababSSExxxxxxnsnsnsSSEnsnsMSEnabnn(8)計算誤差平方和SSE2EESSEMSEdfdfnabE MSE6.2 兩向分組數(shù)據方差分析21112212211111111ijijijnabiijkaiiinnababijkijkijkijkiSSAxxn xnxxxnnSSA表征因素表征因素A效應效應(9)計算因素A平方和SSA021AAHSSAMSAdfdfaEMSA6.2 兩向分組數(shù)據方差分析211121

9、21()()11ijnabiijkaiiiaiiiSSAxxnxxnxxSSAMSAaa(9)計算因素A平方和SSA因素A均方MSA021AAHSSAMSAdfdfaEMSA6.2 兩向分組數(shù)據方差分析22111ijnbiijkjkiiVar xVar xnn 022122121aHiiiaiiiiESSAn E xnE xn Var xExn Var xExa(9)計算因素A平方和SSA021HSSAEa6.2 兩向分組數(shù)據方差分析21112212211111111ijijijnabjijkbjjjnnbaabijkijkjikijkjSSBxxn xnxxxnn SSB表征因素表征因素B效

10、應效應(10)計算因素B平方和SSB021BBHSSBMSBdfdfbEMSB6.2 兩向分組數(shù)據方差分析21112121()()11ijnabjijkbjjjbjjjSSBxxnxxnxxSSBMSBbb (10)計算因素B平方和SSB因素B均方MSB021BBHSSBMSBdfdfbEMSB6.2 兩向分組數(shù)據方差分析22111ijnajijkikjjVar xVar xnn 022122121bHjjjbjjjjESSBn E xnE xn Var xExn Var xExb 210bSSBEH(10)計算因素B平方和SSB6.2 兩向分組數(shù)據方差分析211122111111221111

11、11()1111ijijijijijnabijijijknnababijkijkijkijkijinnbaabijkijkjikijkjSSABxxxxxxnnxxnn (1)(1)ABdfab(11)計算因素AB平方和SSAB6.2 兩向分組數(shù)據方差分析2111211211()()()(1)(1)(1)(1)ijnabijijijkabijijijijabijijijijSSABxxxxn xxxxn xxxxSSABMSABabab (11)計算因素AB平方和SSAB因素AB均方MSAB因素AB的互作效應6.2 兩向分組數(shù)據方差分析)()()()(xxxxxxxxxxxxjiijjiiji

12、jkijk aibjnkjiijaibjnkjaibjnkiaibjnkijijkaibjnkijkijijijijijxxxxxxxxxxxx11121112111211121112SSABSSBSSASSESST(12)總離差平方和分解效應分解平方和代表效應6.2 兩向分組數(shù)據方差分析B1B2BbA1x111 x11,n11x121 x12,n12x1b1 x1b,n1bA2x211 x21,n21x221 x22,n21x2b1 x2b,n2bAaxa11 xa1,na1xa21 xa2,na2xab1 xab,nabAB兩向分組A單向分組6.2 兩向分組數(shù)據方差分析(12)總離差平方和

13、分解B單向分組111111111111總和組和組和組和ijijijijabbaijiijjijijjinnabijkijijkijkknnbaiijkjijkjkiknnnnnnTxijTxiTxjTx (13)平方和算法匯總分組數(shù)據和6.2 兩向分組數(shù)據方差分析aibjijijnTE112 aiiinTA12bjjjnTB1222221111ababijjiijijijijTTTTEABCRnnnn 2111ijnabijkijkRx(13)平方和算法匯總分組數(shù)據平方和2TCRn矯正數(shù)6.2 兩向分組數(shù)據方差分析111(1)(1)TEABABTEABABSSTRCRdfnSSEREdfnab

14、SSAACRdfaSSBBCRdfbSSABEABCRdfabSSTSSESSASSBSSABdfdfdfdfdf，(13)平方和算法匯總平方和及自由度計算6.2 兩向分組數(shù)據方差分析,AEMSAFdfdfMSE拒絕域：AASSAMSAdf因素 均方ESSEMSEdf誤差均方,抽樣觀測概率AEAdfdfMSApP FMSEAMSAFfMSE值：(14)假設檢驗1AEdfadfnab6.2 兩向分組數(shù)據方差分析,BEMSBFdfdfMSE拒絕域：B1SSBMSBb因素 均方ESSEMSEdf誤差均方,抽樣觀測概率BEBdfdfMSBpP FMSEBMSBFfMSE值：(14)假設檢驗1BEdfb

15、dfnab6.2 兩向分組數(shù)據方差分析,ABEMSABFdfdfMSE拒絕域：ABABSSABMSABdf因素均方,抽樣觀測概率ABEABdfdfMSABpP FMSEABMSABFfMSE值：(14)假設檢驗11ABEdfabdfnabESSEMSEdf誤差均方6.2 兩向分組數(shù)據方差分析SourceSSdfMSF ValuePrFFactor AFactor BFactor ABErrorSSASSBSSABSSEa-1b-1(a-1)(b-1)n-abMSAMSBMSABMSEMSA/MSEMSB/MSEMSAB/MSEpApBpABTotalSSTn-1(15)方差分析表The Tab

16、le of ANOVA6.2 兩向分組數(shù)據方差分析SourceSSdfMSF ValueFactor AFactor BFactor ABErrorSSASSBSSABSSEa-1b-1(a-1)(b-1)n-abMSAMSBMSABMSEMSA/MSEMSB/MSEMSAB/MSETotalSSTn-1(15)方差分析表The Table of ANOVAF(df1,df2)F(dfA,dfE)F(dfAB,dfE)F(dfB,dfE)6.2 兩向分組數(shù)據方差分析B1B2B3Ti.T2i./ni.A158.252.6110.8615456.241.297.44855.8865.360.812

17、6.17960.73334.318626.087A249.142.891.94242.6554.150.5104.65477.0651.648.41005005.12296.514652.04A360.158.3118.47010.970.973.2144.110385.0539.240.779.93193.13342.419539.63A475.871.5147.310857.8958.251.0109.25988.2448.741.490.14085.65346.620191.93T.j.468.4455.3396.1T=1319.8A72839.68T2.j./n.j27424.8225

18、912.2612519611.90125B72948.98CR72578.00T2ij./nij28219.0526558.68520201.615E74979.35R.j28265.4426706.2320244.63R75216.30(16)方差分析案例數(shù)據的表格算法6.2 兩向分組數(shù)據方差分析B1B2B3Ti.T2i./ni.A158.252.656.241.265.360.8334.318626.087A249.142.854.150.551.648.4296.514652.04A360.158.370.973.239.240.7342.419539.63A475.871.558.25

19、1.048.741.4346.620191.93T.j.468.4455.3396.1T=1319.8A72839.68T2.j./n.j27424.8225912.2612519611.90125B72948.98CR72578.00T2ij./nij28219.0526558.68520201.615E74979.35R.j28265.4426706.2320244.63R75216.30(16)方差分析案例數(shù)據的表格算法6.2 兩向分組數(shù)據方差分析dfT=n-1=24-1=23；dfE=n-ab=24-43=12dfA=a-1=4-1=3；dfB=b-1=3-1=2dfAB=(a-1)(b-1)=32=6SST=R-CR=75216.3-72578.00167=2638.29833SSE=R-E=7521