第六章方差分析

1、第六章 方差分析第一節(jié) Simple Factorial 過程6.1.1 主要功能6.1.2 實(shí)例操作第二節(jié) General Factorial 過程6.2.1 主要功能6.2.2 實(shí)例操作第三節(jié)Multivarite 過程6.3.1 主要功能6.3.2 實(shí)例操作方差分析是R.A.Fister 發(fā)明的，用于兩個(gè)及兩個(gè)以上樣本均數(shù)差別的顯著性檢驗(yàn)。由于各種因素的影響，研究所得的數(shù)據(jù)呈現(xiàn)波動(dòng)狀，造成波動(dòng)的原因可分成兩類，一是不可控的隨機(jī)因素，另一是研究中施加的對結(jié)果形成影響的可控因素。方差分析的基本思想是：通過分析研究中不同來源的變異對總變異的貢獻(xiàn)大小，從而確定可控因素對研究結(jié)果影響力的大小。方差

2、分析主要用于：1、均數(shù)差別的顯著性檢驗(yàn)，2、 分離各有關(guān)因素并估計(jì)其對總變異的作用，3、分析因素間的交互作用，4、方差齊性檢驗(yàn)。第一節(jié) Simple Factorial 過程6.1.1 主要功能調(diào)用此過程可對資料進(jìn)行方差分析或協(xié)方差分析。在方差分析中可按用戶需要作單因素方差分析（其結(jié)果將與第五章第四節(jié)相同）或多因素方差分析（包括醫(yī)學(xué)中常用的配伍組方差分析） ；當(dāng)觀察因素中存在有很難或無法人為控制的因素時(shí)，則可對之加以指定以便進(jìn)行協(xié)方差分析。返回目錄返回全書目錄6.1.2實(shí)例操作例6-1下表為運(yùn)動(dòng)員與大學(xué)生的身高（ cm）與肺活量（cm3）的數(shù)據(jù)，考慮到身高與 肺活量有關(guān)，而一般運(yùn)動(dòng)員的身高高于

3、大學(xué)生，為進(jìn)一步分析肺活量的差異是否由于體育鍛煉所致，試作控制身高變量的協(xié)方差分析。運(yùn)動(dòng)員大學(xué)生身高肺活量身高肺活1184.94300168.73450167.93850170.84100171.04100165.03800171.04300169.73300188.04800171.53450179.04000166.53250177.05400165.03600179.54000165.03200187.04800173.03950187.04800169.04000169.04500173.84150188.04780174.03450176.73700170.53250179.0525

4、0176.04100183.04250169.53650180.54800176.33950179.05000163.03500178.03700172.53900164.03600177.03450174.04050173.0|_ 38506.1.2.1數(shù)據(jù)準(zhǔn)備激活數(shù)據(jù)管理窗口，定義變量名：組變量為 group （運(yùn)動(dòng)員=1,大學(xué)生=2）,身高為x, 肺活量為y,按順序輸入相應(yīng)數(shù)值，建立數(shù)據(jù)庫，結(jié)果見圖6.1。6.1.2.2統(tǒng)計(jì)分析激活 Statistics ANOV A對話框（圖 組變量group,點(diǎn)擊圖6.1 原始數(shù)據(jù)的輸入菜單選 ANOVA Models 中的 Simple Facto

5、rial.項(xiàng),彈出 Simple Factorial 6.2）。在變量列表中選變量 y,點(diǎn)擊？鈕使之進(jìn)入Dependent框；選分 ? 鈕使之進(jìn)入 FactoKs）框中，并點(diǎn)擊Define Range.鈕在彈出的SimpleFactorial ANOV A:Define Range框中確定分組變量 group的起止值（1,2）;選協(xié)變量 x,點(diǎn)擊 ?鈕使之進(jìn)入 Covariate（s）框中。圖6.2協(xié)方差分析對話框點(diǎn)擊Options.框，彈出Simple Factorial ANOV A:Options對話框。系統(tǒng)在協(xié)方差分析的 方法（Method）上有三種選項(xiàng)：1、Unique :同時(shí)評價(jià)所

6、有的效應(yīng)；2、Hierarchical :除主效應(yīng)外，逐一評價(jià)各因素的效應(yīng)；3、Experimental:評價(jià)因素干預(yù)之前的主效應(yīng)。本例選Unique方法，之后點(diǎn)擊 Continue鈕返回Simple Factorial ANOV A對話框，再點(diǎn) 擊OK鈕即可。6.1.2.3 結(jié)果解釋在結(jié)果輸出窗口中可見如下統(tǒng)計(jì)數(shù)據(jù)：先輸出肺活量總均數(shù)和兩組的肺活量均數(shù)，總均數(shù)為4033.25,運(yùn)用員組均數(shù)為 4399.00,大學(xué)生組為3667.50。接著協(xié)方差分析表明，混雜因素X (身高)兩組間是有差異的(F=10.679, P=0.002),控制其影響后，兩組間肺活量的差別依然存在(F=9.220, P=

7、0.004),故可以認(rèn)為兩組間肺活量的均數(shù)在消除了身高因素的影響之后仍有差別，運(yùn)動(dòng)員的肺活量大于大學(xué)生，即體育鍛煉會(huì)提高肺活量。最后系統(tǒng)輸出公共回歸系數(shù)，=36.002 ,該值可用于求修正均數(shù)：本例為=4399.00 - 36.002 X (178.175 - 174.3325) = 4260.6623=3667.50 - 36.002 X (170.49 - 174.3325 ) = 3805.8377Y by GROUP Total Population 4033.25 (40)GROUP124399.003667.50(20) (20)Y by GROUPwith XUNIQUE sum

8、s of squaresAll effects entered simultaneouslySum ofMeanSigSource of VariationSquaresDFSquareF of FCovariates163076311630762.63510.679 .002X163076311630762.63510.679 .002Main Effects140784711407847.0959.220 .004GROUP140784711407847.0959.220 .004Explained698168523490842.56822.860 .000Residual56499923

9、7152702.496Total40 cases were processed.0 cases (.0 pct) were missing1263167839323889.167Covariate Raw Regression CoefficientX36.002返回目錄返回全書目錄第二節(jié) General Factorial 過程6.2.1 主要功能調(diào)用此過程可對完全隨機(jī)設(shè)計(jì)資料、配伍設(shè)計(jì)資料、析因設(shè)計(jì)資料、正交設(shè)計(jì)資料等等進(jìn)行多因素方差分析或協(xié)方差分析。返回目錄返回全書目錄6.2.2 實(shí)例操作例6-2下表為三因素析因?qū)嶒?yàn)的資料，請用方差分析說明不同基礎(chǔ)液與不同血清種類 對鉤端螺旋體的培養(yǎng)計(jì)數(shù)

10、的影響。基礎(chǔ)液 （A）血清種類（B）兔血清濃度（C）胎盤血清濃度（C）5%8%5%8%緩沖液6481144830578124618778536691398167144164390918451030|_1002蒸儲(chǔ)水1763144792093312411883709102413811896848109224211926574|_742自來水580178911266851026121511765461026143412805958301651121215666.2.2.1 數(shù)據(jù)準(zhǔn)備激活數(shù)據(jù)管理窗口，定義變量名：基礎(chǔ)液為base,血清種類為sero,血清濃度為pct,鉤端螺旋體的培養(yǎng)計(jì)數(shù)為X,按順序

11、輸入相應(yīng)數(shù)值，建立數(shù)據(jù)庫。6.2.2.2 統(tǒng)計(jì)分析激活 Statistics 菜單選 ANOVA Models 中的 General Factorial項(xiàng),彈出 General Factorial ANOVA對話框（圖6.3）。在對話框左側(cè)的變量列表中選變量x,點(diǎn)擊？鈕使之進(jìn)入DependentVariable框；選要控制的分組變量base sero和pct,點(diǎn)？ 鈕使之進(jìn)入Factor（s）框中，并分別點(diǎn)擊Define Range鈕，在彈出的 General Factorial ANOV A:Define Range對話框中確定各變 量的起止值，本例變量base的起止值為1、3,變量sero

12、的起止值為1、2,變量pct的起止值為1、2。之后點(diǎn)擊 OK鈕即可。圖6.3析因方差分析對話框6.2.2.3 結(jié)果解釋在結(jié)果輸出窗口中，系統(tǒng)顯示48個(gè)觀察值進(jìn)入統(tǒng)計(jì)，三個(gè)因素按其各自水平共產(chǎn)生12種組合。分析表明，模型總效應(yīng)的F值為10.55, P值< 0.001,說明三因素間存在有交互作用。單因素效應(yīng)和交互效應(yīng)導(dǎo)致的組間差別比較結(jié)果是：單因素組間比較：A:基礎(chǔ)液（BASE）F = 4.98, P = 0.012,說明三種培養(yǎng)基培養(yǎng)鉤體的計(jì)數(shù)有差別；B:血清種類（SERO）F = 61.265, P < 0.001,說明兩種血清培養(yǎng)鉤體的計(jì)數(shù)有差別；C：血清濃度（PCT）F = 3

13、.49, P = 0.070,說明兩種血清濃度培養(yǎng)鉤體的計(jì)數(shù)無差別。兩因素構(gòu)成的一級交互作用：AXB:基礎(chǔ)液（BASE）清種類（SERO）F = 5.16, P = 0.011,交互作用明顯；BXC：血清種類（SERO）清濃度（PCT）F = 15.96, P < 0.001 ,交互作用明顯；A>C：基礎(chǔ)液（BASE） X血清濃度（PCT）F = 0.78, P = 0.465,交互作用不明顯。三因素構(gòu)成的二級交互作用：AXBXC：基礎(chǔ)液（BASE） X清種類（SERO） X清濃度（PCT）F = 6.75, P = 0.003,交互作用明顯。48 cases accepted.0

14、 cases rejected because of out-of-range factor values.0 cases rejected because of missing data.12 non-empty cells.1 design will be processed.Univariate Homogeneity of Variance TestsVariable . XCochrans C(3,12) =.34004, P = .036 (approx.)Bartlett-Box F(11,897) =1.69822, P = .069* * * * * * a n a l y

15、s i s o f V a r i a n c e - design 1 * * * * * * Tests of Significance for X using UNIQUE sums of squaresSource of VariationSSDFMSFSig of FWITHIN+RESIDUAL2459233.753668312.05BASE679967.382339983.694.98.012PCT238713.021238713.023.49.070SERO4184873.521 4184873.561.26.000BASE BY PCT107005.54253502.77.7

16、8.465BASE BY SERO705473.042352736.525.16.011PCT BY SERO1089922.691 1089922.715.96.000BASE BY PCT BY SERO922307.372461153.696.75.003(Model)7928262.5611720751.1410.55.000(Total)10387496.3147221010.56R-Squared =.763Adjusted R-Squared = .691返回目錄返回全書目錄第三節(jié)Multivarite 過程6.3.1 主要功能調(diào)用此過程可進(jìn)行多元方差分析。此外，對于一元設(shè)計(jì)，如

17、涉及混合模型的設(shè)計(jì)、分割設(shè)計(jì)（又稱列區(qū)設(shè)計(jì)）、重復(fù)測量設(shè)計(jì)、嵌套設(shè)計(jì)、因子與協(xié)變量交互效應(yīng)設(shè)計(jì)等，此過程 均能適用。返回目錄返回全書目錄6.3.2 實(shí)例操作例6-3甲地區(qū)為大城市，乙地區(qū)為縣城，丙地區(qū)為農(nóng)村。某地分別調(diào)查了上述三類地 區(qū)8歲男生三項(xiàng)身體生長發(fā)育指標(biāo)：身高、體重和胸圍，數(shù)據(jù)見下表，問：三類地區(qū)之間男 生三項(xiàng)身體生長發(fā)育指標(biāo)的差異有無顯著性？學(xué)生 編P甲地區(qū)乙地區(qū)丙地區(qū)身高|"體重胸圍身高胸圍身高體重胸圍1119.8022.6060.50125.1023.0062.00118.3020.4054.402121.7021.5055.50127.0021.5059.00121

18、.3020.0054.303121.4019.1056.50125.7023.4061.50121.8026.6061.104124.4021.8060.50114.9017.5052.50124.2022.1058.605120.0021.4057.70124.9023.5058.50123.5023.2060.206117.0020.1057.00117.6018.9057.00123.0022.9058.207118.1018.8057.10124.2020.8058.50134.9032.3064.808118.8022.0061.70117.9020.3061.00123.7022.

19、7059.909124.2021.3058.40120.4020.0056.00105.2020.2054.5010124.9024.0060.80115.0019.7056.50112.2020.8057.5011124.7023.3060.00126.2021.2056.50118.6021.0057.6012123.0022.5060.00125.1022.1058.50112.0023.2058.2013125.3022.9065.20114.9019.7056.00121.5024.0060.3014124.2019.5053.80121.5022.0057.00124.5021.5

20、055.6015127.4022.9059.50114.0019.0054.50119.5020.5055.5016128.2022.3060.00118.7019.1054.50122.5023.0056.7017126.1022.7057.40120.6020.0055.50115.5019.0054.2018128.7023.5060.40122.9018.5056.00122.5022.5057.6019129.5024.5051.00119.6019.5059.50124.5025.0057.9020126.9025.5061.50112.3020.0058.00125.0025.5

21、060.3021126.5025.0063.90121.3020.0058.00117.5023.0059.0022128.201 26.101 63.001 121.2021.2059.00127.3022.5058.9023131.4027.9063.10120.2023.1059.50122.3022.0058.2024130.8026.8061.50120.3021.0059.50121.3021.0055.6025133.9027.2065.80120.0022.2059.50120.5022.0055.1026130.4024.4062.60123.3020.1056.50116.

22、0019.0053.5027131.3024.4059.50122.1021.0057.50120.5020.0054.4028130.2023.0062.60123.3021.5061.00114.5019.0053.4029136.0026.3060.00109.9017.8056.50131.0025.5058.3030141.0031.9063.70125.6023.3060.50122.5024.5058.706.3.2.1數(shù)據(jù)準(zhǔn)備激活數(shù)據(jù)管理窗口，定義變量名：地區(qū)為 G,身高為X1 ,體重為X2,胸圍為X3,按 順序輸入相應(yīng)數(shù)值，變量G的數(shù)值是：甲地區(qū)為 1,乙地區(qū)為2,丙地區(qū)為3

23、。6.3.2.2統(tǒng)計(jì)分析激活 Statistics 菜單選 ANOVA Models 中的 Multivarite項(xiàng)，彈出 Multivarite ANOV A 對 話框(圖6.8)。首先指定供分析用的變量 x1、x2、x3,故在對話框左側(cè)的變量列表中選變 量x1、x2、x3,點(diǎn)擊？鈕使之進(jìn)入 Dependent Variable框；然后選變量 g (分組變量)點(diǎn)擊 ?鈕使之進(jìn)入 Factor(s)框中，并點(diǎn)擊 Define Range鈕，確定g的起始值和終止值。點(diǎn)擊Options鈕，彈出 Multivarite ANOV A:Options對話框，選擇需要計(jì)算的指標(biāo)。在 Factor(s)欄

24、內(nèi)選變量g,點(diǎn)擊？鈕使之進(jìn)入 Display Means for框，要求計(jì)算平均值指標(biāo)；在 Matriced Within Cell欄內(nèi)選 Correlation、Covariance、SSCP項(xiàng)，要求計(jì)算單元內(nèi)的相關(guān)矩陣、 方差協(xié)方差矩陣和離均差平方和交叉乘積矩陣；在Error Matrices欄內(nèi)也選上述三項(xiàng)，要求計(jì)算誤差的相關(guān)矩陣、方差協(xié)方差矩陣和離均差平方和交叉乘積矩陣；在Diagnostics欄內(nèi)選Homogeneity test項(xiàng)，要求作變量的方差齊性檢驗(yàn)。之后點(diǎn)擊 Continue鈕返回Multivarite ANOVA對話框，最后點(diǎn)擊 OK鈕即可。6.3.2.3 結(jié)果解釋在結(jié)果

25、輸出窗口中將看到如下分析結(jié)果：系統(tǒng)首先顯示共90個(gè)觀察值進(jìn)入統(tǒng)計(jì)分析，因分組變量g為三個(gè)地區(qū)，故分析的單元數(shù)為3。然后輸出3個(gè)應(yīng)變量（x1、x2、x3）的方差齊性檢驗(yàn)結(jié)果，分別輸出了 Cochran C 檢驗(yàn)值及其顯著性水平P值、Bartlett-Box F檢驗(yàn)值及其顯著性水平P值。其中身高：C = 0.39825, P = 0.540； F = 1.01272, P = 0.363；體重：C = 0.43787 , P = 0.227; F = 4.48624, P = 0.011;胸圍：C = 0.47239, P = 0.089; F = 2.06585, P = 0.127;可見3項(xiàng)指

26、標(biāo)的方差基本整齊（P值均大于0.05）。90 cases accepted.0 cases rejected because of out-of-range factor values.0 cases rejected because of missing data.3 non-empty cells.1 design will be processed.VariableGCELL NUMBER123123Univariate Homogeneity of Variance TestsVariable . X1Cochrans C(29,3) =.39825,Bartlett-Box F(2,

27、17030) = 1.01272,Variable . X2Cochrans C(29,3) =.43787,Bartlett-Box F(2,17030) = 4.48624,Variable . X3Cochrans C(29,3) =.47239,Bartlett-Box F(2,17030) = 2.06585,P = .540 (approx.)P = .363P = .227 (approx.)P = .011P = .089 (approx.)P = .127Cochran C檢驗(yàn)和Bartlett-Box F檢驗(yàn)對考查協(xié)方差矩陣的相等性比較方便，但還不夠。于是系統(tǒng)接著分別輸出了

28、三類地區(qū)（即各個(gè)單元）各生長發(fā)育指標(biāo)的離均差平方和交叉乘積 矩陣和方差協(xié)方差矩陣。之后作Box M檢驗(yàn)，Box M檢驗(yàn)提供矩陣一致性的多元測試，本例Boxs M = 36.93910 ,在基于方差分析的顯著性檢驗(yàn)中F = 2.92393 ;在基于 2的顯著性檢驗(yàn)中2 = 35.09922,兩者P < 0.001,故認(rèn)為矩陣一致性不佳。Cell Number . 1Sum of Squares and Cross-Products matrix X1X2X3X1861.187444.983546.0980463.906404.15742X2X3380.137215.937230.51915

29、6.559314.859Variance-Covariance matrixX1X129.696X213.108X37.446X27.9495.399X310.857Cell Number . 1 (Cont.)Correlation matrix with Standard Deviations on DiagonalX1X2X3X15.449X2.8532.819X3.415.5813.295Determinant of Covariance matrix of dependent variables = LOG(Determinant)=Cell Number . 2Sum of Squ

30、ares and Cross-Products matrixX1X2X3X1565.368X2147.22278.910X3139.43079.337147.967Variance-Covariance matrixX1X2X3X119.495X25.0772.721X34.8082.7365.102Correlation matrix with Standard Deviations on DiagonalX1X2X3X14.415X2.6971.650X3.482.7342.259Determinant of Covariance matrix of dependent variables

31、 = LOG(Determinant)=Cell Number . 3Sum of Squares and Cross-Products matrixX1X2X3X1944.128X2307.722217.030X3261.130186.252203.702Variance-Covariance matrixX1X2X3X132.556X210.6117.484X39.0046.4227.024Correlation matrix with Standard Deviations on DiagonalX1X2X3X15.706X2.6802.736X3.595.8862.650Determi

32、nant of Covariance matrix of dependent variables =LOG(Determinant)=198.135075.28895X1X2X3X127.249X29.5996.051X37.0864.8527.661Pooled within-cells Variance-Covariance matrixDeterminant of pooled Covariance matrix of dependent vars.= LOG(Determinant)=272.069065.60606Multivariate test for Homogeneity o

33、f Dispersion matricesBoxs M =F WITH (12,36680) DF =Chi-Square with 12 DF =36.939102.92393, P =.000 (Approx.)35.09922, P =.000 (Approx.)卜面系統(tǒng)輸出將三類地區(qū)看成一個(gè)大樣本時(shí)的離均差平方和交叉乘積矩陣。如X1、X2和X3的離均差平方和分別為662.884、121.562和114.902。在此基礎(chǔ)上，進(jìn)行多元差異的檢驗(yàn)。通常有四種方法：1、Pillai 軌跡:V =2、Wilks 入值：W =3、Hotelling 軌跡：T =4、Roy最大根：R =式中Zmax

34、為最大特征值，力為第i個(gè)特征值，s為非零特征值個(gè)數(shù)。根據(jù)這些值變換的F檢驗(yàn)均有顯著性（P<0.001）,說明三類地區(qū)各生長發(fā)育指標(biāo)之間的差別有高度顯著性。這一計(jì)算結(jié)果對上述三項(xiàng)生長發(fā)育指標(biāo)進(jìn)行了單因素的方差分析，可見：X1: SS = 662.88356, F = 12.16335X2: SS = 121.56200, F = 10.04439X3: SS = 114.90200, F = 7.49893差別均有顯著性，說明三項(xiàng)生長發(fā)育指標(biāo)各地區(qū)間的差別均有顯著性。Combined Observed Means for GVariable .X1G123WGT.UNWGT.WGT.UNW

35、GT.WGT.UNWGT.126.46667126.46667120.52000120.52000120.92000120.92000Variable .X2G1WGT.23.50667UNWGT.23.506672WGT.20.69667UNWGT.20.696673WGT.22.49667Variable .X3GUNWGT.22.496671WGT.60.00667UNWGT.60.006672WGT.57.86667UNWGT.57.866673WGT.57.41667_|UNWGT. 57.41667WITHIN+RESIDUAL Correlations with Std. Dev

36、s. on DiagonalX1X2X3X15.220X2.7472.460X3.490.7132.768Statistics for WITHIN+RESIDUAL correlationsLog(Determinant) =.00000Bartlett test of sphericity =. with 3 D. F.Significance =F(max) criterion =4.50308 with (3,87) D. F.WITHIN+RESIDUAL Variances and CovariancesX1X2X3X127.249X29.5996.051X37.0864.8527

37、.661WITHIN+RESIDUAL Sum-of-Squares and Cross-ProductsX1X2X3X12370.683X2835.081526.458X3616.497422.147666.527EFFECT . GAdjusted Hypothesis Sum-of-Squares and Cross-ProductsX1X2X3X1662.884X2230.323121.562X3269.11778.193114.902Multivariate Tests of Significance (S = 2, M = 0, N = 41 1/2)Test NameValueA

38、pprox.FHypoth. DFError DFSig. of FPillais.512279.870806.00172.00.000Hotellings.704279.859786.00168.00.000Wilks.550149.866436.00170.00.000Roys.31265Note. F statistic for WILKS' Lambda is exact.EFFECT . G (Cont.)Univariate F-tests with (2,87) D. F.VariableHypoth. SSError SS Hypoth. MSError MSFSig.

39、 of FX1662.88356 2370.68267331.4417827.2492312.16335.000X2121.56200526.4580060.781006.0512410.04439.000X3114.90200666.5270057.451007.661237.49893.001之后按單元輸出各項(xiàng)指標(biāo)的觀察值均數(shù)(Obs.Mean)、調(diào)整均數(shù)(Adj.Mean )、估計(jì)均數(shù)(Est.Mean)、粗誤差(Raw Resid)、標(biāo)準(zhǔn)化誤差(Std.Resid)以及不分地區(qū)的總均數(shù) (Comined Adjusted Means for G )。Adjusted and Estim

40、ated MeansVariable . X1CELLObs. MeanAdj. MeanEst. MeanRaw Resid. Std. Resid.1126.467126.467126.467.000.0002120.520120.520120.520.000.0003120.920120.920120.920.000.000Adjusted and Estimated Means (Cont.)Variable . X2CELL123Obs. Mean 23.507 20.697 22.497Adj. Mean23.50720.69722.497Est. Mean 23.507 20.697 22.497Raw Resid. Std