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1、第六章 方差分析第一节 Simple Factorial 过程6.1.1 主要功能6.1.2 实例操作第二节 General Factorial 过程6.2.1 主要功能6.2.2 实例操作第三节Multivarite 过程6.3.1 主要功能6.3.2 实例操作方差分析是R.A.Fister 发明的，用于两个及两个以上样本均数差别的显著性检验。由于各种因素的影响，研究所得的数据呈现波动状，造成波动的原因可分成两类，一是不可控的随机因素，另一是研究中施加的对结果形成影响的可控因素。方差分析的基本思想是：通过分析研究中不同来源的变异对总变异的贡献大小，从而确定可控因素对研究结果影响力的大小。方差
2、分析主要用于：1、均数差别的显著性检验，2、 分离各有关因素并估计其对总变异的作用，3、分析因素间的交互作用，4、方差齐性检验。第一节 Simple Factorial 过程6.1.1 主要功能调用此过程可对资料进行方差分析或协方差分析。在方差分析中可按用户需要作单因素方差分析（其结果将与第五章第四节相同）或多因素方差分析（包括医学中常用的配伍组方差分析） ；当观察因素中存在有很难或无法人为控制的因素时，则可对之加以指定以便进行协方差分析。返回目录返回全书目录6.1.2实例操作例6-1下表为运动员与大学生的身高（ cm）与肺活量（cm3）的数据，考虑到身高与 肺活量有关，而一般运动员的身高高于
3、大学生，为进一步分析肺活量的差异是否由于体育锻炼所致，试作控制身高变量的协方差分析。运动员大学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数据准备激活数据管理窗口，定义变量名：组变量为 group （运动员=1,大学生=2）,身高为x, 肺活量为y,按顺序输入相应数值，建立数据库，结果见图6.1。6.1.2.2统计分析激活 Statistics ANOV A对话框（图 组变量group,点击图6.1 原始数据的输入菜单选 ANOVA Models 中的 Simple Facto
5、rial.项,弹出 Simple Factorial 6.2）。在变量列表中选变量 y,点击？钮使之进入Dependent框；选分 ? 钮使之进入 FactoKs）框中，并点击Define Range.钮在弹出的SimpleFactorial ANOV A:Define Range框中确定分组变量 group的起止值（1,2）;选协变量 x,点击 ?钮使之进入 Covariate（s）框中。图6.2协方差分析对话框点击Options.框，弹出Simple Factorial ANOV A:Options对话框。系统在协方差分析的 方法（Method）上有三种选项：1、Unique :同时评价所
6、有的效应；2、Hierarchical :除主效应外，逐一评价各因素的效应；3、Experimental:评价因素干预之前的主效应。本例选Unique方法，之后点击 Continue钮返回Simple Factorial ANOV A对话框，再点 击OK钮即可。6.1.2.3 结果解释在结果输出窗口中可见如下统计数据：先输出肺活量总均数和两组的肺活量均数，总均数为4033.25,运用员组均数为 4399.00,大学生组为3667.50。接着协方差分析表明，混杂因素X (身高)两组间是有差异的(F=10.679, P=0.002),控制其影响后，两组间肺活量的差别依然存在(F=9.220, P=
7、0.004),故可以认为两组间肺活量的均数在消除了身高因素的影响之后仍有差别，运动员的肺活量大于大学生，即体育锻炼会提高肺活量。最后系统输出公共回归系数，=36.002 ,该值可用于求修正均数：本例为=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返回目录返回全书目录第二节 General Factorial 过程6.2.1 主要功能调用此过程可对完全随机设计资料、配伍设计资料、析因设计资料、正交设计资料等等进行多因素方差分析或协方差分析。返回目录返回全书目录6.2.2 实例操作例6-2下表为三因素析因实验的资料，请用方差分析说明不同基础液与不同血清种类 对钩端螺旋体的培养计数
10、的影响。基础液 （A）血清种类（B）兔血清浓度（C）胎盘血清浓度（C）5%8%5%8%缓冲液6481144830578124618778536691398167144164390918451030|_1002蒸储水1763144792093312411883709102413811896848109224211926574|_742自来水580178911266851026121511765461026143412805958301651121215666.2.2.1 数据准备激活数据管理窗口，定义变量名：基础液为base,血清种类为sero,血清浓度为pct,钩端螺旋体的培养计数为X,按顺序
11、输入相应数值，建立数据库。6.2.2.2 统计分析激活 Statistics 菜单选 ANOVA Models 中的 General Factorial项,弹出 General Factorial ANOVA对话框（图6.3）。在对话框左侧的变量列表中选变量x,点击？钮使之进入DependentVariable框；选要控制的分组变量base sero和pct,点？ 钮使之进入Factor（s）框中，并分别点击Define Range钮，在弹出的 General Factorial ANOV A:Define Range对话框中确定各变 量的起止值，本例变量base的起止值为1、3,变量sero
12、的起止值为1、2,变量pct的起止值为1、2。之后点击 OK钮即可。图6.3析因方差分析对话框6.2.2.3 结果解释在结果输出窗口中，系统显示48个观察值进入统计，三个因素按其各自水平共产生12种组合。分析表明，模型总效应的F值为10.55, P值 0.001,说明三因素间存在有交互作用。单因素效应和交互效应导致的组间差别比较结果是：单因素组间比较：A:基础液（BASE）F = 4.98, P = 0.012,说明三种培养基培养钩体的计数有差别；B:血清种类（SERO）F = 61.265, P 0.001,说明两种血清培养钩体的计数有差别；C：血清浓度（PCT）F = 3.49, P =
13、0.070,说明两种血清浓度培养钩体的计数无差别。两因素构成的一级交互作用：AXB:基础液（BASE）清种类（SERO）F = 5.16, P = 0.011,交互作用明显；BXC：血清种类（SERO）清浓度（PCT）F = 15.96, P C：基础液（BASE） X血清浓度（PCT）F = 0.78, P = 0.465,交互作用不明显。三因素构成的二级交互作用：AXBXC：基础液（BASE） X清种类（SERO） X清浓度（PCT）F = 6.75, P = 0.003,交互作用明显。48 cases accepted.0 cases rejected because of out-of
14、-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 s i s o f V a r i a n c e - desig
15、n 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 PCT107005.54253502.77.78.465BASE BY SERO705473.042352736
16、.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返回目录返回全书目录第三节Multivarite 过程6.3.1 主要功能调用此过程可进行多元方差分析。此外，对于一元设计，如涉及混合模型的设计、分割设计（又称列区设计）、重复测量设计、嵌套设
17、计、因子与协变量交互效应设计等，此过程 均能适用。返回目录返回全书目录6.3.2 实例操作例6-3甲地区为大城市，乙地区为县城，丙地区为农村。某地分别调查了上述三类地 区8岁男生三项身体生长发育指标：身高、体重和胸围，数据见下表，问：三类地区之间男 生三项身体生长发育指标的差异有无显著性？学生 编P甲地区乙地区丙地区身高|体重胸围身高胸围身高体重胸围1119.8022.6060.50125.1023.0062.00118.3020.4054.402121.7021.5055.50127.0021.5059.00121.3020.0054.303121.4019.1056.50125.7023.
18、4061.50121.8026.6061.104124.4021.8060.50114.9017.5052.50124.2022.1058.605120.0021.4057.70124.9023.5058.50123.5023.2060.206117.0020.1057.00117.6018.9057.00123.0022.9058.207118.1018.8057.10124.2020.8058.50134.9032.3064.808118.8022.0061.70117.9020.3061.00123.7022.7059.909124.2021.3058.40120.4020.0056.0
19、0105.2020.2054.5010124.9024.0060.80115.0019.7056.50112.2020.8057.5011124.7023.3060.00126.2021.2056.50118.6021.0057.6012123.0022.5060.00125.1022.1058.50112.0023.2058.2013125.3022.9065.20114.9019.7056.00121.5024.0060.3014124.2019.5053.80121.5022.0057.00124.5021.5055.6015127.4022.9059.50114.0019.0054.5
20、0119.5020.5055.5016128.2022.3060.00118.7019.1054.50122.5023.0056.7017126.1022.7057.40120.6020.0055.50115.5019.0054.2018128.7023.5060.40122.9018.5056.00122.5022.5057.6019129.5024.5051.00119.6019.5059.50124.5025.0057.9020126.9025.5061.50112.3020.0058.00125.0025.5060.3021126.5025.0063.90121.3020.0058.0
21、0117.5023.0059.0022128.201 26.101 63.001 121.2021.2059.00127.3022.5058.9023131.4027.9063.10120.2023.1059.50122.3022.0058.2024130.8026.8061.50120.3021.0059.50121.3021.0055.6025133.9027.2065.80120.0022.2059.50120.5022.0055.1026130.4024.4062.60123.3020.1056.50116.0019.0053.5027131.3024.4059.50122.1021.
22、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数据准备激活数据管理窗口，定义变量名：地区为 G,身高为X1 ,体重为X2,胸围为X3,按 顺序输入相应数值，变量G的数值是：甲地区为 1,乙地区为2,丙地区为3。6.3.2.2统计分析激活 Statistics 菜单选 ANOVA Mod
23、els 中的 Multivarite项，弹出 Multivarite ANOV A 对 话框(图6.8)。首先指定供分析用的变量 x1、x2、x3,故在对话框左侧的变量列表中选变 量x1、x2、x3,点击？钮使之进入 Dependent Variable框；然后选变量 g (分组变量)点击 ?钮使之进入 Factor(s)框中，并点击 Define Range钮，确定g的起始值和终止值。点击Options钮，弹出 Multivarite ANOV A:Options对话框，选择需要计算的指标。在 Factor(s)栏内选变量g,点击？钮使之进入 Display Means for框，要求计算平
24、均值指标；在 Matriced Within Cell栏内选 Correlation、Covariance、SSCP项，要求计算单元内的相关矩阵、 方差协方差矩阵和离均差平方和交叉乘积矩阵；在Error Matrices栏内也选上述三项，要求计算误差的相关矩阵、方差协方差矩阵和离均差平方和交叉乘积矩阵；在Diagnostics栏内选Homogeneity test项，要求作变量的方差齐性检验。之后点击 Continue钮返回Multivarite ANOVA对话框，最后点击 OK钮即可。6.3.2.3 结果解释在结果输出窗口中将看到如下分析结果：系统首先显示共90个观察值进入统计分析，因分组变
25、量g为三个地区，故分析的单元数为3。然后输出3个应变量（x1、x2、x3）的方差齐性检验结果，分别输出了 Cochran C 检验值及其显著性水平P值、Bartlett-Box F检验值及其显著性水平P值。其中身高：C = 0.39825, P = 0.540； F = 1.01272, P = 0.363；体重：C = 0.43787 , P = 0.227; F = 4.48624, P = 0.011;胸围：C = 0.47239, P = 0.089; F = 2.06585, P = 0.127;可见3项指标的方差基本整齐（P值均大于0.05）。90 cases accepted.0
26、 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(2,17030) = 1.01272,Variable . X2Cochrans
27、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检验和Bartlett-Box F检验对考查协方差矩阵的相等性比较方便，但还不够。于是系统接着分别输出了三类地区（即各个单元）各生长发育指标的离均差平方和交叉乘积 矩阵和方差协方差矩
28、阵。之后作Box M检验，Box M检验提供矩阵一致性的多元测试，本例Boxs M = 36.93910 ,在基于方差分析的显著性检验中F = 2.92393 ;在基于 2的显著性检验中2 = 35.09922,两者P 0.001,故认为矩阵一致性不佳。Cell Number . 1Sum of Squares and Cross-Products matrix X1X2X3X1861.187444.983546.0980463.906404.15742X2X3380.137215.937230.519156.559314.859Variance-Covariance matrixX1X129
29、.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 Squares and Cross-Products matrixX1X2X3X1565.36
30、8X2147.22278.910X3139.43079.337147.967Variance-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 = LOG(Determinant)=Cell Number . 3Sum of Sq
31、uares and Cross-Products matrixX1X2X3X1944.128X2307.722217.030X3261.130186.252203.702Variance-Covariance matrixX1X2X3X132.556X210.6117.484X39.0046.4227.024Correlation matrix with Standard Deviations on DiagonalX1X2X3X15.706X2.6802.736X3.595.8862.650Determinant of Covariance matrix of dependent varia
32、bles =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 of Dispersion matricesBoxs M =F WITH (12,3668
33、0) DF =Chi-Square with 12 DF =36.939102.92393, P =.000 (Approx.)35.09922, P =.000 (Approx.)卜面系统输出将三类地区看成一个大样本时的离均差平方和交叉乘积矩阵。如X1、X2和X3的离均差平方和分别为662.884、121.562和114.902。在此基础上，进行多元差异的检验。通常有四种方法：1、Pillai 轨迹:V =2、Wilks 入值：W =3、Hotelling 轨迹：T =4、Roy最大根：R =式中Zmax为最大特征值，力为第i个特征值，s为非零特征值个数。根据这些值变换的F检验均有显著性（P
34、0.001）,说明三类地区各生长发育指标之间的差别有高度显著性。这一计算结果对上述三项生长发育指标进行了单因素的方差分析，可见：X1: SS = 662.88356, F = 12.16335X2: SS = 121.56200, F = 10.04439X3: SS = 114.90200, F = 7.49893差别均有显著性，说明三项生长发育指标各地区间的差别均有显著性。Combined Observed Means for GVariable .X1G123WGT.UNWGT.WGT.UNWGT.WGT.UNWGT.126.46667126.46667120.52000120.5200
35、0120.92000120.92000Variable .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. Devs. on DiagonalX1X2X3X15.220X2.7472.460X3.490.713
36、2.768Statistics 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 CovariancesX1X2X3X127.249X29.5996.051X37.0864.8527.661WITHIN+RESIDUAL Sum-of-Squares and Cross-Pro
37、ductsX1X2X3X12370.683X2835.081526.458X3616.497422.147666.527EFFECT . 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 NameValueApprox.FHypoth. DFError DFSig. of FPillais.512279
38、.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. of FX1662.88356 2370.68267331.4417827.2492312.16335.
39、000X2121.56200526.4580060.781006.0512410.04439.000X3114.90200666.5270057.451007.661237.49893.001之后按单元输出各项指标的观察值均数(Obs.Mean)、调整均数(Adj.Mean )、估计均数(Est.Mean)、粗误差(Raw Resid)、标准化误差(Std.Resid)以及不分地区的总均数 (Comined Adjusted Means for G )。Adjusted and Estimated MeansVariable . X1CELLObs. MeanAdj. MeanEst. Mea
40、nRaw 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. Resid.000.000.000.000.000.000Adjusted and Estimated Means (Cont.)Variable . X3CELLObs. Mean Adj. MeanEst. MeanRaw Resid. Std. Resid.160.00760.00760.007.0