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1、多元统计分析方法多元统计分析方法The Methods of Multivariate Statistical Analysis第四章第四章 多变量方差分析多变量方差分析 什么是多变量方差分析？多变量方差分析在医学中的应用方差分析的分类方差分析的分类单反应变量(y)多反应变量(y1,y2yk)单效应因子(A)双效应因子(A,B)多效应因子(A,B,C)无交互效应有交互效应2）根据效应因子的随机性：固定模型(fixed model)：效应因子是专门指定的。随机模型(random model)：效应因子是从很多的因子中随机抽取出来的。混合模型(mixed model)：效应因子包含两种类型因子。1
2、）根据变量的个数：什么是多变量方差分析？什么是多变量方差分析？MANOVA分析一个或多个效应因子是如何影响一组反应变量的。身高：y1体重：y2胸围：y3=父母SES舒张压：y1收缩压：y2=职务生活方式+反应变量 效应因子 多变量方差分析在医学中的应用实例多变量方差分析在医学中的应用实例1、单组设计资料的MANOVA2、配对设计资料的MANOVA3、成组设计资料的MANOVA4、多因子的MANOVA5、重复设计资料的MANOVA6、有协变量的MANOVA【例【例4-1】单组设计资料的】单组设计资料的MANOVA实例实例 为了了解某地在不同时期的儿童生长发育情况，调查了20名8岁男童身高(x1)
3、、体重(x2)、胸围(x3)，cm、21.54kg、57.98cm。试问：本次调查结果与10年前结果是否相同？表4-6 儿童生长发育情况调查数据【SAS程序】data eg4_1；input id x1 x2 x3；y1=x1-121.57；y2=x2-21.54；y3=x3-57.98；cards；run；proc means；var y1-y3；run；proc glm；model y1 y2 y3=/ss3 nouni；manova h=intercept/printe printh；run；【SASSAS输出的结果】输出的结果】The MEANS Procedure The MEANS
4、 Procedure Variable N Mean Std Dev Minimum Maximum Variable N Mean Std Dev Minimum Maximum-The GLM Procedure The GLM Procedure Number of observations 20 Number of observations 20 Multivariate Analysis of Variance Multivariate Analysis of VarianceMANOVA Test Criteria and Exact F Statistics for the Hy
5、pothesis of No MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept EffectOverall Intercept EffectStatistic Value F Value Num DF Den DF Pr FStatistic Value F Value Num DF Den DF Pr FWilks Lambda 0.20656246 21.77 3 17 .0001Wilks Lambda 0.20656246 21.77 3 17 .0001Pill
6、ais Trace 0.79343754 21.77 3 17 .0001Pillais Trace 0.79343754 21.77 3 17 .0001Hotelling-Lawley Trace 3.84115073 21.77 3 17 .0001Hotelling-Lawley Trace 3.84115073 21.77 3 17 .0001Roys Greatest Root 3.84115073 21.77 3 17 .0001Roys Greatest Root 3.84115073 21.77 3 17 .0001结论：因为P FStatistic Value F Valu
7、e Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHotelling-Lawley【例【例4-3】成组设计资料的】成组设计资料的MANOVA实例实例为了研究某种疾病的治疗，观察了24个病人使用三种不同药品后的两个指标，每种药品观察了4个男性和4个女性，数据列在表4-8中。试比较药品对两个指标所起的作用。表4-8 三种不同药品用药后的观察数据【SAS程序】程序】data eg4_3；input sex$drug$；input y1 y2；output；input y1 y2；output；input y1 y2；output；in
8、put y1 y2；output；cards；M A 5 6 5 4 9 9 7 6 F C 14 13 12 12 12 10 8 7run;proc glm manova；classes drug；model y1 y2=drug/nouni；contrast Drug A vs B drug 1-1 0；contrast Drug A vs C drug 1 0-1；contrast Drug B vs C drug 0 1-1；manova h=drug ；means drug；run；【SASSAS部分部分 输出结果】输出结果】General Linear Models Proced
9、ureGeneral Linear Models ProcedureClass Level InformationClass Level InformationClass Levels ValuesClass Levels ValuesSEX 2 Female MaleSEX 2 Female MaleDRUG 3 A B CDRUG 3 A B CNumber of observations in data set=24Number of observations in data set=24Multivariate Analysis of VarianceMultivariate Anal
10、ysis of VarianceManova Test Criteria and F Approximations for the Hypothesis of Manova Test Criteria and F Approximations for the Hypothesis of no Overall DRUG no Overall DRUG EffectEffectStatistic Value F Num DF Den DF Pr FStatistic Value F Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHo
11、telling-LawleyManova Test Criteria and Exact F Statistics for the Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall no Overall Drug A vs B EffectDrug A vs B EffectStatistic Value F Num DF Den DF Pr FStatistic Value F Num DF Den DF Pr FWilksWilksPillaiPillaiHo
12、telling-LawleyHotelling-LawleyManova Test Criteria and Exact F Statistics for the Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall no Overall Drug A vs C EffectDrug A vs C EffectStatistic Value F Num DF Den DF Pr FStatistic Value F Num DF Den DF Pr FWilksWil
13、ksPillaiPillaiHotelling-LawleyHotelling-LawleyManova Test Criteria and Exact F Statistics for the Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall Drug no Overall Drug B vs C EffectB vs C EffectStatistic Value F Num DF Den DF Pr FStatistic Value F Num DF Den
14、 DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHotelling-LawleyLevel of -Y1-Y2-Level of -Y1-Y2-DRUG N Mean SD Mean SDDRUG N Mean SD Mean SD【例【例4-4】析因设计资料的】析因设计资料的MANOVA实例实例为了研究某种疾病的治疗，观察了24个病人使用三种不同药品后的两个指标，每种药品观察了4个男性和4个女性，数据列在表4-8中。试分析性别和药品对两个指标所起的作用。表4-8 三种不同药品用药后的观察数据【SAS 程序】proc glm data=eg4_3 m
15、anova；classes sex drug；model y1 y2=sex drug sex*drug/nouni；contrast Drug A vs B drug 1-1 0；contrast Drug A vs C drug 1 0-1；contrast Drug B vs C drug 0 1-1；contrast Drug A vs B/sex=m drug 1-1 0 sex*drug 1-1 0 0 0 0；contrast Drug A vs B/sex=f drug 1-1 0 sex*drug 0 0 0 1-1 0；contrast Drug A vs C/sex=m
16、drug 1 0-1 sex*drug 1 0-1 0 0 0；contrast Drug A vs C/sex=f drug 1 0-1 sex*drug 0 0 0 1 0-1；contrast Drug B vs C/sex=m drug 0 1-1 sex*drug 0 1-1 0 0 0；contrast Drug B vs C/sex=f drug 0 1-1 sex*drug 0 0 0 0 1-1；manova h=sex drug sex*drug ；means sex drug；run；【SASSAS主要输出结果】主要输出结果】General Linear Models P
17、rocedureGeneral Linear Models ProcedureMultivariate Analysis of VarianceMultivariate Analysis of VarianceManova Test Criteria and Exact F Statistics for the Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no Overall SEX no Overall SEX EffectEffectStatistic Value F Num
18、 DF Den DF Pr FStatistic Value F Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHotelling-LawleyManova Test Criteria and F Approximations for the Hypothesis of Manova Test Criteria and F Approximations for the Hypothesis of no Overall DRUG no Overall DRUG EffectEffectStatistic Value F Num D
19、F Den DF Pr FStatistic Value F Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHotelling-Lawley【例【例4-5】重复测量设计资料的】重复测量设计资料的MANOVA实例实例欲比较两种治疗（胸腔切开术grp=1和胸腔镜检术grp=2）方案的效果，将40个病人随机分成两组，分别在术前、术后2天、术后7天测定患者的T细胞结果列在表4-9中。表4-9 胸腔切开术和胸腔镜检术患者的T细胞测定结果【SAS程序】程序】data eg4_5；do id=1 to 22；do grp=1 to 2；inp
20、ut d1 d2 d3；output；end；end；cards；74 68 71 46 61 58 .83 87 83run；proc glm；class grp；model d1 d2 d3=grp/nouni ss3；repeated time 3 contrast(1)/printe summary；lsmeans grp/stderr；run；【SAS输出结果】输出结果】General Linear Models ProcedureClass Level InformationClass Levels ValuesGRP 2 1 2Number of observations in
21、data set=44NOTE:Observations with missing values will not be included in this analysis.Thus,only 41 observations can be used in this analysis.Test for SphericityTest for Sphericity:Mauchly:MauchlyChisquare Approximation=15.725084 with 2 df Chisquare Approximation=15.725084 with 2 df Prob ChisquarePr
22、ob ChisquareApplied to Orthogonal Components:Applied to Orthogonal Components:Test for Sphericity:MauchlyTest for Sphericity:MauchlyChisquare Approximation=1.3937694 with 2 df Chisquare Approximation=1.3937694 with 2 df Prob ChisquareProb Chisquare Manova Test Criteria and Exact F Statistics for the
23、 Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no time no time EffectEffectStatistic Value F Value Num DF Den DF Pr FStatistic Value F Value Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-Lawley Trace 0.18957956 3.60 2 38 Hotelling-Lawley Trace 0.18957956 3.60 2
24、38 Manova Test Criteria and Exact F Statistics for the Hypothesis of Manova Test Criteria and Exact F Statistics for the Hypothesis of no time*grp no time*grp EffectEffectStatistic Value F Value Num DF Den DF Pr FStatistic Value F Value Num DF Den DF Pr FWilksWilksPillaiPillaiHotelling-LawleyHotelli
25、ng-LawleyGeneral Linear Models ProcedureGeneral Linear Models ProcedureRepeated Measures Analysis of VarianceRepeated Measures Analysis of VarianceTests of Hypotheses for Between Subjects EffectsTests of Hypotheses for Between Subjects EffectsSource DF Type III SS Mean Square F Value Pr FSource DF T
26、ype III SS Mean Square F Value Pr FGeneral Linear Models ProcedureGeneral Linear Models ProcedureRepeated Measures Analysis of VarianceRepeated Measures Analysis of VarianceUnivariate Tests of Hypotheses for Within Subject EffectsUnivariate Tests of Hypotheses for Within Subject EffectsSource:TIMESo
27、urce:TIME Adj Pr F Adj Pr F DF Type III SS Mean Square F Value Pr F G-G H-F DF Type III SS Mean Square F Value Pr F G-G H-F 2 169.29178823 84.64589411 3.07 0.0519 0.0538 2 169.29178823 84.64589411 3.07 0.0519 0.0538 Source:TIME*GRPSource:TIME*GRP Adj Pr F Adj Pr F DF Type III SS Mean Square F Value
28、Pr F G-G H-F DF Type III SS Mean Square F Value Pr F G-G H-FSource:Error(TIME)Source:Error(TIME)DF Type III SS Mean Square DF Type III SS Mean SquareGreenhouse-GeisserGreenhouse-Geisser Huynh-Feldt Huynh-FeldtGeneral Linear Models ProcedureGeneral Linear Models ProcedureRepeated Measures Analysis of
29、 VarianceRepeated Measures Analysis of VarianceAnalysis of Variance of Contrast VariablesAnalysis of Variance of Contrast VariablesTIME.N represents the contrast between the nth level of TIME and the 1stTIME.N represents the contrast between the nth level of TIME and the 1stSource DF Type III SS Mea
30、n Square F Value Pr FSource DF Type III SS Mean Square F Value Pr FSource DF Type III SS Mean Square F Value Pr FSource DF Type III SS Mean Square F Value Pr FMEAN 1 321.68887852 321.68887852 6.48 MEAN 1 321.68887852 321.68887852 6.48 Least Squares Means Least Squares Means Standard Standard grp d1
31、LSMEAN Error Pr|t|grp d1 LSMEAN Error Pr|t|1 71.0000000 2.3785943 .0001 1 71.0000000 2.3785943 .0001 2 70.7272727 2.2104761 .0001 2 70.7272727 2.2104761|t|grp d2 LSMEAN Error Pr|t|1 70.4210526 2.4944288 .0001 1 70.4210526 2.4944288 .0001 2 73.0000000 2.3181235 .0001 2 73.0000000 2.3181235|t|grp d3 L
32、SMEAN Error Pr|t|1 73.5263158 1.9411064 .0001 1 73.5263158 1.9411064 .0001 2 73.8181818 1.8039098 .0001 2 73.8181818 1.8039098 0.05,单变量方差分析，将时间作为一个效应因子。P Chisquare=Applied to Orthogonal Components:Test for Sphericity:MauchlyChisquare Approximation=17.631505 with 5 dfProb Chisquare=SAS输出结果输出结果相关分析结果说
33、明重复测量变量之间高度相关。球形检验结果说明该数据应当使用多变量方差分析方法。Manova Test Criteria and Exact F Statistics forthe Hypothesis of no TIME EffectH=Type III SS&CP Matrix for TIME E=Error SS&CP MatrixS=1 M=0.5 N=7Statistic Value F Num DF Den DF Pr FWilksPillaiHotelling-Lawley Tra 8.795432 46.909 3 16 SAS输出结果输出结果多变量方差分析结果说明时间对体温
34、有显著性影响。Manova Test Criteria and Exact F Statistics forthe Hypothesis of no TIME*A EffectH=Type III SS&CP Matrix for TIME*A E=Error SS&CP MatrixS=1 M=0.5 N=7Statistic Value F Num DF Den DF Pr FWilksPillaiHotelling-Lawley 0.10778626 0.57486 3 16 SAS输出结果输出结果多变量方差分析结果说明时间与处理方法之间对体温的交互影响不显著。General Linea
35、r Models ProcedureRepeated Measures Analysis of VarianceUnivariate Tests of Hypotheses for Within Subject EffectsSource:TIME Adj Pr FDF Type III SS Mean Square F Value Pr F G-G H-F 3 20.57537500 6.85845833 77.23 0.0001 0.0001 Source:TIME*A Adj Pr FDF Type III SS Mean Square F Value Pr F G-G H-F 3 0.
36、15637500 0.05212500 0.59 0.6262 0.5746 SAS输出结果输出结果多变量重复资料的方差分析结果说明时间对体温有显著性影响；时间与处理方法之间对体温的交互影响不显著。nth level of TIME and the 2ndSource DF Type III SS F Value Pr FMEAN 1 31.25000000 107.92 Source DF Type III SS F Value Pr FMEAN 1 25.08800000 138.78 Source DF Type III SS F Value Pr FMEAN 1 25.31250000
37、 123.91 SAS输出结果输出结果时间点2和时间点1,3,4都有显著性差异。General Linear Models ProcedureRepeated Measures Analysis of VarianceTests of Hypotheses for Between Subjects EffectsSource DF Type III SS F Value Pr FA 1 11.93512500 12.51 SAS输出结果输出结果多变量重复资料的方差分析结果说明处理方法对体温有显著性影响。General Linear Models Procedure Least Squares
38、MeansA T1 Pr|T|H0:LSMEAN LSMEAN1=LSMEAN21 31.3300000 A T2 Pr|T|H0:LSMEAN LSMEAN1=LSMEAN21 32.5100000 A T3 Pr|T|H0:LSMEAN LSMEAN1=LSMEAN21 31.3600000 A T4 Pr|T|H0:LSMEAN LSMEAN1=LSMEAN21 31.3400000 SAS输出结果输出结果各个时间点上两个处理组之间体温有显著性差异。且随着时间的延长，差异的显著性越来越强。医学研究中经常需要根据不同的要求制订出不同类型的研究设计，每一种研究设计收集到的数据资料能否找到一个恰当的统计分析方法来处理，这是很关键的一个问题。若处理不好，得到的结论可能不可靠，或没有实际意义，将会造成某种程度的浪费。因此，在选择用哪一种方差分析方法来处理实验或试验设计研究资料时，要非常认真考虑，并要结合本专业基础知识和实际经验，而不能生搬硬套。注意事项注意事项总总 结结什么是方差分析以及方差分析的基本原理？方差分析对数据的假设条件？方差分析的分类？完全随机设计资料的单因子方差分析方法？随机区组设计资料的双因子方差分析方法？析因设计资料的多因子方差分析方法？拉丁方设计资料的三因子方差分析方法？嵌套设计、裂区设计和重复测量设计？Thanks!THANK