协方差分析 完整版

协方差分析

协方差分析是把直线回归法与方差分析法结合起来的一种方法，其目的是把与y值呈直线关系的x值化成相等后，再来检验各组y均数（即修正均数）间差别的统计意义。1一元完全随机设计资料的协方差分析

1.1例11－1

下列数据是研究镉作业工人暴露于烟尘的年数与肺活量的关系（表1）。按暴露年数将工人分为两组：甲组暴露≥10年，乙组暴露<10年。两组工人年龄未经控制。问该两组暴露于镉作业工人平均肺活量是否相同？表1镉作业工人接触烟尘年数与肺活量的关系研究

甲组（暴露≥10年）

乙组（暴露<10年）x1(年龄)y1(肺活量，L)x2(年龄)y2(肺活量，L)394.62405.29415.52413.71454.02495.09522.70474.31612.70653.03582.73593.67434.61394.73384.58425.12433.89434.62374.30502.70503.50453.06484.06514.51464.66582.88383.64385.09

由于肺活量的大小除与接触烟尘的年数有关，也与人的年龄（协变量）有关。因此在作两组肺活量大小的比较时首先要校正年龄因素。这也是协方差分析的基本思想。协方差分析对数据有一定要求，请参见统计书籍，这里只介绍SAS分析方法。本例为两组比较，协方差分析亦可用于多组比较。协方差分析对数据的要求：正态分布两条回归线平行yx1.2数据集

协方差分析需调用SAS的GLM过程来分析。如需用infile语句调入数据。应先在当前目录中建立数据集anocov1.dat如下：

12394.62405.29415.25413.71454.02495.09522.70474.31612.7653.03582.73593.6716434.61394.73384.58425.12433.89434.62374.30502.70503.50453.06484.06514.51464.66582.88383.64385.091.3程序11-1

dataanocov1;infile‘anocov1.dat’;doc=1to2;inputn;doi=1ton;inputxy@@;output;end;end;procglm;classc;modely=xc/solution;lsmeansc/stderrpdiff;run;

数据用循环语句读入。首先读入第一组的样 本含量n1＝12，再逐对读入第一组的x,y值，读入第一组后读入第二组样本含量n2=16，再逐对读入第二组各x,y值。程序中需有output语句。用一般线性模型过程procglm作协方差分析，语句classc；指定分组变量为c；用modely=xc说明模型。由于在模型中已指定c为分组变量，x与c的次序不影响结果。但class语句必须在model语句之前。选择项solution是要输出回归方程中各参数的估计值，Lsmeans语句（leastsquaremeans）是要输出各组（c）修正均数，选择项为标准误（stderr）及均值之间差异比较的概率（pdiff）。1.4结果与解释GeneralLinearModelsProcedureDependentVariable:YSourceDFSumofSquaresMeanSquareFValuePr>FModel29.484012864.74200643210.0003Error2510.574797860.42299191CorrectedTotal2720.05881071R-SquareC.V.RootMSEYMean0.47281016.105590.650378294.03821429

这一部分是对数据的总的方差分析，其中Error部分即一般统计书籍中所介绍的公共部分的估计误差的自由度，平方和及均方。这里误差均方为0.43895596。SourceDFTypeISSMeanSquareFValuePr>FX19.136990469.1369904621.600.0001C10.347022400.347022400.820.3737SourceDFTypeIIISSMeanSquareFValuePr>FX19.222712569.2227125621.800.0001C10.347022400.347022400.820.3737

这部分出现了TypeISS及TypeIIISS，TypeISS已在第6章中说明，TypeIIISS即为我们在一般方差分析中计算的平方和，这里c行中是修正均数的平方和均方、F及概率。其中：

F＝c的均方/Error的均方＝0.44375368/0.43895596＝1.01公式即为统计书籍中的以下公式：

F＝修正均数均方/公共误差均方本例自由度为1及25，P＝0.3243，说明两组修正均数间差别无统计意义。TforH0:Pr>|T|StdErrorofParameterEstimateParameter=0EstimateINTERCEPT7.671016715B9.870.00010.77727765X-0.080093466-4.670.00010.01715275C10.240299888B0.910.37370.2653022020.000000000B...NOTE:TheX'Xmatrixhasbeenfoundtobesingularandageneralizedinversewasusedtosolvethenormalequations.Estimatesfollowedbytheletter'B'arebiased,andarenotuniqueestimatorsoftheparameters.

这一部分截矩，变量c及x的参数估计及其检验结果。其中x的回归系数－0.081736730为公共回归线的斜率。注释（note)：中说明矩阵x’x为奇异矩阵，正规方程用广义逆矩阵求解。后面附有字母‘B’的估计值是有偏的，不是参数的唯一估计值。经过t检验截矩和斜率P皆<0.001，而分组变量t=1.01,P＝0.3243，无统计意义。GeneralLinearModelsProcedureLeastSquaresMeansCYStdErrPr>|T|Pr>|T|H0:LSMEANLSMEANH0:LSMEAN=0LSMEAN1=LSMEAN214.175528510.195166070.00010.37373.935228620.167435830.0001

最后部分是修正均数，两者与零比较的t值皆有高度统计意义，而两修正均数总体无效假设检验的概率为0.3243，无统计意义。这里P＝0.3243一共出现了三次皆说明同一问题。修正均数间的差别无统计意义。2随机区组设计资料的协方差分析2.1例11－2

在“核黄素缺乏对于蛋白质利用的影响研究”中，将体重相近，出生3周左右的大鼠36只，按照窝别，性别等条件分成12个配伍组，每组3只，随机分到三个不同饲料组进行喂养。观察核黄素缺乏对体重增长的影响，结果如表2所示表2三组大鼠的进食量（x,g）与所增体重（y,g）区组

核黄素缺乏组限食量组不限食量组xyxyxy1256.927.0260.332.0544.7160.32271.641.7271.147.7481.296.13210.225.0214.736.7418.9114.64300.152.0300.165.0566.6134.85262.214.5269.739.0394.576.36304.448.8307.537.9426.672.87272.448.0278.951.5416.199.48248.29.5256.226.7549.9133.79242.837.0240.841.0580.5147.010342.956.5340.761.3608.3165.811356.976.0356.3102.1559.6169.812198.29.2199.28.1371.954.31.2程序

程序11－2dataa;doa=1to12;dob=1to3;inputxy@@;output;end;end;cards;256.927.0260.332.0544.7160.3271.641.7271.147.7481.296.1210.225.0214.736.7418.9114.6300.152.0300.165.0566.6134.8262.214.5269.739.0394.576.3304.448.8307.537.9426.672.8272.448.0278.951.5416.199.4248.29.5256.226.7549.9133.7242.837.0240.841.0580.5147.0342.956.5340.761.3608.3165.8356.976.0356.3102.1559.6169.8198.29.2199.28.1371.954.3

procglm;classab;modely=xab/solution;lsmeansb/stderrpdiff;run;x,y数据是按区组配对的，可用随机区组的协方差分析。这时只要在模型中加入区组变量a即可。2.3结果（解释略）GeneralLinearModelsProcedureDependentVariable:YSourceDFSumofSquaresMeanSquareFValuePr>FModel1473431.563870035245.1117050047.250.0001Error212331.04585219100218344CorrectedTotal3575762.60972222R-SquareC.V.RootMSEYMean0.96923215.6788610.5357573767.19722222SourceDFTypeISSMeanSquareFValuePr>FX168955.8201692468955.82016924621.210.0001A114022.75020308365.704563923.290.0091B2452.99349771226.496748862.040.1550SourceDFTypeIIISSMeanSquareFValuePr>FX16077.123592256077.1235922554.750.0001A113766.10034965342.372759063.080.0128B2452.99349771226.496748862.040.1550TforH0:Pr>|T|StdErrorofParameterEstimateParameter=0EstimateINTERCEPT-86.79904269B-3.790.001122.88610688X0.402537117.400.00010.05440297A19.97254764B0.990.335010.1072345123.80468431B0.390.70079.76310076324.90366188B2.860.00948.7078514046.73050018B0.600.5551223297275-1.67952643B-0.190.85479.061881746-6.82099627B-0.690.49809.89105853715.85246646B1.700.10389.322385078-5.47435836B-0.550.591210.03556677957735376B1.140.265910.12823581100.54470288B0.040.966512.799021441124.54085578B1.960.063912.54471628120.00000000B.．B17.31903366B0.570.572612.76927063215.00629909B1.190.248612.6468364130.00000000B...NOTE:TheX'Xmatrixhasbeenfoundtobesingularandageneralizedinversewasusedtosolvethenormalequations.Estimatesfollowedbytheletter'B'arebiased,andarenotuniqueestimatorsoftheparameters.GeneralLinearModelsProcedureLeastSquaresMeansBYStdErrPr>|T|Pr>|T|H0:SMEAN(i)=LSMEAN(j)LSMEANLSMEANH0:LSMEAN=0i/j123167.07447835.06569300.00011.0.08850.5726274.76174374.96225530.000120.0885.0.2486359.75544468.53246570.000130.57260.2486.NOTE:Toensureoverallprotectionlevel,onlyprobabilitiesassociatedwithpre-plannedcomparisonsshouldbeused.

以上结果说明每一个调整均数都有统计意义．对于各组间的比较结果由最后三行表示．可见两两之间差别皆无统计意义。3多元完全随机设计的协方差分析例11－1中的分组变量c为两个水平的单一因素。如果还有另外的一个或几个因素，则为多元协方差分析。如果仍是完全随机设计资料，只要在模型语句中加入这些变量即可。如有因素a和b,则语句可写成：modely=aba*bx/solution;。3.1例11－3

某地测量了30名初生至3周岁的身高、体重和体表面积（表3），男女各15名，考虑男女身高、体重相同时体表面积是否相同，即能否将男女有身高、体重推算体表面积的推算公式合并为一个，进行了协方差分析。表330名婴幼儿身高（cm)、体重（kg）及体表面积（cm2）

男(=1)

女(=2)身高(x1)体重(x2)体表面积(y)身高(x1)体重(x2)体表面积(y)5432446.25432117.350.52.251928.4532.252200.2512.52094.551.52.52200.256.53.52506.75131950.35232121.05131632.5769.53845.9777.53934.08094380.877104180.4749.54314.2779.54246.18094078.47493358.87684143.5737.53809.7（续表）9613.55830.291125358.497146013.691135601.799166410.694156074.992115283.392125299.494156101.69112.55291.53.2程序

程序11－3datab;dosex=1to2;doi=1to15;inputx1x2y@@;output;end;end;cards;5432446.250.52.251928.4512.52094.556.53.52506.75232121.0769.53845.98094380.8749.54314.28094078.47684143.59613.55830.297146013.699166410.692115283.394156101.65432117.3532.252200.251.52.52200.25131950.35131632.5777.53934.077104180.4779.54246.17493358.8737.53809.7

91125358.491135601.794156074.992125299.412.55291.5;procglm;classsex;modely=x1x2sex/solution;lsmeanssex/stderrpdiff;quit;run;2.3结果（解释略）GeneralLinearModelsProcedureClassLevelInformation

ClassLevelsValuesSEX212Numberofobservationsindataset=29DependentVariable:YSourceDFSumofSquaresMeanSquareFValuePr>FModel365081193.4566943021693731340.0001Error251145778.0702024545831.12280810CorrectedTotal2866226971.52689670R-SquareC.V.RootMSEYMean0.9826995.471731214.082046913912.51034483SourceDFTypeISSMeanSquareFValuePr>FX1164635484.6818889064635484.681888901410.300.0001X21360428.63327691360428.633276917.860.0096SEX