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正交因子分析（设计性实验）(Orthogonalfactoranalysis)实验原理：因子分析是主成分分析的推广和发展，其目的是用少数几个不可观测的隐变量，即因子，来解释原始变量之间的相关关系，它也是属于多元分析中处理降维的一种统计方法。因子分析的基本思想是通过变量间的协方差矩阵（或相关系数矩阵）内部结构的研究，寻找能控制所有变量的少数几个因子去描述多个变量之间的相关关系。因子分析中最常用的数学模型是正交因子模型，其特点是模型中的因子相互之间正交。实验题目一：下表中给出了二战以来奥运会运动员十项运动成绩的相关系数矩阵：(E9a6)100米1.00.........跳远0.591.00........铅球0.350.421.00.......跳高0.340.510.381.00......400米0.630.490.190.291.00.....110米跨栏0.400.520.360.460.341.00....铁饼0.280.310.730.270.170.321.00...撑竿跳高0.200.360.240.390.230.330.241.00..标枪0.110.210.440.170.130.180.340.241.00.1500米-0.070.09-0.080.180.390.00-0.020.17-0.001.00实验要求：（1）试由相关系数矩阵作因子分析；covmat（2）试根据因子载荷，并结合题目背景知识，对公共因子进行命名。实验题目二：下表中给出了不同国家及地区的女子径赛记录：(t1a7)Country100m(s)200m(s)400m(s)800m(min)1500m(min)3000m(min)Marathon(min)argentin11.6122.9454.52.154.439.79178.52australi11.222.3551.081.984.139.08152.37austria11.4323.0950.621.994.229.34159.37belgium11.4123.045224.148.88157.85bermuda11.4623.0553.32.164.589.81169.98brazil11.3123.1752.82.14.499.77168.75burma12.1424.47552.184.459.51191.02canada1122.2550.0624.068.81149.45chile1224.5254.92.054.239.37171.38china11.9524.4154.972.084.339.31168.48columbia11.62453.262.114.359.46165.42cookis12.927.160.42.34.8411.1233.22costa11.9624.658.252.214.6810.43171.8czech11.0921.9747.991.894.148.92158.85denmark11.4223.5253.62.034.188.71151.75domrep11.7924.0556.052.244.749.89203.88finland11.1322.3950.142.034.18.92154.23france11.1522.5951.7324.148.98155.27gdr10.8121.7148.161.933.968.75157.68frg11.0122.3949.751.954.038.59148.53gbni1122.1350.461.984.038.62149.72greece11.7924.0854.932.074.359.87182.2guatemal11.8424.5456.092.284.8610.54215.08hungary11.4523.0651.52.014.148.98156.37india11.9524.2853.62.14.329.98188.03indonesi11.8524.2455.342.224.6110.02201.28ireland11.4323.5153.242.054.118.89149.38israel11.4523.5754.92.14.259.37160.48italy11.292352.011.963.988.63151.82japan11.732453.732.094.359.2150.5kenya11.7323.8852.724.159.2181.05korea11.9624.4955.72.154.429.62164.65dprkorea12.2525.7851.21.974.259.35179.17luxembou12.0324.9656.12.074.389.64174.68malaysia12.2324.2155.092.194.6910.46182.17mauritiu11.7625.0858.12.274.7910.9261.13mexico11.8923.6253.762.044.259.59158.53netherla11.2522.8152.381.994.069.01152.48nz11.5523.1351.62.024.188.76145.48norway11.5823.3153.122.034.018.53145.48png12.2525.0756.962.244.8410.69233philippi11.7623.5454.62.194.610.16200.37poland11.1322.2149.291.953.998.97160.82portugal11.8124.2254.32.094.168.84151.2rumania11.4423.4651.21.923.968.53165.45singapor12.32555.082.124.529.94182.77spain11.823.9853.592.054.149.02162.6sweden11.1622.8251.792.024.128.84154.48switzerl11.4523.3153.112.024.078.77153.42taipei11.2222.6252.52.14.389.63177.87thailand11.7524.4655.82.24.7210.28168.45turkey11.9824.4456.452.154.379.38201.08usa10.7921.8350.621.963.958.5142.72ussr11.0622.1949.191.893.878.45151.22wsamoa12.7425.8558.732.335.8113.04306（数据来源：1984年洛杉机奥运会IAAF/AFT径赛与田赛统计手册）ussr11.0622.1949.191.893.878.45151.22rumania11.4423.4651.21.923.968.53165.45实验要求：（1）根据以上数据对女子径赛项目作因子分析；（2）对公共因子进行解释；（3）计算各个国家的第一因子得分并进行排名。要求列出排名前10的国家或地区，并给出中国的名次。实验题目一分析报告：R程序：record-read.table(data4.txt,head=F)#导入数据record-record[,-1]#删除第一列record-as.matrix(record)#将原数据矩阵化options(digits=2)#保留两位小数pca.data1-princomp(covmat=record)#以相关系数矩阵作为基础，建立主成分分析summary(pca.data1)#输出主成分分析报表fact1.st-factanal(covmat=record,factors=5,rotation=none)#作因子分析，不旋转fact1.ro-factanal(covmat=record,factors=5,rotation=varimax)#作因子分析，旋转fact1.st#输出不旋转的结果fact1.ro#输出旋转的结果apply((fact1.ro$loadings)^2,1,sum)#计算共同度fact2.ro-factanal(covmat=record,factors=4,rotation=varimax)#作因子分析，旋转fact2.ro#输出旋转的结果apply((fact2.ro$loadings)^2,1,sum)#计算共同度输出结果及分析：（1）试由相关系数矩阵作因子分析；record-read.table(data4.txt,head=F)#导入数据record-record[,-1]#删除第一列record-as.matrix(record)#将原数据矩阵化options(digits=2)#保留两位小数pca.data1-princomp(covmat=record)#以相关系数矩阵作为基础，建立主成分分析summary(pca.data1)#输出主成分分析报表为了确定因子分析中因子的数目，我们先对相关系数矩阵做主成分分析表1主成分分析报表Comp.1Comp.2Comp.3Comp.4Comp.5Comp.6Comp.7Comp.8Comp.9Comp.10Standarddeviation1.951.231.060.9560.8490.7710.7260.6190.4850.456ProportionofVariance0.380.150.110.0910.0720.0590.0530.0380.0240.021CumulativeProportion0.380.530.640.7330.8050.8650.9170.9560.9791.000由方差累计贡献率得到，在第五主成分，累积贡献率达到了80%以上，并趋于稳定。我们确定因子分析中因子数目为5.fact1.st-factanal(covmat=record,factors=5,rotation=none)#作因子分析，不旋转fact1.ro-factanal(covmat=record,factors=5,rotation=varimax)#作因子分析，旋转fact1.st#输出不旋转的结果fact1.ro#输出旋转的结果apply((fact1.ro$loadings)^2,1,sum)#计算共同度做因子分析，得到未旋转的因子载荷以及旋转的因子载荷表2未旋转的因子载荷FactorFactor1Factor2Factor3Factor4Factor5100米0.2080.7910.301-0.167跳远0.3780.5950.2460.242铅球0.6440.761跳高0.4150.3440.1570.471-0.139400米0.4460.688-0.113-0.2030.116110米跨栏0.2650.4350.2610.343铁饼0.5030.534撑竿跳高0.3070.2400.4020.214标枪0.3130.3140.3781500米0.707-0.704累积贡献率0.20.380.550.6160.640表3旋转的因子载荷FactorFactor1Factor2Factor3Factor4Factor5Communalities100米0.1710.8150.276-0.1410.79跳远0.2230.4800.5800.62铅球0.9550.1390.2411.00跳高0.2110.1520.6870.1170.56400米0.7600.1930.3260.1260.74110米跨栏0.1870.2780.5650.45铁饼0.6930.1250.1940.1110.55撑竿跳高0.1120.5010.1190.2820.36标枪0.4080.1400.4010.351500米0.9891.00累积贡献率0.170.340.500.610.640观察表格中被标注为