第十章因子分析

高級(jí)生物統(tǒng)計(jì)第十章因子分析因子分析介紹：1、因子分析的概念2、因子分析的過(guò)程因子分析的概念由于實(shí)測(cè)的變量間存在一定的相關(guān)關(guān)系，因此有可能用較少數(shù)的綜合指標(biāo)分別綜合存在于各變量中的各類信息，而綜合指標(biāo)之間彼此不相關(guān)，即各指標(biāo)代表的信息不重疊。綜合指標(biāo)稱為因子研究相關(guān)陣或協(xié)方差陣的內(nèi)部依賴關(guān)系，將多個(gè)變量綜合為少數(shù)幾個(gè)因子把這個(gè)數(shù)據(jù)的N個(gè)變量用一兩個(gè)綜合變量特征值>1累計(jì)貢獻(xiàn)率>0.8R型分析和Q型分析R型分析：研究指標(biāo)之間的相關(guān)關(guān)系，通過(guò)對(duì)變量的相關(guān)陣或協(xié)差陣內(nèi)部結(jié)構(gòu)的研究，找出控制所有變量的公共因子Q型分析：研究樣品之間的相關(guān)關(guān)系，通過(guò)對(duì)樣品的相似矩陣內(nèi)部結(jié)構(gòu)的研究，找出控制所有樣品的幾個(gè)主要因素。因子分析的作用一.尋求幾個(gè)控制所有變量的公共因子，因子數(shù)量少于變量數(shù)量二.所得到的公共因子進(jìn)一步分析（聚類分析）因子分析與主成分分析的區(qū)別1.主成分分析是通常的變量變換，因子分析需要構(gòu)造因子模型，把一個(gè)變量看成由公共因子和特殊因子構(gòu)成，解釋變量的內(nèi)部關(guān)系2.主成分的個(gè)數(shù)與變量個(gè)數(shù)相等，是將一組相關(guān)的變量轉(zhuǎn)化為不相關(guān)的分量，因子的個(gè)數(shù)少于變量個(gè)數(shù)因子分析模型一般地，設(shè)X=(x1,x2,…,xp)’為可觀測(cè)的隨機(jī)變量，且有f=(f1,f2,…,fm)’為公共（共性）因子(commonfactor)，簡(jiǎn)稱因子(factor)e=(e1,e2,…,ep)’為特殊因子（specificfactor）μ=(μ1,μ2,…,μp)’為隨機(jī)變量x的總體均值A(chǔ)=(aij)p*m為因子負(fù)荷（載荷）（factorloading）矩陣通常先對(duì)x作標(biāo)準(zhǔn)化處理，使標(biāo)準(zhǔn)化得到的新變量均值為零，方差為１．這樣就有假定（１）fi的均數(shù)為０，方差為１；（２）ei的均數(shù)為０，方差為δi；（３）fi與ei相互獨(dú)立．則稱x為具有m個(gè)公共因子的因子模型因子載荷（負(fù)荷）aij是隨機(jī)變量xi與公共因子fj的相關(guān)系數(shù)。設(shè)稱gj2為公共因子fj對(duì)x的“貢獻(xiàn)”，是衡量公共因子fj重要性的一個(gè)指標(biāo)。因子分析的步驟輸入原始數(shù)據(jù)xn*p，計(jì)算樣本均值和方差，進(jìn)行標(biāo)準(zhǔn)化計(jì)算（處理）；求樣本相關(guān)系數(shù)矩陣R=(rij)p*p；求相關(guān)系數(shù)矩陣的特征根λi

(λ1,λ2,…,λp>0)和相應(yīng)的標(biāo)準(zhǔn)正交的特征向量li；12個(gè)玉米雜交種10項(xiàng)指標(biāo)觀察值：品種代號(hào)平均畝產(chǎn)（x1）穗長(zhǎng)（x2）穗行數(shù)（x3）行粒數(shù)（x4）穗粒重（x5）出粒率（x6）千粒重（x7）蛋白質(zhì)（x8）全籽粒狀賴氨酸（x9）百克賴氨酸（x10）194723.414.845.30.4685.23739.540.373.88293523.216.241.70.485.33057.90.384.813918.220.914.843.30.3882.63209.510.430.524910.723.416.1440.4685.23388.60.333.84590522.91739.80.4580.43489.530.424.46890.622.315.7440.4185.42868.670.394.57853.420.915.941.60.3585.42739.790.424.298837.820.214.437.30.3382.53267.620.364.739833.322.215.238.30.3782.23107.840.45.110760.920.415.540.70.3284.22687.750.354.5211760.320.815.144.80.3579.53728.910.455.0512742.523.414.743.10.3579.53109.130.44.3610項(xiàng)指標(biāo)的相關(guān)系數(shù)矩陣變量x1x2x3x4x5x6x7x8x9x10X11X20.421X30.36160.34871X40.16280.3523-0.04741X50.77810.76560.48470.40431X60.51740.02610.14820.16990.28501X70.61560.5711-0.00990.04090.72660.02641X80.23700.21820.13540.54240.3751-0.04370.24111X9-0.1717-0.2255-0.00340.1266-0.1962-0.5585-0.27610.53531x10-0.4096-0.4491-0.1622-0.4569-0.5906-0.5142-0.5138-0.50320.45871相關(guān)系數(shù)矩陣的特征向量U1U2U3U4U5U6U7U8U9U100.3885-0.09190.15100.17630.5149-0.3694-0.0204-0.5885-0.18990.04850.36660.06530.2755-0.175-0.5215-0.1606-0.6687-0.09220.0228-0.03130.19630.00940.38010.7053-0.26060.28350.2403-0.08510.3236-0.03860.22750.3499-0.4828-0.0156-0.3456-0.48260.3790-0.13690.2773-0.02480.46470.05370.20510.028-0.0221-0.16850.29020.6049-0.50130.08990.2274-0.3908-0.45010.36180.2247-0.1371-0.37680.39920.2883-0.09170.3556-0.03670.277-0.51590.29910.12480.18550.13070.5978-0.12450.22110.5625-0.21050.06320.22460.3774-0.22170.02250.04940.5843-0.16160.62430.14040.1730.2806-0.1394-0.19560.1720.004-0.6071-0.39410.0410.37200.09410.0775-0.5473-0.05570.21630.2960.5027相關(guān)系數(shù)矩陣的特征值及累積百分?jǐn)?shù)特征值特征值占總體百分?jǐn)?shù)特征值的累積百分?jǐn)?shù)14.09320.40930.409321.93410.19340.602731.30030.13000.732841.09710.10970.842650.78050.0780.920560.54310.05430.974870.18480.01850.993380.06010.0060.999390.00690.00071.000100.0000.0001.000

因子載荷矩陣

A=(U1*,u2*,…….).

Hi2=A112+A122+A132+A142+A152Ai2為第i個(gè)公共因子對(duì)X1變量的方差。Hi2為各公共因子對(duì)變量X1的方差總和Y1Y2Y3Y4Y5Hi2δ210.7860-0.12790.17210.18460.45490.90480.095220.74170.09070.3141-0.1833-0.46070.90290.097130.39720.01310.43340.7387-0.23030.94450.055540.46020.4866-0.5506-0.0164-0.30530.84520.154850.94010.07470.23880.0293-0.01950.94530.054760.4601-0.5435-0.51320.3790.19850.95350.046570.7194-0.05100.3158-0.54040.26420.98170.018380.44730.7823-0.24000.06620.19840.91340.08669-0.32700.86820.16010.18120.24790.98060.019410-0.79730.0570.42420.09860.06850.83330.1667gi24.09321.93411.30031.09710.7805累積貢獻(xiàn)0.40930.19340.13000.10970.0781X1=0.786Y1-0.1279Y2+0.1721Y3+0.1846Y4+0.4549Y5X2=X3=X4=X5=X6=X7=X8=X9=X10=應(yīng)用數(shù)據(jù)集SOCECON為洛杉基12個(gè)地區(qū)統(tǒng)計(jì)的五個(gè)社會(huì)經(jīng)濟(jì)指標(biāo)：人口總數(shù)（POP），教育程度（SCHOOL），就業(yè)數(shù)（EMPLOY），服務(wù)業(yè)人數(shù)（SERVICES），中等的房?jī)r(jià)（HOUSE）。用FACTOR過(guò)程可以進(jìn)行主分量分析。DATASOCECON;TITLE'五個(gè)經(jīng)濟(jì)指標(biāo)的分析';INPUTPOPSCHOOLEMPLOYSERVICESHOUSE;CARDS;570012.8250027025000100010.9600101000034008.81000109000380013.6170014025000400012.816001402500082008.326006012000120011.44001016000910011.533006014000990012.5340018018000960013.736003902500096009.633008012000940011.4400010013000;PROCFACTORDATA=SOCECONSIMPLECORR;TITLE2'主分量分析';RUN;為了得到好的因子解釋，我們?cè)谏厦娴腜ROCFACTOR語(yǔ)句中再加上一個(gè)ROTATE=PROMAX旋轉(zhuǎn)選項(xiàng)，這樣將在得到主因子分析后先產(chǎn)生方差最大正交預(yù)旋轉(zhuǎn)（VARIMAX）然后進(jìn)行斜交旋轉(zhuǎn)，并加了一個(gè)REORDER選項(xiàng)使輸出時(shí)把原始變量受相同因子影響的放在一起：PROCFACTORDATA=SOCECONPRIORS=SMCROTATE=PROMAXREORDER;TITLE2'主因子分析及PROMAX斜交旋轉(zhuǎn)';RUN;

五個(gè)經(jīng)濟(jì)指標(biāo)的分析11:29Monday,March20,20001

主分量分析

TheFACTORProcedureMeansandStandardDeviationsfrom12ObservationsVariableMeanStdDevPOP6241.6673439.9943SCHOOL11.4421.7865EMPLOY2333.3331241.2115SERVICES120.833114.9275HOUSE17000.0006367.5313CorrelationsPOPSCHOOLEMPLOYSERVICESHOUSEPOP1.000000.009750.972450.438870.02241SCHOOL0.009751.000000.154280.691410.86307EMPLOY0.972450.154281.000000.514720.12193SERVICES0.438870.691410.514721.000000.77765HOUSE0.022410.863070.121930.777651.00000

InitialFactorMethod:PrincipalComponentsPriorCommunalityEstimates:ONEEigenvaluesoftheCorrelationMatrix:Total=5Average=1EigenvalueDifferenceProportionCumulative12.873313591.076653500.57470.574721.796660091.581823210.35930.934030.214836890.114902830.04300.977040.099934050.084678680.02000.996950.015255370.00311.00002factorswillberetainedbytheMINEIGENcriterion.FactorPatternFactor1Factor2POP0.580960.80642SCHOOL0.76704-0.54476EMPLOY0.672430.72605SERVICES0.93239-0.10431HOUSE0.79116-0.55818VarianceExplainedbyEachFactorFactor1Factor22.87331361.7966601FinalCommunalityEstimates:Total=4.669974POPSCHOOLEMPLOYSERVICESHOUSE0.987826290.885105550.979305830.880235620.93750041按照缺省的選擇因子個(gè)數(shù)的準(zhǔn)則MINEIGEN，取大于1的特征值，所以取兩個(gè)因子。它們是用公因子預(yù)報(bào)原始變量的回歸系數(shù)。第一主分量（因子）在所有五