第講多元線性回歸

第2講多元線性回歸1主要內(nèi)容1多元線性回歸模型簡介2回歸系數(shù)的估計3方程的假設(shè)檢驗4決定系數(shù)與剩余標(biāo)準(zhǔn)差5偏回歸系數(shù)的假設(shè)檢驗6指標(biāo)的量化7回歸與t檢驗、方差分析的關(guān)系8標(biāo)準(zhǔn)偏回歸系數(shù)與自變量的貢獻

文獻導(dǎo)讀2某地13歲男童身高，體重，肺活量的實測數(shù)據(jù)(部分)編號身高(cm)x1體重(kg)x2肺活量(L)y1135.132.01.753163.646.22.755156.237.12.757167.841.52.759145.033.02.5011165.549.53.0013153.341.02.7515160.547.22.2517147.640.52.0019155.144.72.7521143.031.51.7523160.840.42.7525158.237.52.0027144.534.72.2529156.532.01.753問題：身高、體重與肺活量有無線性關(guān)系？用身高和體重預(yù)測肺活量有多高的精度？單獨用身高、或體重是否也能達到同樣效果？身高的貢獻大，還是體重的貢獻大？41多元線性回歸模型簡介多元回歸multipleregressionmultiplelinearregression因變量dependentvariableresponsevariable(響應(yīng)變量)自變量independentvariableexplanatoryvariable(解釋變量)5回歸模型因變量y,自變量為x1,x2,

,xma為截距(intercept)，又稱常數(shù)項(constant),表示各自變量均為0時y的估計值bi稱為偏回歸系數(shù)(partialregressioncoefficient)，簡稱為回歸系數(shù)稱為y的估計值或預(yù)測值(predictedvalue)6例：根據(jù)某地29名13歲男童的身高x1(cm)，體重x2(kg)和肺活量y(L)建立的回歸方程為：當(dāng)x1=150，x2=32時，=1.9168，表示對所有身高為150cm，體重為32kg的13歲男童，估計平均肺活量為1.9168(L)。72回歸系數(shù)的估計最小二乘法(leastsquare,LS)基本思想殘差平方和(sumofsquaresforresiduals)最小8估計值與殘差9估計值與殘差有下列性質(zhì)：103Y的總變異分解未引進回歸時的總變異：(sumofsquaresaboutthemeanofY)引進回歸以后的變異(剩余):(sumofsquaresaboutregression)回歸的貢獻，回歸平方和：(sumofsquaresduetoregression)11回歸方程的方差分析表12例3.1資料回歸方程的方差分析

134決定系數(shù)與剩余標(biāo)準(zhǔn)差決定系數(shù)(determinationcoefficient)14R2可用于檢驗多元回歸方程的顯著性：H0：

2=0；H1：

2

0。檢驗統(tǒng)計量為：15復(fù)相關(guān)系數(shù)的性質(zhì)0≤R≤1。當(dāng)只有一個因變量y與一個自變量x時，R就等于y與x的簡單相關(guān)系數(shù)之絕對值：R=|ryx

|當(dāng)有多個自變量x1,x2,…,xm時，R的值比任何一個自變量與因變量的簡單相關(guān)系數(shù)之絕對值大，即：16剩余標(biāo)準(zhǔn)差剩余標(biāo)準(zhǔn)差17剩余標(biāo)準(zhǔn)差的用途剩余標(biāo)準(zhǔn)差可用于偏回歸系數(shù)的假設(shè)檢驗y的容許區(qū)間估計y的可信區(qū)間估計自變量的選擇等

因此，剩余標(biāo)準(zhǔn)差在回歸分析中是一個非常重要的統(tǒng)計量185偏回歸系數(shù)的假設(shè)檢驗H0：

i=0；H1：

i

0。19STATA的輸出結(jié)果.regyx1x2

Source|SSdfMSNumberofobs=29-------------+------------------------------F(2,26)=15.63Model|3.0757339421.53786697Prob>F=0.0000Residual|2.5578867526.098380259R-squared=0.5460-------------+------------------------------AdjR-squared=0.5110Total|5.6336206928.201200739RootMSE=.31366------------------------------------------------------------------------------y|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------x1|.0050165.01057540.470.639-.0167216.0267547x2|.0540611.01598383.380.002.021206.0869162_cons|-.56566431.240127-0.460.652-3.1147821.983454------------------------------------------------------------------------------206標(biāo)準(zhǔn)偏回歸系數(shù)與自變量的貢獻21STATA的輸出結(jié)果.regyx1x2,beta

Source|SSdfMSNumberofobs=29-------------+------------------------------F(2,26)=15.63Model|3.0757339421.53786697Prob>F=0.0000Residual|2.5578867526.098380259R-squared=0.5460-------------+------------------------------AdjR-squared=0.5110Total|5.6336206928.201200739RootMSE=.31366------------------------------------------------------------------------------y|Coef.Std.Err.tP>|t|Beta-------------+----------------------------------------------------------------x1|.0050165.01057540.470.639.0935215x2|.0540611.01598383.380.002.6668242_cons|-.56566431.240127-0.460.652.------------------------------------------------------------------------------22一元回歸分析的結(jié)果.regyx1------------------------------------------------------------------------------y|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------x1|.0315609.00834713.780.001.0144341.0486878_cons|-2.6085411.275414-2.050.051-5.225474.008393------------------------------------------------------------------------------.regyx2------------------------------------------------------------------------------y|Coef.Std.Err.tP>|t|[95%Conf.Interval]-------------+----------------------------------------------------------------x2|.0596878.01055875.650.000.0380232.0813524_cons|-.0091673.3961987-0.020.982-.8221.8037653------------------------------------------------------------------------------為什么單變量分析時都有統(tǒng)計學(xué)意義，而同時放入方程則一個有統(tǒng)計學(xué)意義，另一個無統(tǒng)計學(xué)意義？23自變量的作用X1YX224自變量作用的分解自變量中間變量直接貢獻間接貢獻與y的相關(guān)riy身高x1x2b1

=0.09352b2

r12=0.666820.7421=0.49480.5884體重x2x1b2

=0.66682b1

r12=0.093520.7421=0.06940.7362253.8指標(biāo)的量化性別26例t檢驗與回歸的關(guān)系正常人與矽肺患者血清粘蛋白合理(mg/100mg)27資料重新整理

ygroup1.64.2602.42.8403.52.4804.48.1905.80.2206.69.6107.18.1908.50.909.74.97110.88.06111.93.47112.95.1113.100.67114.101.14115.113.52128t檢驗結(jié)果.ttesty,by(group)Two-samplettestwithequalvariances----------------------------------------------------------------------------Group|ObsMeanStd.Err.Std.Dev.[95%Conf.Interval]---------+------------------------------------------------------------------0|853.336256.66210218.8432737.5828869.089621|795.275714.53563112.0001584.17742106.374---------+------------------------------------------------------------------combined|1572.9086.87165826.6138258.1697687.64624---------+------------------------------------------------------------------diff|-41.939468.307497-59.88672-23.99221----------------------------------------------------------------------------Degreesoffreedom:13Ho:mean(0)-mean(1)=diff=0Ha:diff<0Ha:diff~=0Ha:diff>0t=-5.0484t=-5.0484t=-5.0484P<t=0.0001P>|t|=0.0002P>t=0.999929與方差分析結(jié)果等價.anovaygroupNumberofobs=15R-squared=0.6622RootMSE=16.0516AdjR-squared=0.6362Source|PartialSSdfMSFProb>F-----------+----------------------------------------------------Model|6566.6291816566.6291825.490.0002|group|6566.6291816566.6291825.490.0002|Residual|3349.5038913257.654145-----------+----------------------------------------------------Total|9916.1330714708.2952230與回歸分析結(jié)果的比較.regygroupSource|SSdfMSNumberofobs=15----------+-----------------------------F(1,13)=25.49Model|6566.6291816566.62918Prob>F=0.0002Residual|3349.5038913257.654145R-squared=0.6622----------+-----------------------------AdjR-squared=0.6362Total|9916.1330714708.29522RootMSE=16.052-------------------------------------------------------------------y|Coef.Std.Err.tP>|t|[95%Conf.Interval]---------+---------------------------------------------------------group|41.939468.3074975.050.00023.9922159.88672_cons|53.336255.6751019.400.00041.0759465.59656------------------------------------------------------------31回歸系數(shù)與各組均數(shù)的關(guān)系32指標(biāo)的量化血型(A，B，AB，O)x1=0,x2=0,x3=0表示O型x1=1,x2=0,x3=0表示A型x1=0,x2=1,x3=0表示B型x1=0,x2=0,x3=1表示AB型啞變量(dummy)又稱指示變量(indicatorvariables)33方差分析與回歸分析血清粘蛋白合理(mg/100mg)34各組均數(shù).tabgroup,sum(y)|Summaryofygroup|MeanStd.Dev.Freq.------------+------------------------------------0|53.33625118.8432781|80.05000114.76619882|95.27571312.0001537------------+------------------------------------Total|75.39217423.0696052335指標(biāo)的量化組別(0，1，2)x1=0,x2=0表示0組(正常人)x1=1,x2=0表示1組(矽肺I期)x1=0,x2=1表示2組(矽肺II期)啞變量(dummy)又稱指示變量(indicatorvariables)36資料整理血清粘蛋白含量(mg/100mg)37方差分析的結(jié)果.anovaygNumberofobs=23R-squared=0.5836RootMSE=15.6138AdjR-squared=0.5419Source|PartialSSdfMSFProb>F-----------+----------------------------------------------------Model|6832.758823416.379414.010.0002|group|6832.758823416.379414.010.0002|Residual|4875.7881520243.789407-----------+----------------------------------------------------Total|11708.546922532.206679

38回歸分析的結(jié)果.regyg2g3Source|SSdfMSNumberofobs=23------------+------------------------------F(2,20)=14.01Model|6832.758823416.3794Prob>F=0.0002Residual|4875.7881520243.789407R-squared=0.5836------------+------------------------------AdjR-squared=0.5419Total|11708.546922532.206679RootMSE=15.614----------------------------------------------------------------------y|Coef.Std.Err.tP>|t|[95%Conf.Interval]---------+------------------------------------------------------------g2|26.713757.8068783.420.00310.4288942.99861g3|41.939468.0808875.190.00025.0830358.7959_cons|53.336255.5202979.660.00041.8211164.85139----------------------------------------------------------------------39系數(shù)與均數(shù)40協(xié)方差分析與回歸分析41heightweightygenderhwygender543.0024461543.0021170502.2519281532.2522000512.5020941512.5019060563.5025061513.0018500523.0021211513.0016320769.5038451777.5039340809.00438017710.0041800749.5043141779.5042460809.0040781749.0033580768.0041341737.50380909613.50583019112.00535809714.00601319113.00561009916.00641019415.00607409211.00528319212.00529009415.00610119112.5052910資料整理42協(xié)方差分析.anovayheightweightgender,cate(gender)

Numberofobs=30R-squared=0.9845RootMSE=203.667AdjR-squared=0.9827Source|PartialSSdfMSFProb>F-----------+----------------------------------------------------Model|68508456.5322836152.2550.530.0000|height|925956.9041925956.90422.320.0001weight|374288.7521374288.7529.020.0058gender|144515.8411144515.8413.480.0733|Residual|1078488.662641480.3332-----------+----------------------------------------------------Total|69586945.2292399549.83

43.regywhgSource|