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1、Chapter8Chapter8LinearRegressionLinearRegressionZhangJin-xin（张晋昕）2011.04.111HowdoesthevalueofonevariableHowdoesthevalueofonevariabledependonthatofanotherone?dependonthatofanotherone?Howdoestheson’sheightdependonthefather’sheight?Howdoesthedeathrateofanimaldependonthedrugdosage?----Toexplorelineard
2、ependencequantitativelybetweentwocontinuousvariables.28.1StatisticalDescriptionofLinearRegression8.1.1LinearregressionequationYˆ=a+bX3Independentvariable(explanatoryvariable),XrandomlychangingorfixedbytheresearcherDependentvariable(responsevariable),Yrandomlyfollowingalinearequation4Whatisregressi
3、oninstatistics?Whatisregressioninstatistics?220200180160Son’sheight(cm)140120100100120140160180200220Father’sheight（cm）Tofindoutthetrackofthemeans5µ=αβ+Xyx
4、°GiventhevalueofX,Yvariesaroundacenter(µ)y
5、x°Allthecenterslocateonaline--regressionline.°TherelationshipbetweenthecenterµandXisy
6、xdescribedbya
7、linearequation6Linearregressionµ=αβ+Xyx
8、Trytoestimateαandβ,gettingYˆ=a+bXWherea--estimateofα,intercept（截距）b--estimateofβ,slop（斜率）--estimateofµYˆy
9、x78.1.2RegressioncoefficientanditscalculationTofindastraightlinetobestfitthepoints.Residual:yˆ−y2Fitnessoftheregressionline:∑(yˆ−y)Principleofleastsquar
10、es:Tofindastraightlinethatminimizesthesumofsquaredresiduals.Undersuchaprinciple,itiseasytogettheformulasforaandbbyfirst-orderpartialderivative（一阶偏导数）:lxy∑(xi−x)(yi−y)(8.3)b==2lxx∑(xi−x)a=y−bx(8.4)8Example8.1CalculatetheregressionequationExample8.1Calculatetheregressionequationoftheheightofsonofthe
11、heightofsonYYontheheightoffatherontheheightoffatherXX..No.12345678910Father’sheight,X150153155158161164165167168169Son’sheight,Y159157163166169170169167169170No.11121314151617181920Father’sheight,X170171172174175177178181183185Son’sheight,Y173170170176178174173178176180x=168.8y=170.35lxx=1859.2lxy
12、=1059.4lxy1059.4b===0.5698a=170.35−(0.5698)(168.8)=74.17l1859.2xxYˆ=74.17+0.5698X9108.2StatisticalInferenceonRegression8.2.1Hypothesistests8.2.1.1Thet-testforregressioncoefficientbisthesampleregressioncoefficient
