2Note that1r1otal Sum of Squares TSS of the data set asTSSni1Syiy2note thatTSSSyygression SSR asRSSni1Syiy2and note that sinceyiwill not be exactly on the regression lineTSSx0000RSSunless the poi ID: 867197
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1 Linear Regression and Correlation NotesS
Linear Regression and Correlation NotesSuppose there is a data set of n data points (xi,yive plotted these using a scatterplot and it appears that a linear relationship between them is reasonable. Then the least-squaresline (regression line) that best ts these data,^y=^+^bgression coefcients ^mand^bchosen so as to minimize the sum of the square errorsni=1S(yi-^yi)2=ni=1S(yi-i+^b2This says that the regression line that "best ts" the data is the line chosen so as to provide thesmallest aference between the data points (yialues predicted by theregression line (^yi).The values of the regression coefcients are calculated from^m=SxySxxwhereSxx=ni=1Sxi2-(ni=1Sxi)2n=ni=1S(xi-x)2andSxy=ni=1Sxiyi-(ni=1Sxini=1Syin=ni=1S(xi-xyi-y)and^b=y-^andxandyare the means dened byx=ni=1Sxinandy=ni=1SyinThe correlation coefcient is dened to be^r=SxyÖ` `SxxSyywhereSyy=ni=1Syi2-(ni=1Syi)2n=ni=1S(yi-y)2 -2-Note that-
2 1£Ãr£1.otal Sum of Squares (TSS) of the
1£Ãr£1.otal Sum of Squares (TSS) of the data set asTSS=ni=1S(yi-y)2(note thatTSS=Syygression (SSR) asRSS=ni=1Syi-y)2and note that sinceyiwill not be exactly on the regression line,TSS�RSS(unless the points areactly on a line in which caseTSS=RSS). Then the closer the points are to the regression line,the closer TSS is to RSS. The Coef®cient of Determination is de®ned to ber2=RSS/TSSoasthe data points get close to being exactly on a line, RSS gets close to TSS and sor2gets close to1. Whenr2is close to 1, the points are said to be highly correlated which means that a very largeproportion ot the Total Sum of Squares is accounted for by the regression (SSR). Thus the Coef-®cient of Determination is a measure of the strength of the straight-line relationship.RSS=Sxy2Sxxand so thatr2=RSSTSS=Sxy2SxxSyy=Ãr2so that the correlation coef®cient can be thought of as measuring hogression line ®tsata set