lecture7 - ISYE6414 Summer 2010 Lecture 7 Multiple...

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ISYE6414 Summer 2010 Lecture 7 Multiple Regression More Features Dr. Kobi Abayomi June 22, 2010 1 Introduction Sophisticated users of linear models appreciate the simple way in which many predictors (or covariates) can be included with the response variable. Model diagnosis is the setting where (the sometimes inevitable) violations of the linear model are detected; it is the exploration of these departures that are the rich work of statisticians. Of course, you will never just estimate parameters and then go on your merry way. .. 2 Extra Sums of Squares Let’s use a multiple regression model on 3 covariates as the illustration Y = β 0 + 3 X i =1 β i X i + ± i We call the extra sum of squares of regression, for X 3 given X 1 ,X 2 SSR ( X 3 | X 1 ,X 2 ) = SSR ( X 1 ,X 2 ,X 3 ) - SSR ( X 1 ,X 2 ) or 1
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SSR ( X 3 | X 1 ,X 2 ) = SSE ( X 1 ,X 2 ) - SSE ( X 1 ,X 2 ,X 3 ) The extra sum of squares is the marginal change (in error, SSE , or regression, SSR ) in sum of squares when X 3 is added to X 1 ,X 2 . Remember that the SSR increases as the difference between the model fit and a null fit increases; the SSR is just measure of the deviation ˆ y - y . Remember that SST = SSR + SSE so an increase in SSR is matched by a decrease in SSE as SST — the natural variation in the data ( y - y ) — is fixed. Let’s look at an example in R gpagos<-read.csv(galapagos.csv) ybar<-mean(gpagos$Species) lmfull<-lm(Species~ Endemics + Area + Elevation, data=gpagos) lmreduced<-lm(Species~Endemics + Area, data=gpagos) afull<-(lmfull$fitted.values -rep(ybar,length(lmfull$fitted.values))) SSRfull<- sum(afull^2) t(afull)%*%(afull) ared<-(lmreduced$fitted.values -rep(ybar,length(lmreduced$fitted.values))) SSRreduced<- sum(ared^2) t(ared)%*%(ared) SSRfull-SSRreduced SSEfull<-t(lmfull$residuals)%*%lmfull$residuals SSEreduced<-t(lmreduced$residuals)%*%lmreduced$residuals 2
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SSEreduced-SSEfull f<-((SSEreduced-SSEfull)/1) / (SSEfull/26) qf(.95,1,26) sqrt(f) summary(lmfull) In general, we can write SSR ( X i ,X j | X k ) = SSR ( X i ,X j ,X k ) - SSR ( X k ) for the extra sum of squares between a three covariate and one covariate model. Implicitly, the hypothesis test i H 0 : β i = β k = 0 which we reject for large values of F . 2.1 ANOVA decomposition Take a look at this ANOVA decomposition: Source/SS df MS F SSR k SSR/k SSR k / SSE n - k SSR ( X 1 ) 1 SSR ( X 1 ) / 1 SSR ( X 1 ) 1 / SSE n - k SSR ( X 2 | X 1 ) 1 SSR ( X 2 | X 1 ) / 1 SSR ( X 2 | X 1 ) 1 / SSE n - k SSR ( X 3 | X 1 ,X 2 ) 1 SSR ( X 3 | X 1 ,X 2 ) / 1 SSR ( X 3 | X 1 ,X 2 ) 1 / SSE n - k SSE n - ( k + 1) SSE n - ( k +1) SST n - 1 Of course this relationship holds SSR = SSR ( X 1 ) + SSR ( X 2 | X 1 ) + SSR ( X 3 | X 1 ,X 2 ) = SSR ( X 1 ) + SSR ( X 2 ,X 1 ) - SSR ( X 1 ) + SSR ( X 1 ,X 2 ,X 3 ) - SSR ( X 1 ,X 2 ) = SSR ( X 1 ,X 2
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This note was uploaded on 09/01/2011 for the course ISYE 6414 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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lecture7 - ISYE6414 Summer 2010 Lecture 7 Multiple...

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