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Statistics Diagnostics May 2, 2008 1. Plot the standardized residuals: e∗i = ei /sei . 2. Leverage: points far away from x pull the regression line toward them. A way to measure leverage is by the hat values 1 (xi − x)2 . hi = + n Sxx 3. Influence: the amount that a point affects the regression line. (a) One way to measure influence is to refit the line with the point removed. (R function dfbeta().) (b) A summary statistic is Cook’s distance: Di = e∗i hi . 2 1 − hi 4. Multiple regression: the standard linear model. Y = β0 + β1 x1 + · · · + βk xk + 5. Assumptions: has mean 0, variance σ 2 and is normally distributed. 6. Data: (x1,1 , x1,2 , . . . , x1,k , y1 ), . . . , (xn,1 , xn,2 , . . . , xn,k , yn ) 7. Do all the same stuff! Example. Homework - due Tuesday, May 6, 2008 1. Read Section 8.5. Useful R > > > > > > > > l=lm(ohms~juice,data=fo) plot(residuals(l)~fitted(l)) library(MASS) l=lm(brain~body,data=mammals) plot(rstandard(l)~fitted(l)) dfbeta(l) library(faraway) l=lm(brozek~weight+abdom,data=fat)