Multiple Linear Regression

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How to do it with RThe important thing is to double check that the qualitative variable in question is stored as a factor.The way to check is with the class command. For example,> class(trees$Tall)[1] "factor"If the qualitative variable is not yet stored as a factor then we may convert it to one with thefactor command. Other than this we perform MLR as we normally would.> treesdummy.lm summary(treesdummy.lm)Call:lm(formula = Volume ~ Girth + Tall, data = trees)Residuals:Min 1Q Median 3Q Max-5.7788 -3.1710 0.4888 2.6737 10.0619Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) -34.1652 3.2438 -10.53 3.02e-11 ***Girth 4.6988 0.2652 17.72 < 2e-16 ***Tallyes 4.3072 1.6380 2.63 0.0137 *---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 3.875 on 28 degrees of freedom<strong>Multiple</strong> R-squared: 0.9481, Adjusted R-squared: 0.9444F-statistic: 255.9 on 2 and 28 DF,p-value: < 2.2e-16From the output we see that all parameter estimates are statistically significant and we concludethat the mean response differs for trees with Tall = yes and trees with Tall = no.We were somewhat disingenuous when we defined the dummy variable Tallyes because,in truth, R defines Tallyes automatically without input from the user 3 . Indeed, I fit the model3 That is, R by default handles contrasts according to its internal settings which may be customized by the user forfine control. Given that we will not investigate contrasts further in this book it does not serve the discussion to delveinto those settings, either. The interested reader should check ?contrasts for details.22
eforehand and wrote the discussion afterward with the knowledge of what R would do so thatthe output the reader saw would match what (s)he had previously read. The way that R handlesfactors internally is part of a much larger topic concerning contrasts. The interested reader shouldsee Neter et al [?] or Fox [?] for more.In general, if an explanatory variable foo is qualitative with n levels bar1, bar2, . . . , barnthen R will by default automatically define n − 1 indicator variables in the following way:⎧⎧⎪⎨ 1, if foo = ”bar2”,⎪⎨ 1, if foo = ”barn”,foobar2 =, . . . , foobarn =⎪⎩ 0, otherwise.⎪⎩ 0, otherwise.The level bar1 is represented by foobar2 = · · · = foobarn = 0. We just need to make sure thatfoo is stored as a factor and R will take care of the rest.Graphing the <strong>Regression</strong> LinesWe can see a plot of the two regression lines with the following mouthful of code.> treesTall treesTall[["yes"]]$Fit treesTall[["no"]]$Fit plot(Volume ~ Girth, data = trees, type = "n")> points(Volume ~ Girth, data = treesTall[["yes"]], pch = 1)> points(Volume ~ Girth, data = treesTall[["no"]], pch = 2)> lines(Fit ~ Girth, data = treesTall[["yes"]])> lines(Fit ~ Girth, data = treesTall[["no"]])Here is how it works: first we split the trees data into two pieces, with groups determinedby the Tall variable. Next we add the Fitted values to each piece via predict. Then we setup a plot for the variables Volume versus Girth, but we do not plot anything yet (type = n)because we want to use different symbols for the two groups. Next we add points to the plot forthe Tall = yes trees and use an open circle for a plot character (pch = 1), followed by pointsfor the Tall = no trees with a triangle character (pch = 2). Finally, we add regression lines tothe plot, one for each group.There are other – shorter – ways to plot regression lines by groups, namely the scatterplotfunction in the car package and the xyplot function in the lattice package. We elected tointroduce the reader to the above approach since many advanced plots in R are done in a similar,consecutive fashion.23
Page 5: We see from the output that for the
Page 10 and 11: predict(trees.lm, newdata = new, in
Page 12 and 13: 3.3 Overall F-TestAnother way to as
Page 14 and 15: ●Volume10 30 50 70● ●●●
Page 16 and 17: Coefficients:Estimate Std. Error t
Page 18 and 19: F-statistic: 350.5 on 2 and 28 DF,p
Page 21: Height by a new variable Tall which
Page 25 and 26: for this hypothesis test there are
Page 27 and 28: ---Signif. codes: 0 ‘***’ 0.001
Page 29 and 30: In the trees example we may take th
Page 31 and 32: Here is an example of how it works,

Multiple Linear Regression

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