Multiple Linear Regression

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●Volume10 30 50 70● ● ●●●●●●●●● ● ●8 10 12 14 16 18 20GirthFigure 5: A dummy variable model for the trees data7 Partial F StatisticWe saw in Section 3.3 how to test H 0 : β 0 = β 1 = · · · = β p = 0 with the overall F statistic andwe saw in Section 3.5 how to test H 0 : β i = 0 that a particular coefficient β i is zero. Here we testwhether a certain part of the model is significant. Consider the regression modelY = β 0 + β 1 x 1 + · · · + β j x j + β j+1 x j+1 + · · · + β p x p + ɛ, (52)where j ≥ 1 and p ≥ 2. Now we wish to test the hypothesisH 0 : β j+1 = β j+2 = · · · = β p = 0 (53)versus the alternativeH 1 : at least one of β j+1 , β j+2 , , . . . , β p 0. (54)The interpretation of H 0 is that none of the variables x j+1 , . . . ,x p is significantly related to Y andthe interpretation of H 1 is that at least one of x j+1 , . . . ,x p is significantly related to Y. In essence,24
for this hypothesis test there are two competing models under consideration:the full model: y = β 0 + β 1 x 1 + · · · + β p x p + ɛ, (55)the reduced model: y = β 0 + β 1 x 1 + · · · + β j x j + ɛ, (56)Of course, the full model will always explain the data better than the reduced model, but does thefull model explain the data significantly better than the reduced model? This question is exactlywhat the partial F statistic is designed to answer.We first calculate S S E f , the unexplained variation in the full model, and S S E r , the unexplainedvariation in the reduced model. We base our test on the difference S S E r − S S E f which measuresthe reduction in unexplained variation attributable to the variables x j+1 , . . . ,x p . In the full modelthere are p + 1 parameters and in the reduced model there are j + 1 parameters, which gives adifference of p − j parameters (hence degrees of freedom). The partial F statistic isF = (S S E r − S S E f )/(p − j). (57)S S E f /(n − p − 1)It can be shown when the regression assumptions hold under H 0 that the partial F statistic has anf(df1 = p − j, df2 = n − p − 1) distribution. We calculate the p-value of the observed partial Fstatistic and reject H 0 if the p-value is small.How to do it with RThe key ingredient above is that the two competing models are nested in the sense that the reducedmodel is entirely contained within the complete model. The way to test whether the improvementis significant is to compute lm objects both for the complete model and the reduced model thencompare the answers with the anova function.For the trees data, let us fit a polynomial regression model and for the sake of argument wewill ignore our own good advice and fail to rescale the explanatory variables.> treesfull.lm summary(treesfull.lm)Call:lm(formula = Volume ~ Girth + I(Girth^2) + Height + I(Height^2),data = trees)Residuals:Min 1Q Median 3Q Max25
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 and 22: Height by a new variable Tall which
Page 23: eforehand and wrote the discussion
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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