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7.8 Lab: Non-linear Modeling 289 poly(age, 4, raw = T)3 6.81e-03 3.07e-03 2.22 0.026398 poly(age, 4, raw = T)4 -3.20e-05 1.64e-05 -1.95 0.051039 There are several other equivalent ways of fitting this model, which showcase the flexibility of the formula language in R. For example > fit2a=lm(wage∼age+I(age^2)+I(age^3)+I(age^4),data=Wage) > coef(fit2a) (Intercept ) age I(age^2) I(age^3) I(age^4) -1.84e+02 2.12e+01 -5.64e-01 6.81e-03 -3.20e-05 This simply creates the polynomial basis functions on the fly, taking care to protect terms like age^2 via the wrapper function I() (the ^ symbol has wrapper a special meaning in formulas). > fit2b=lm(wage∼cbind(age,age^2,age^3,age^4),data=Wage) This does the same more compactly, using the cbind() function for building a matrix from a collection of vectors; any function call such as cbind() inside a formula also serves as a wrapper. We now create a grid of values for age at which we want predictions, and then call the generic predict() function, specifying that we want standard errors as well. > agelims=range(age) > age.grid=seq(from=agelims[1],to=agelims [2]) > preds=predict(fit,newdata=list(age=age.grid),se=TRUE) > se.bands=cbind(preds$fit +2*preds$se.fit,preds$fit -2*preds$se. fit) Finally, we plot the data and add the fit from the degree-4 polynomial. > par(mfrow=c(1,2),mar=c(4.5,4.5,1,1) ,oma=c(0,0,4,0)) > plot(age,wage,xlim=agelims ,cex=.5,col="darkgrey ") > title("Degree -4 Polynomial ",outer=T) > lines(age.grid,preds$fit ,lwd=2,col="blue") > matlines(age.grid,se.bands ,lwd=1,col="blue",lty=3) Here the mar and oma arguments to par() allow us to control the margins of the plot, and the title() function creates a figure title that spans both title() subplots. We mentioned earlier that whether or not an orthogonal set of basis functionsisproducedinthepoly() function will not affect the model obtained in a meaningful way. What do we mean by this? The fitted values obtained in either case are identical: > preds2=predict(fit2,newdata=list(age=age.grid),se=TRUE) > max(abs(preds$fit -preds2$fit )) [1] 7.39e-13 In performing a polynomial regression we must decide on the degree of the polynomial to use. One way to do this is by using hypothesis tests. We now fit models ranging from linear to a degree-5 polynomial and seek to determine the simplest model which is sufficient to explain the relationship
290 7. Moving Beyond Linearity between wage and age. Weusethe anova() function, which performs an anova() analysis of variance (ANOVA, using an F-test) in order to test the null analysis of variance hypothesis that a model M 1 is sufficient to explain the data against the alternative hypothesis that a more complex model M 2 is required. In order to use the anova() function, M 1 and M 2 must be nested models: the predictors in M 1 must be a subset of the predictors in M 2 .Inthiscase, we fit five different models and sequentially compare the simpler model to the more complex model. > fit.1=lm(wage∼age,data=Wage) > fit.2=lm(wage∼poly(age ,2),data=Wage) > fit.3=lm(wage∼poly(age ,3),data=Wage) > fit.4=lm(wage∼poly(age ,4),data=Wage) > fit.5=lm(wage∼poly(age ,5),data=Wage) > anova(fit.1,fit.2,fit.3,fit.4,fit.5) Analysis of Variance Table Model 1: wage ∼ age Model 2: wage ∼ poly(age, 2) Model 3: wage ∼ poly(age, 3) Model 4: wage ∼ poly(age, 4) Model 5: wage ∼ poly(age, 5) Res.Df RSS Df Sum of Sq F Pr(>F) 1 2998 5022216 2 2997 4793430 1 228786 143.59 |t|) (Intercept ) 111.70 0.7288 153.2780 0.000e+00 poly(age, 5)1 447.07 39.9161 11.2002 1.491e-28 poly(age, 5)2 -478.32 39.9161 -11.9830 2.368e-32 poly(age, 5)3 125.52 39.9161 3.1446 1.679e-03
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Springer Texts in Statistics Gareth
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Gareth James • Daniela Witten •
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To our parents: Alison and Michael
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viii Preface disciplines who wish t
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x Contents 3 Linear Regression 59 3
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xii Contents 6.6 Lab 2: Ridge Regre
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xiv Contents 10.5 Lab 2: Clustering
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2 1. Introduction Wage 50 100 200 3
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4 1. Introduction Predicted Probabi
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8 1. Introduction in order to contr
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10 1. Introduction In general, we w
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12 1. Introduction of A and B is de
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3 Linear Regression This chapter is
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3.1 Simple Linear Regression 3.1 Si
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3.1 Simple Linear Regression 63 3 2
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Quantity Value Residual standard er
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3.2 Multiple Linear Regression 71 a
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3.6 Lab: Linear Regression 109 p=1
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3.6 Lab: Linear Regression 111 Coef
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4 Classification The linear regress
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4.2 Why Not Linear Regression? 129
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4.3 Logistic Regression 131 0 500 1
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4.3 Logistic Regression 133 β 1 do
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4.3 Logistic Regression 135 Coeffic
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4.7 Exercises 169 we will use obser
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176 5. Resampling Methods 5.1 Cross
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180 5. Resampling Methods LOOCV 10
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6.2 Shrinkage Methods 223 Mean Squa
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9.3 Support Vector Machines 349 X2
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10.3 Clustering Methods 387 K=2 K=3
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10.3 Clustering Methods 389 Data St
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10.3 Clustering Methods 391 X2 −2
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10.3 Clustering Methods 393 0.0 0.5
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Algorithm 10.2 Hierarchical Cluster
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10.3 Clustering Methods 397 Average
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10.3 Clustering Methods 399 0 2 4 6
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10.4 Lab 1: Principal Components An
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10.5 Lab 2: Clustering 405 Cluster
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Index C p , 78, 205, 206, 210-213 R
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Index 421 Wage, 1, 2, 9, 10, 14, 26
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Index 423 noise, 22, 228 non-linear
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Index 425 read.table(), 48, 49 regs
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