Multiple Linear Regression

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2.5 % 97.5 %(Intercept) 0.0002561078 0.006783093log(Girth) 6.2276411645 8.468066317log(Height) 2.0104387829 4.645475188(Note that we did not update the row labels of the matrix to show that we exponentiated andso they are misleading as written.) We do predictions just as before. Remember to transform theresponse variable back to the original scale after prediction.> new exp(predict(treesNonlin.lm, newdata = new, interval = "confidence"))fit lwr upr1 11.90117 11.25908 12.579892 20.82261 20.14652 21.521393 28.93317 27.03755 30.96169The predictions and intervals are slightly different from those calculated earlier, but they areclose. Note that we did not need to transform the Girth and Height arguments in the dataframenew. All transformations are done for us automatically.9.2 Real Nonlinear <strong>Regression</strong>We saw with the trees data that a nonlinear model might be more appropriate for the data basedon theoretical considerations, and we were lucky because the functional form of µ allowed us totake logarithms to transform the nonlinear model to a linear one. The same trick will not work inother circumstances, however. We need techniques to fit general models of the formY = µ(X) + ɛ, (64)where µ is some crazy function that does not lend itself to linear transformations.There are a host of methods to address problems like these which are studied in advancedregression classes. The interested reader should see Neter et al [?] or Tabachnick and Fidell [?].It turns out that John Fox has posted an Appendix to his book [?] which discusses some of themethods and issues associated with nonlinear regression; seehttp://cran.r-project.org/doc/contrib/Fox-Companion/appendix.html30
Here is an example of how it works, based on a question from R-help. We would like to fit amodel that looks like> set.seed(1)( ) 2 2πY = α + β sin360 x − γ + ɛ.> x y acc.nls summary(acc.nls)Formula: y ~ a + b * (sin((2 * pi * x/360) - c))^2Parameters:Estimate Std. Error t value Pr(>|t|)a 0.95884 0.23097 4.151 4.92e-05 ***b 2.22868 0.37114 6.005 9.07e-09 ***c 3.04343 0.08434 36.084 < 2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 1.865 on 197 degrees of freedomNumber of iterations to convergence: 3Achieved convergence tolerance: 6.554e-0831
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 and 24: eforehand and wrote the discussion
Page 25 and 26: for this hypothesis test there are
Page 27 and 28: ---Signif. codes: 0 ‘***’ 0.001
Page 29: In the trees example we may take th

Multiple Linear Regression

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