Multiple Linear Regression

Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 27.7452 0.8161 33.996 < 2e-16 ***scale(Girth) 14.5995 0.6773 21.557 < 2e-16 ***I(scale(Girth)^2) 2.5067 0.5729 4.376 0.000152 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 3.335 on 28 degrees of freedomMultiple R-squared: 0.9616, Adjusted R-squared: 0.9588F-statistic: 350.5 on 2 and 28 DF, p-value: < 2.2e-16We see that the F statistic indicates the overall model including Girth and Girth^2 is significant.Further, there is strong evidence that both Girth and Girth^2 are significantly related toVolume. We may examine a scatterplot together with the fitted quadratic function using the linesfunction, which adds a line to the plot tracing the estimated mean response.> plot(Volume ~ scale(Girth), data = trees)> lines(fitted(treesquad.lm) ~ scale(Girth), data = trees)The plot is shown in Figure 4. Pay attention to the scale on the x-axis: it is on the scale of thetransformed Girth data and not on the original scale.We do estimation/prediction as above, except we do not need a Height column in the dataframenew since the variable is not included in the quadratic model.> new predict(treesquad.lm, newdata = new, interval = "prediction")fit lwr upr1 11.56982 4.347426 18.792212 20.30615 13.299050 27.313253 25.92290 18.972934 32.87286The predictions and intervals are slightly different from what they were previously. Notice thatit was not necessary to rescale the Girth prediction data before input to the predict function; themodel did the rescaling for us automatically.We have mentioned on several occasions that it is important to rescale the explanatory variablesfor polynomial regression. Watch what happens if we ignore this advice:16