Multiple Linear Regression

3.3 Overall F-TestAnother way to assess the model’s utility is to to test the hypothesisH 0 : β 1 = β 2 = · · · = β p = 0 versus H 1 : at least one β i 0.The idea is that if all β i ’s were zero, then the explanatory variables X 1 , . . . , X p would be worthlesspredictors for the response variable Y. We can test the above hypothesis with the overall F statistic,which in MLR is defined byF =S S R/pS S E/(n − p − 1) . (37)When the regression assumptions hold and under H 0 then F ∼ f(df1 = p, df2 = n − p − 1). Wereject H 0 when F is large, that is, when the explained variation is large relative to the unexplainedvariation.3.4 How to do it with RThe overall F statistic and its associated p-value is listed at the bottom of the summary output,or we can access it directly by name; it is stored in the fstatistic component of the summaryobject.> treesumry$fstatisticvalue numdf dendf254.9723 2.0000 28.0000For the trees data, we see that F = 254.972337410669 with a p-value < 2.2e-16. Consequentlywe reject H 0 , that is, the data provide strong evidence that not all β i ’s are zero.3.5 Student’s t TestsWe know thatand we now test(b ∼ mvnorm mean = β, sigma = σ ( 2 X T X ) ) −1(38)H 0 : β i = 0 versus H 1 : β i 0, (39)where β i is the coefficient for the i th independent variable. If H 0 is rejected, then we conclude thatthere is a significant relationship between Y and x i in the regression model Y ∼ (x 1 , . . . , x p ). This12