1. Faculty of Arts Atkinson College MATH 3330 M W 2003 Welcome Seventh Lecture for MATH 3330 M Professor G.E. Denzel

2. Faculty of Arts Atkinson College MATH 3330 M W 2003 Agenda  Begin discussion of multi-predictor models

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5. Faculty of Arts Atkinson College MATH 3330 M W 2003 Learning Objectives  How to find slope and intercept to minimize the error sum of squares (the ‘least squared error’ estimators).  Properties of these estimators  Linear functions of the data  Caution on conclusions possible without further knowledge of population

6. Faculty of Arts Atkinson College MATH 3330 M W 2003 Sums of Squares  SSE and SSTOT (= SSY) play same role as for one- predictor models SSTOT=SSE + (SSTOT-SSE) = SSE + SSR SSE now has N-k-1 degrees of freedom (df), where k=number of predictors in the model. SSR now has k df F*=MSR/MSE=(SSR/k)/MSE will again have an F distribution under the H 0: all predictors have coefficient 0, with df =k for numerator and N-k-1 for denominator. We reject H 0 for large values of F*. The alternative hypothesis is that AT LEAST ONE OF THE COEFFICIENTS IS NON-ZERO

7. Faculty of Arts Atkinson College MATH 3330 M W 2003 Example using Anscombe data  We will step through the process of fitting the model after the data has been put into a SAS workspace.  First here is a part of the data, along with a description of the variables. Note that this data does not really represent a random sample of 51 observations, except perhaps in the sense of one year’s data sampled from many years. However, we can still fit models as long as we think about what the hypothetical population which we are making inferences about might be.

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9. Faculty of Arts Atkinson College MATH 3330 M W 2003 Here is the input menu after selecting spend as the ‘Y’ and income, prop18, and propurban as predictors.:

10. Faculty of Arts Atkinson College MATH 3330 M W 2003 And here is the Output screen:

11. Faculty of Arts Atkinson College MATH 3330 M W 2003 The first output tables

12. Faculty of Arts Atkinson College MATH 3330 M W 2003 The plot of residuals vs predicted values

13. Faculty of Arts Atkinson College MATH 3330 M W 2003  Next slide shows what the dataset now looks like with the residual and predicted variables added to the data (using default names; they can be changed).

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15. Faculty of Arts Atkinson College MATH 3330 M W 2003 The Type I SS table:

16. Faculty of Arts Atkinson College MATH 3330 M W 2003 What happens if we change the order?