Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Welcome

Seventh Lecture for MATH 3330 M Professor G.E. Denzel

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Agenda Begin discussion of multi-predictor models

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Faculty of ArtsAtkinson 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

Faculty of ArtsAtkinson 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 H0: all predictors have coefficient 0, with df =k for numerator and N-k-1 for denominator.

• We reject H0 for large values of F*.• The alternative hypothesis is that AT LEAST ONE OF THE

COEFFICIENTS IS NON-ZERO

Faculty of ArtsAtkinson 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.

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Here is the input menu after selecting spend as the ‘Y’ and income, prop18, and propurban as

predictors.:

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

And here is the Output screen:

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

The first output tables

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

The plot of residuals vs predicted values

Faculty of ArtsAtkinson 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).

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

The Type I SS table:

Faculty of ArtsAtkinson College

MATH 3330 M W 2003

What happens if we change the order?