Class Notes, Chapter 11, second part; Chapter 12

Reading 11.6-11.12, 12.1-12.5

Things you might want to do with regression: Fitting model to data (estimating &betas) vs. Tests about &betas

GLM and fitting:

Using the residuals to assess model assumptions of normality and equal variances

Normality: Normal quantile-quantile plot, sorted residuals plotted vs. standard normal quantile, q

Equal variances: Plot residuals vs. x-values or vs. predicted y values (y-hats)

GLM and interval estimates or tests of individual &betas

GLM and the test "of Regression"

GLM and the test "of Fit"

Regression models: examples of the General Linear Model

Predictor variables have been added to our schema

Regression models use quantitative predictor variables (regressors)

ANOVA predictor variables code group membership - Robust models

Regression models: A model is an assumption; a test is only as good as the assumed model

If you have the wrong model, you can't interpret tests of parameters

Detail to remember that can trip a person up:

p is often used as the total number of predictor variables, including the intercept

k is often used as the total number of regressors, that is predictor variables other than the intercept (so k = p - 1)

but not always!

The statistical packages and discussion in virtually all textbooks regard the intercept parameter (&beta₀) as a given, the most basic model, but this is not necessary, as we have seen. However, if your real underlying model "goes through zero", you will have to do something extra when you are using software packages to get rid of the intercept in the model. For instance, in excel there is a box to check called "constant is zero".

Example, based on the data from Exercise 12.4, using the model

This is a common form for what is called a "Response Surface Model"; there are two predictor variables, x₁ and x₂, their squares and their product, forming 5 predictors (as you see) with, of course, the intercept, for a total of 6 coefficients, &beta₀ to &beta₅. To re-iterate, 6 parameters with 5 regressors.

Here are the needed numbers:

And here is the GLM written out:

And here are the useful calculations:

These are the data that will be used in the HW exercise