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 -
- 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