NOTE: Except in cases of complex calculations, we use brackets [ ] to indicate “functions of” and parentheses ( ) to indicate “multiplication.”
SE[Beta_1-hat] = the standard error of the estimated slope Beta_1-hat. There is no multiplication!
Beta_1-hat(AGE) = the multiplication of estimated slope Beta_1-hat and the variable AGE.
Introduction and Links to Materials
In Unit 3 we will spend a significant amount of time covering the details regarding multiple linear regression. Many of the basic ideas are direct extensions of simple linear regression and many skills in interpretation and modeling will translate, at least partially, to other regression models we will cover and that you may learn in the future.
IMPORTANT: In this section we will discuss INTERACTION effects also called EFFECT MODIFICATION or MODERATION.
- This is a difficult subject to master conceptually and mathematically.
- We will need to carefully work with models in order to properly test for and estimate effects of interest.
- Understanding and correct use of interactions in both multiple linear and multiple logistic regression is an important objective of this course!
As additional video material, we recommend the following three videos. We suggest you review these first to get the overall picture and then review the reading and videos created for this course to learn the specifics of putting these ideas into practice.
We will be using PROC GLM instead of PROC REG in this course although many analyses can be completed using PROC REG if you are willing to code all categorical variables manually. We will learn in this unit how PROC GLM can be used to correctly code categorical variables for us in our analyses which is the main reason we prefer PROC GLM for multiple linear regression in this course.
Useful SAS Procedures
- PROC GLM
- CLASS statement in PROC GLM
- CONTRAST, ESTIMATE, and TEST statements in PROC GLM
- Other procedures as needed for exploratory data analysis
Consider the following materials from Penn State STAT 501 as support for our lecture materials for this content.
Review Penn State materials with a focus on the definitions, concepts, and interpretations. You do not need to understand the mathematical details or be able to calculate regression models by hand (although you are expected to be able to work with models and ANOVA tables requiring simple mathematical calculations).They don’t cover multiple linear regression in exactly the same was we do so focus on using their materials to SUPPORT our lectures.
NOTE: The material provided in our lectures and Learn by Doing activities is the most important content to understand.
One difference in their materials of importance is that they let p = the number of parameters in the model (including the intercept) and we let p = the number of x-variables in our model. You should convince yourself that these approaches are equivalent and stick with whichever you like best for your own work. We will not ask any questions which will require one method over the other.
PENN STATE STAT 501 Materials – required textbook reading for this material
Note: If you click on “Printer Friendly Version” in the main lesson page it will show all pages in that lesson. These are provided here after the main lesson link.
We will be using PROC GLM for simple and multiple linear regression so let’s look at some PROC GLM documentation including examples.
SAS Documentation: Here are links to documentation about GLM including common statements along with a new example of a Two-way ANOVA with interaction.
PHC 6053 Videos
We have two sets of materials for interactions. Please review our videos very carefully – likely a few times – as this topic is extremely important and often difficult to fully grasp even after considerable study.
The first is our scripted and produced lecture using a motivating example. You will notice we abuse the fact that we can find this interaction between age and gender in most datasets which have systolic blood pressure!
The second is edited version of a live lecture using numerous examples.
Examples and Learn by Doing Activities
Now let’s look at some exercises on working with models.
EXAMPLE: Working With Models – Part 2
In this activity we will look at multiple linear regression including an interaction term.