Overview of this Lesson

In this lesson, we make our first (and last?!) major jump in the course. We move from the simple linear regression model with one predictor to the multiple linear regression model with two or more predictors. That is, we use the adjective "simple" to denote that our model has only predictor, and we use the adjective "multiple" to indicate that our model has at least two predictors.

In the multiple regression setting, because of the potentially large number of predictors, it is more efficient to use matrices to define the regression model and the subsequent analyses. This lesson considers some of the more important multiple regression formulas in matrix form. If you're unsure about any of this, it may be a good time to take a look at this Matrix Algebra Review.

The good news is that everything you learned about the simple linear regression model extends — with at most minor modification — to the multiple linear regression model. Think about it — you don't have to forget all of that good stuff you learned! In particular:

• The models have similar "LINE" assumptions. The only real difference is that whereas in simple linear regression we think of the distribution of errors at a fixed value of the single predictor, with multiple linear regression we have to think of the distribution of errors at a fixed set of values for all the predictors. All of the model checking procedures we learned earlier are useful in the multiple linear regression framework, although the process becomes more involved since we now have multiple predictors. We'll explore this issue further in Lesson 6.
• The use and interpretation of r² (which we'll denote R² in the context of multiple linear regression) remains the same. However, with multiple linear regression we can also make use of an "adjusted" R² value, which is useful for model building purposes. We'll explore this measure further in Lesson 11.
• With a minor generalization of the degrees of freedom, we use t-tests and t-intervals for the regression slope coefficients to assess whether a predictor is significantly linearly related to the response, after controlling for the effects of all the opther predictors in the model.
• With a minor generalization of the degrees of freedom, we use confidence intervals for estimating the mean response and prediction intervals for predicting an individual response. We'll explore these further in Lesson 6.

For the simple linear regression model, there is only one slope parameter about which one can perform hypothesis tests. For the multiple linear regression model, there are three different hypothesis tests for slopes that one could conduct. They are:

• a hypothesis test for testing that one slope parameter is 0
• a hypothesis test for testing that all of the slope parameters are 0
• a hypothesis test for testing that a subset — more than one, but not all — of the slope parameters are 0

In this lesson, we also learn how to perform each of the above three hypothesis tests.