# Lesson 6: MLR Model Evaluation

### Introduction

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 learn how to perform each of the above three hypothesis tests. Along the way, however, we have to take two asides — one to learn about the "**general linear F-test**" and one to learn about** **"**sequential sums of squares.**"** **Knowledge about both are necessary in performing the three hypothesis tests.

### Learning objectives and outcomes

Upon completion of this lesson, you should be able to do the following:

- Translate research questions involving slope parameters into the appropriate hypotheses for testing.
- Understand the general idea behind the general linear test.
- Calculate a sequential sums of squares using either of the two definitions.
- Know how to obtain a two (or more)-degree-of-freedom sequential sum of squares.
- Know how to calculate a partial
*F*-statistic from sequential sums of squares. - Understand the decomposition of a regression sum of squares into a sum of sequential sums of squares.
- Understand that the
*t*-test for a slope parameter tests the*marginal*significance of the predictor after adjusting for the other predictors in the model (as can be justified by the equivalence of the*t*-test and the corresponding partial*F*-test for one slope). - Perform a general hypothesis test using the general linear test and relevant Minitab output.
- Know how to specify the null and alternative hypotheses and be able to draw a conclusion given appropriate Minitab output for the
*t*-test or partial*F*-test for*H*_{0}:*β*_{k}= 0. - Know how to specify the null and alternative hypotheses and be able to draw a conclusion given appropriate Minitab output for the overall
*F*-test for*H*_{0}:*β*_{1}= ... =*β*_{p-1}= 0. - Know how to specify the null and alternative hypotheses and be able to draw a conclusion given appropriate Minitab output for the partial
*F*-test for any subset of the slope parameters. - Calculate and understand partial R
^{2}.