A previous chapter of the book demonstrated that the simple linear regression model is very useful. In some cases, however, the simple linear model can be improved upon by considering additional explanatory variables. In other words, it is sometimes possible to explain more of the variation in the response variable by including more than one explanatory variable in the regression model. In this chapter, we show how to generalize the simple linear regression model, involving just one explanatory variable, to a multiple linear regression model, involving more than one explanatory variable.

The focus in this chapter is on the use of multiple quantitative explanatory variables. It is possible to include qualitative explanatory variables in multiple regression models as well. We deal with qualitative (both nominal and ordinal) explanatory variables in Chapter 5.

Learning objectives of this chapter include:

Knowledge

• The goal of multiple linear regression.
• Interpreting the parameters in a multiple linear regression model.
• The concepts of main effects, interaction effects, and quadratic effects.
• The principle of least squares regression and how to derive the least squares estimator for multiple linear regression.
• The assumptions behind the statistical inference in the context of multiple linear regression and how to derive the properties of the least squares estimator that form the basis for statistical inference.
• Interpreting the elements of a variance-covariance matrix.
• Testing the significance of the parameters in a multiple linear regression model.
• Testing hypotheses concerning one or more linear combinations of model parameters.
• Making predictions using a multiple linear regression model.
• Evaluating the quality of a multiple linear regression model using the (adjusted) coefficient of determination, various information criteria, and the global F-test.
• The techniques for verifying whether the assumptions behind the multiple linear regression model hold.
• Understanding the mathematical derivations listed in the technical appendices.

Skills

• Calculating the least squares regression model manually for small, well-structured data sets.
• Computing the least squares regression model with JMP software for any given data set.
• Interpreting the multiple linear regression output produced by the Prediction Profiler and the Surface Profiler in JMP.
• Reconstructing most pieces of JMP output for multiple linear regression.
• Conducting tailor-made hypothesis tests for one or more linear combinations of model parameters in JMP.
• Building an analysis of variance (ANOVA) table corresponding to a given multiple linear regression model.
• Interpreting residual plots created to check the model assumptions.

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