6: Dood! Check the Requirements
- Page ID
- 57730
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There are several requirements for ordinary least squares regression to be applicable. In this chapter, we cover them, their tests, and their relative importance. Focus on the relationship between the assumption/requirement and the tests used. As these assumptions (requirements) are used for some other fitting methods, not just ordinary least squares, the lessons learned here are helpful in the future.
However, realize that each fitting method has its own set of assumptions/requirements. For instance, median regression requires only the constant expected residual value. Maximum likelihood for Poisson regression requires that, plus a specific relationship between the expected value and variance.
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In Chapters 3: Intro to Linear Regression and 4: Matrices and Linear Regression, we created the ordinary least squares regression technique. The mathematics of the situation required \(\det \mathbf{X}^{\prime} \mathbf{X} \neq 0\). Because this corresponds to the requirement that the design matrix \(\mathbf{X}\) be of full column rank, the requirement is met when the independent variables are not linear combinations of each other.
Chapter 5: Improved! Now with Probabilities added the requirement
\begin{equation}
\varepsilon_i \stackrel{\text{iid}}{\sim} N\left( 0,\ \sigma^2 \right)
\end{equation}
This allowed us to move beyond the pure mathematics of Chapters 3: Intro to Linear Regression and 4: Matrices and Linear Regression, and begin making inferences about the population based on the sample.
The requirement that \(\varepsilon_i \stackrel{\text{iid}}{\sim} N\left( 0,\ \sigma^2 \right)\), equivalently \(\mathbf{E} \sim N\left(\mathbf{0},\ \sigma^2\mathbf{I}\right)\), has several parts to it: Normality, constant expected value (of zero), constant variance, and independence. This chapter takes these assumptions and explores them. Both graphical and numeric tests are presented. Throughout it all, we will rely heavily on R to generate the residuals, create the graphics, and perform the tests.
Some R functions require an additional package to be loaded. When such is the case, I indicate that in a special font. For example, the runs test (Bradley 1968) is implemented in the lawstat package as the function runs.test. The lmtest package has the bptest, which performs the Breusch-Pagan test.
The KnoxStats package extends the runs.test to something more useful for us. That package also provides some other useful functions for regression analysis. Review The Appendix of R for how to load a package.