8.2.1 - Assumptions for the SLR Model

Before we get started in interpreting the output it is critical that we address specific assumptions for regression.  Not meeting these assumptions is a flag that the results are not valid (the results cannot be interpreted with any certainty because the model may not fit the data).  

In this section, we will present the assumptions needed to perform the hypothesis test for the population slope:

\(H_0\colon \ \beta_1=0\)

\(H_a\colon \ \beta_1\ne0\)

We will also demonstrate how to verify if they are satisfied. To verify the assumptions, you must run the analysis in Minitab first.

Assumptions for Simple Linear Regression Section

  1. Linearity: The relationship between \(X\) and \(Y\) must be linear.

    Check this assumption by examining a scatterplot of x and y.

  2. Independence of errors: There is not a relationship between the residuals and the Y variable; in other words, Y is independent of errors.

    Check this assumption by examining a scatterplot of “residuals versus fits”; the correlation should be approximately 0. In other words, there should not look like there is a relationship.

  3. Normality of errors: The residuals must be approximately normally distributed.

    Check this assumption by examining a normal probability plot; the observations should be near the line. You can also examine a histogram of the residuals; it should be approximately normally distributed.

  4. Equal variances: The variance of the residuals is the same for all values of \(X\).

    Check this assumption by examining the scatterplot of “residuals versus fits”; the variance of the residuals should be the same across all values of the x-axis. If the plot shows a pattern (e.g., bowtie or megaphone shape), then variances are not consistent, and this assumption has not been met.