Example: SAT Averages by State

• This data set contains the average SAT score (out of 1600) and other variables that may be associated with SAT performance for each of the 50 U.S. states. The data is based on test takers for the 1982 exam.

• Response variable:

  • SAT: average total SAT score

Data comes from case1201 data set in the Sleuth3 package

SAT Averages: Predictors

• Takers: percentage of high school seniors who took exam
• Income: median income of families of test-takers ($ hundreds)
• Years: average number of years test-takers had formal education in social sciences, natural sciences, and humanities
• Public: percentage of test-takers who attended public high schools
• Expend: total state expenditure on high schools ($ hundreds per student)
• Rank: median percentile rank of test-takers within their high school classes

Model

Model conditions

1. Linearity: There is a linear relationship between the response and predictor variables.

2. Constant Variance: The variability about the least squares line is generally constant.

3. Normality: The distribution of the residuals is approximately normal.

4. Independence: The residuals are independent from each other.

Residuals vs. predicted values

Linearity: Residuals vs. predicted

Linearity: Residuals vs. each predictor

If there is some pattern in the plot of residuals vs. predicted values, you can look at individual plots of residuals vs. each predictor to try to identify the issue.

Checking linearity

✅ The plot of residuals vs. predicted shows no distinguishable pattern

✅ The plots of residuals vs. each predictor variable are generally fine; perhaps look into Years more closely.

The linearity condition is generally satisfied.

Checking constant variance

✅ The vertical spread of the residuals is relatively constant across the plot. The constant variance condition is satisfied.

Checking normality

⚠️ Normality is not satisfied. However, \(n>30\), so by the Central Limit Theorem, we can still do inference about the model parameters.

Checking independence

• We can often check the independence condition based on the context of the data and how the observations were collected.

• If the data were collected in a particular order, examine a scatterplot of the residuals versus order in which the data were collected.

• If there is a grouping variable lurking in the background, check the residuals based on that grouping variable.

Checking independence

Since the observations are US states, let's take a look at the residuals by region.

Checking independence

❌ The model tends to overpredict for states in the South and underpredict for states in the North Central, so the independence condition is not satisfied.

Multiple linear regression is not robust to violations of independence, so before moving forward, we should try fitting a model that includes region to account for these differences by region.