However, the ANOVA method works whether or not we used orthogonal polynomials; it also works when we have other terms in the model as well. For example, we can use anova() to compare these three models:

fit.1 <- lm(wage ~ education + age, data = Wage)
fit.2 <- lm(wage ~ education + poly(age, 2), data = Wage)
fit.3 <- lm(wage ~ education + poly(age, 3), data = Wage)
anova(fit.1, fit.2, fit.3)

Questions

• Create polynomial regression models of degree-1 until degree-3 on the Boston dataset with medv as dependent variable and crim as independent variable. Use the orthogonal polynomials. Also add rm and lstat as additional predictors (only linear functions). The model of degree-1 for crim should be stored in the variable fit.1, model of degree-2 in fit.2, etc.
• Perform an ANOVA to test the null hypothesis that model \(\mathcal{M}_1\) is sufficient to explain the data against the alternative hypothesis that a more complex model \(\mathcal{M}_2\) is required. Store the result in anova.medv.
• MC1: Interpret the ANOVA and decide which order polynomial is sufficient to explain the data. Use a significance level of 0.05.
  • 1: Degree-1 for crim + rm + lstat
  • 2: Degree-2 for crim + rm + lstat
  • 3: Degree-3 for crim + rm + lstat

Assume that:

• The MASS library has been loaded
• The Boston dataset has been loaded and attached