Material

🎥 Watch Models with Multiple Predictors 2

• Slides

Today's Goal

• Use functions in R to fit a linear model with multiple predictors
• Model interactions between variables
• Understand what's linear in linear regressions
• Understand and implement CI and HT for regression parameters
• Understand model diagnostics and how to handle common model violations

Quiz

Suppose a dataset called mydata has variables y, x1, and x2. The variable x1 is numeric and x2 is categorical. For the following questions, write out a regression model and the code to fit the model.

Q - Same slope and same intercept between x1 and y for different levels of x2. $$y={\beta }_0+{\beta }_1{x}_1+ϵ$$

lm(y ~ x1, data = mydata)

Quiz

Q - Same slope and different intercept (parallel lines) for different levels of x2.

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+ϵ$$

linear_reg(engine = "lm") %>% 
  fit(y ~ x1 + x2, data = mydata)

Quiz

Q - Different slope and different intercept (non-parallel lines) for different levels of x2.

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3(x_1\ast x_2)+ϵ$$

lm(y ~ x1*x2, data = mydata)
lm(y ~ x1 + x2 + x1:x2, data = mydata)

Quiz

Q - Write separate fitted models for non-living artists (artistliving = 0) and for living artists (artistliving = 1) using the following result. Your fitted models should include log_price and surface only.

$$\widehat{log\_price}=4.91+0.00021surface-0.126artistliving$$ $$+0.00048surface\ast artistliving$$

• Non-living artists: $\widehat{log\_price}=4.91+0.00021surface$
• Living artists: $\widehat{log\_price}=4.784+0.00069surface$
• Non-parallel lines due to the interaction effect!

Model Conditions

• Linearity: There is a linear relationship between the response and predictor variables.
• Independence: The errors are independent from each other.
• Normality (optional): The errors follow a normal distribution.
• Equal variance: The variability of the errors is equal for all values of the predictor variable.

• For multiple regression, the predictors should not be too correlated with each other.

Linearity and Equal Variance

• Linearity: The residuals vs. fitted values plot should show a random scatter of residuals around 0.

  • No distinguishable pattern or structure along the x or y axes.

  • Why do we want a complete random scatter?

    • It means that my model is good and captures any interesting (linear) relationship in the data.
    • Remaining patterns in residuals vs. fitted values suggest that the linear model is not the best assumption for the data.

• Equal variance: The vertical spread of the residuals should be relatively constant across the plot.

Linearity and Equal Variance

This is what we look for

Linearity and Equal Variance

We don't want
increasing / decreasing variability in residuals as predicted value increases

Linearity and Equal Variance

We don't want
any groups of residuals

Linearity and Equal Variance

We don't want
residuals correlated with predicted values

Linearity and Equal Variance

We don't want
any patterns 1

Linearity and Equal Variance

We don't want
any patterns 2

Normality

Independence

• We can often check the independence assumption 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.

When Model Conditions Are Violated

Linearity and equal variance seem violated.

When Model Conditions Are Violated

Log Transformation

This is still a linear model with $\log \left(y\right)$ as the response: $$\log \left(y\right)={\beta }_0+{\beta }_{1}x+ϵ\Rightarrow \widehat{\log \left(y\right)}={\hat{\beta }}_0+{\hat{\beta }}_{1}x$$

logy <- log(y)
lm2 <- lm(logy ~ x)

Let's Practice Together!

Go to AE 22: Models with Multiple Predictors 2 + Model Diagnostics

Bulletin

• Watch videos for Prepare: June 14

• Project draft due tonight at 11:59pm

• HW02, HW04 due Thursday, June 16 at 11:59pm

• Submit Part 1 and Part 2 of ae22