Multiple Linear Regression

STAT 20: Introduction to Probability and Statistics

Concept Questions

Estimate the coefficient

01:00

m1 <- lm(bill_depth_mm ~ bill_length_mm, data = penguins)

What will be the sign of the coefficient for bill_length_mm?

This one warms students up to the idea of working between the formula, the equation of the line, and a plot. It will also likely surprise them: without controlling for the species, there’s actually a slight negative relationship here.

m1

Call:
lm(formula = bill_depth_mm ~ bill_length_mm, data = penguins)

Coefficients:
   (Intercept)  bill_length_mm  
      20.78665        -0.08233  

Estimate the coefficient, take two

01:00

m2 <- lm(bill_depth_mm ~ bill_length_mm + species, penguins)

What will be the sign of the coefficient for bill_length_mm? How many coefficients will be in this linear model?

After the first question, some should get the second one correct, but the fact that it can change may surprise some students. They haven’t need an example of a categorical variable with more than k = 2 levels, so it will be challenging for them to think through how many coefficients are required (since there are k = 3 levels, there are k = 3-1 dummy variables required, for a total of 4 coefficients.)

You may want to follow this question up by drawing on a board a data frame with bill_length_mm and species and show how you need to mutate two new dummy variables. You can also mention that by default, R leaves as the references level the one which comes first alphabetically.

m2

Call:
lm(formula = bill_depth_mm ~ bill_length_mm + species, data = penguins)

Coefficients:
     (Intercept)    bill_length_mm  speciesChinstrap     speciesGentoo  
         10.5653            0.2004           -1.9331           -5.1033  

Dummy Variable

A variable that is 1 if an observation takes a particular level of a categorical variable and 0 otherwise. A categorical variable with \(k\) levels can be encoded using \(k - 1\) dummy variables.

Interpreting coefficients

01:00

m2

Call:
lm(formula = bill_depth_mm ~ bill_length_mm + species, data = penguins)

Coefficients:
     (Intercept)    bill_length_mm  speciesChinstrap     speciesGentoo  
         10.5653            0.2004           -1.9331           -5.1033  

Which is the correct interpretation of the coefficient in front of Gentoo?

This one assess their ability to use a conditional interpretation of a regression coefficient (controlling for the other variables in the model…)