Multiple Linear Regression

STAT 20: Introduction to Probability and Statistics

Agenda

• Announcements
• Concept Questions
• Break
• Appendix: more practice
• Worksheet: Multiple Linear Regression

Announcements

• Lab 2: Flights due tomorrow at 11:59pm.
  • Part 1: Submit like your portfolios - scan and upload
  • Part 2: Render to PDF, download, and upload - make sure to assign pages

• No reading questions for the next six lectures leading up to the quiz (traditional-style lecture).

Concept Questions

What are \(b_0, b_1, ..., b_p\) in the multiple linear regression formula called?

• A: Terms
• B: Coefficients
• C: Variables
• D: Predictors

02:00

What values can the variable \(geowest\) in the shown formula take?

\[ \widehat{price} = -15.97+2.87\times food-1.45 \times geowest\]

• A: any value on the real number line
• B: either 0 or 1
• C: "east" or "west"

02:00

What name is given to a level of a categorical variable which is not given an indicator in a linear model?

• A: Indicator level
• B: Coefficient
• C: Reference level
• D: Primary level

02:00

If you include a numerical variable as an input variable to a linear model, how many terms for it will appear in the model?

• A: 1
• B: 2
• C: 3
• D: 4

02:00

If you include a categorical variable with 3 levels as an input variable in a linear model, how many terms for it will appear in the model?

• A: 1
• B: 2
• C: 3
• D: 4

02:00

If I fit a model of bill_length_mm based on bill_depth_mm, and then a second model which includes body_mass_grams as an explanatory variable, the coefficient of bill_depth_mm will always be the same in both models

• True
• False

01:00

Break

05:00

Appendix - More practice!

Question 1

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

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

How many more coefficients does the second model have than the first?

Remind students that they need to remember whether species and body_mass_g are numerical or categorical. Students should know how many species there are (three).

One addl. coefficient for body mass; two addl. for species (one less than the number of species, which is three). This gives a total of three more coefficients.

Questions 2-4

Consider the following multiple linear regression model, which will be the subject of the next three review questions.

Question 2

01:00

m2

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

Coefficients:
     (Intercept)    bill_length_mm       body_mass_g  speciesChinstrap  
        10.33083           0.09484           0.00117          -0.90748  
   speciesGentoo  
        -5.80117  

Which is the correct interpretation of the coefficient in front of bill length? Select all that apply.

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

The correct interpretation controls for both other variables (body mass and species), and has the x variable and the y variable in the correct places.

Question 3

01:00

m2

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

Coefficients:
     (Intercept)    bill_length_mm       body_mass_g  speciesChinstrap  
        10.33083           0.09484           0.00117          -0.90748  
   speciesGentoo  
        -5.80117  

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

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

The correct interpretation needs to involve a comparison to the reference level, Adelie and should not involve anything about bill length or body mass, (other than to say they should be fixed).

Question 4

01:00

m2

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

Coefficients:
     (Intercept)    bill_length_mm       body_mass_g  speciesChinstrap  
        10.33083           0.09484           0.00117          -0.90748  
   speciesGentoo  
        -5.80117  

How would this linear model best be visualized?

Three numerical variables means we will have planes in a 3D space. The coefficients for the dummy variables will shift the planes up and down. There are three species, so three parallel planes is what we are looking for.

Worksheet: Multiple Linear Regression

30:00