Multiple Linear Regression

STAT 20: Introduction to Probability and Statistics

Agenda

• Announcements
• Reading Questions (open-note)
• Break
• Worksheet: Multiple Linear Regression
• Appendix: more practice (quiz-level questions).

Announcements

• Lab 2: Flights due tomorrow at 12pm.
  • 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).

Reading Questions

• Join https://pollev.com/jeremysanchez! You may register with your name or as an anonymous guest. You will not be graded today, but you can get a sense of how you would do by keeping track of the points you score based on your answers.

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

Break

05:00

Worksheet: Multiple Linear Regression

03:00

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.

Question 5

Consider the following linear regression output where the variable school is categorical and the variable hours_studied is numerical.

Coefficients	Estimate
(Intercept)	2.5
hours_studied	.2
schoolCal	1
schoolStanford	-1

Question 5 (cont.)

• Say I wanted to create a data frame from the original edu dataframe which contains the minimum, median, and IQR for hours_studied among each school. In order to do this, I make use of group_by() followed by summarize(). I save this data frame into an object called GPA_summary.

What are the dimensions of GPA_summary?

01:00

The correct answer in the poll should be 3x4. The three rows are for the three levels in the school category (there is an additional level beyond "Cal" and "Stanford", regardless if it is not stated). One column is for the school name, the other three are for each of the statistics calculated; the three rows are for the three levels in the school category.