Multiple Linear Regression

STAT 20: Introduction to Probability and Statistics

Agenda

• Announcements
• Multiple Linear Regression Refresher
• Quiz Review (this week’s notes)
• Break
• Lab 2.2 (extended)

Announcements

• Quiz 1 is Monday, in-class and covers all lectures from the beginning to the semester until today.

• Lab 2.2, Problem Set 4 and Problem Set 5 are due Tuesday 9am
  • Make sure you follow Lab Submission Guidelines on Ed

• RQ: Introducing Probability due on Monday/Tuesday at 11:59pm; Probability unit begins next week

• Extra Practice for Multiple Linear Regression added to the resources tab on the course home page.

Multiple Linear Regression Refresher

• Head to pollev.com for a set of rapid-fire questions on last night’s notes.

Quiz Review

• Head to pollev.com for a set of quiz-level questions pertaining to Summarizing Numerical Associations and Multiple Linear Regression.

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 indicator coefficients 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.

Break

05:00

Lab 2.2 (extended)

40:00

End of Lecture