8  Introduction to multiple regression

Settling In

  • Grab a Card and Sit at that table
    • Introduce yourself
    • Share your names, pronouns, major / minor.
    • Check in about how finishing PS2 went. What resources did you find helpful?
  • Help each other get ready to take notes!
    • Open your notebook.
    • Open the online manual to the “Course Schedule” and click on today’s activity. That brings you here!
    • Download “08-mlr-intro.qmd” and open it in RStudio. Read the “Organizing your files” directions at the top of the file!!





Recap

NoteLearning goals

By the end of this lesson, you should be familiar with:

  • some limitations of simple linear regression
  • the general goals behind multiple linear regression
  • strategies for visualizing and interpreting multiple linear regression models of Y vs 2 predictors, 1 quantitative and 1 categorical
NoteAdditional resources

Today is a day to discover ideas, so no readings or videos to go through before class.









Bivariate Relationships

Quantitative Y, Categorical X

  • Side-by-Side boxplots (y = Y, x = categorical var)
  • Density plots (colored or filled by categorical var)
  • Side-by-Side histogram (faceted by categorical var)





Linear Models with Categorical X

  • Convert X to indicator variables (1/0 values) for each categorical level
  • Incorporate indicator variables for all but 1 level
  • Reference level: the level without an indicator variable in the model





Interpretations

  • Intercept: Average Y for reference level
  • Coefficients: Difference in Average Y between level and reference level





Multiple linear regression model

In general, a multiple linear regression model of Y with multiple predictors (X_1, X_2, ..., X_p) is represented by the following formula:

E[Y \mid X_1, X_2, ..., X_p] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p









NoteOrganize your files

This qmd file is where you’ll type notes, code, etc. Directions:

  • Save this file in the inclass_activities sub-folder of the stat155 folder you created before today’s class. Use a file name related to the activity number and/or today’s date (eg: “activity 8” or “8 multiple linear regression”).

Warm-up

Example 1

Let’s explore some data on penguins. You can find a codebook for these data by typing ?penguins in your console (not qmd).

# Load packages
library(tidyverse)
data(penguins)
penguins <- penguins %>% 
  filter(species != "Adelie", bill_len < 57)

# Check it out
head(penguins)

Our goal is to build a model that we can use to get good predictions of penguins’ flipper (“arm”) lengths. Consider 2 simple linear regression models of flipper_len by penguin bill_len and species:

# Model 1: R-squared = 0.0288
summary(lm(flipper_len ~ bill_len, penguins))

# Model 2: R-squared = 0.7014
summary(lm(flipper_len ~ species, penguins))

How might we improve our predictions of flipper_len using only these 2 predictors? If we improve the predictions, what do you think is a reasonable range of possible values for the new R-squared?

Example 2

Consider a simple linear regression model of flipper_len by bill_len:

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)

Thoughts? What’s going on here? How does this highlight the limitations of a simple linear regression model?

Example 3

The cps dataset contains employment information collected by the U.S. Current Population Survey (CPS) in 2018. We can use these data to explore wages among 18-34 year olds. The original codebook is here.

# Import data
cps <- read_csv("https://mac-stat.github.io/data/cps_2018.csv") %>% 
  select(-education, -hours) %>% 
  filter(age >= 18, age <= 34) %>% 
  filter(wage < 250000)
# Check it out
head(cps)

We can use a simple linear regression model to summarize the relationship of wage with marital status:

# Build the model
wage_mod <- lm(wage ~ marital, data = cps)

# Summarize the model
coef(summary(wage_mod))

What do you / don’t you conclude from this model? How does it highlight the limitations of a simple linear regression model?





Reflection: Why are multiple regression models so useful?

We can put more than 1 predictor into a regression model! Adding predictors to models…

  • Predictive viewpoint: Helps us better predict the response
  • Descriptive viewpoint: Helps us better understand the isolated (causal) effect of a variable by holding constant confounders, other variables that may distort the relationship of interest…

Exercises

In the coming weeks, we’ll explore how to visualize, interpret, build, and evaluate multiple linear regression models. First, we’ll explore some foundations using a model of penguin flipper_len by just 2 predictors: bill_len (quantitative) and species (categorical).

Exercise 1: Visualizing the relationship

We’ve learned how to visualize the relationship of flipper_len by bill_len alone:

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len)) + 
  geom_point()

  1. THINK: How might we change the scatterplot points to also indicate information about penguin species? (There’s more than 1 approach!)

  2. Try out your idea by modifying the code below. If you get stuck, talk with the tables around you!

penguins %>%
  ggplot(aes(y = flipper_len, x = bill_len, ___ = ___)) +
  geom_point()

Exercise 2: Visualizing the model

We’ve also learned that a simple linear regression model of flipper_len by bill_len alone can be represented by a line:

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)

  1. THINK: Reflecting on your plot of flipper_len by bill_len and species in Exercise 1, how do you think a multiple regression model of flipper_len using both of these predictors would be represented?

  2. Check your intuition below by modifying the code below to include species in this plot, as you did in Exercise 1.

penguins %>%
  ggplot(aes(y = flipper_len, x = bill_len, ___ = ___)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE)

Exercise 3: Intuition

Your plot in Exercise 2 demonstrated that the multiple linear regression model of flipper_len by bill_len and species is represented by 2 lines. Let’s interpret the punchlines!

For each question, provide an answer along with evidence from the model lines that supports your answer.

  1. What’s the relationship between flipper_len and bill_len, no matter a penguin’s species? What’s your evidence?

  2. Does the rate of increase in flipper_len with bill_len appear to differ between the two species? What’s your evidence?

  3. What’s the relationship between flipper_len and species, no matter a penguin’s bill_len? What’s your evidence?

  4. The formula for this linear regression model is represented by the below formula:

E[flipper_len | species, bill_len] = \beta_0 + \beta_1 bill_len + \beta_2 speciesGentoo

Based on parts a-c:

  • Will \beta_1 be positive or negative? Why?

  • Will \beta_2 be positive or negative? Why?

Exercise 4: Model formula

Of course, there’s a formula behind the multiple regression model. We can obtain this using the usual lm() function.

# Build the model
penguin_mod <- lm(flipper_len ~ bill_len + species, data = penguins)

# Summarize the model
coef(summary(penguin_mod))

  1. In the lm() function, how did we communicate that we wanted to model flipper_len by both bill_len and species?

  2. Complete the following model formula. (Was your answer in Exercise 3 part d correct?!)

    E[flipper_len | bill_len, speciesGentoo] = ___ + ___ * bill_len + ___ * speciesGentoo

Exercise 5: Sub-model formulas

Ok. We now have a single formula for the model.

And we observed earlier that this formula is represented by two lines: one describing the relationship between flipper_len and bill_len for Chinstrap penguins and the other for Gentoo penguins.

Let’s bring these ideas together.

Utilize the model formula to obtain the equations of these two lines, i.e. to obtain the sub-model formulas for the 2 species. Hint: Plug speciesGentoo = 0 and speciesGentoo = 1.

Chinstrap: E[flipper_len | bill_len] = ___ + ___ bill_len

Gentoo: E[flipper_len | bill_len] = ___ + ___ bill_len

Exercise 6: coefficients – physical interpretation

Reflecting on Exercise 5, let’s interpret what the model coefficients tell us about the physical properties of the two 2 sub-model lines. Choose the correct option given in parentheses:

  1. The intercept coefficient, 127.75, is the intercept of the line for (Chinstrap / Gentoo) penguins.

  2. The bill_len coefficient, 1.40, is the (intercept / slope) of both lines.

  3. The speciesGentoo coefficient, 22.85, indicates that the (intercept / slope) of the line for Gentoo is 22.85mm higher than the (intercept / slope) of the line for Chinstrap. Similarly, since the lines are parallel, the line for Gentoo is 22.85mm higher than the line for Chinstrap at any bill_len.

Exercise 7: coefficients – contextual interpretation

Next, interpret each coefficient in a contextually meaningful way. What do they tell us about penguin flipper lengths?!

  1. Interpret 127.75 (intercept of the Chinstrap line).

  2. Interpret 1.40 (slope of both lines). For both Chinstrap and Gentoo penguins, the expected flipper length…

  3. Interpret 22.85. At any bill_len, the expected flipper length…

Exercise 8: Prediction

Now that we better understand the model, let’s use it to predict flipper lengths! Recall the model and visualization:

E[flipper_len | bill_len, speciesGentoo] = 127.75 + 1.40 * bill_len + 22.85 * speciesGentoo

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)
  1. Predict the flipper length of a Chinstrap penguin with a 50mm long bill. Make sure your calculation is consistent with the plot.
127.75 + 1.40*___ + 22.85*___

To check your work, mark your prediction on the plot using a red dot: Make sure it’s where it should be!

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) + 
  geom_point(aes(x = 50, y = ____), color = "red", size = 4)
  1. Predict the flipper length of a Gentoo penguin with a 50mm long bill. Make sure your calculation is consistent with the plot.
127.75 + 1.40*___ + 22.85*___
penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) + 
  geom_point(aes(x = 50, y = ____), color = "red", size = 4)
  1. Use the predict() function to confirm your predictions in parts a and b.
# Confirm the calculation in part a
predict(penguin_mod,
        newdata = data.frame(bill_len = ___, species = "___"))

# Confirm the calculation in part b
predict(penguin_mod,
        newdata = data.frame(bill_len = ___, species = "___"))

Exercise 9: R-squared

Finally, recall that improving our predictions was one motivation for multiple linear regression (using 2 predictors instead of 1). To this end, consider the R-squared values of the simple linear regression models that use just one predictor at a time:

mod_bill <- lm(flipper_len ~ bill_len, data = penguins)
summary(mod_bill)

mod_species <- lm(flipper_len ~ species, data = penguins)
summary(mod_species)

  1. If you had to use only 1 of our 2 predictors, which would give the better predictions of flipper_len?

  2. What do you guess is the R-squared of our multiple regression model that uses both of these predictors? Why?

  3. Check your intuition. How does the R-squared of our multiple regression model compare to that of the 2 separate simple linear regression models?

summary(penguin_mod)

Reflection

You’ve now explored your first multiple regression model! Thus you likely have a lot of questions about what’s to come. What are they?

Extra exercises

Exercise 10: Challenge Part 1

In this activity, we explored the model of flipper_len by a quantitative predictor (bill_len) and a categorical predictor (species). Challenge yourself to anticipate what visualizations / models might be like with 2 quantitative predictors: bill_len and bill_dep.

Part a

First, let’s visualize the relationship of flipper_len with bill_len and bill_dep. To do so, start with the scatterplot of flipper_len and bill_len, and modify it. THINK: How might you change the points to reflect their corresponding bill_dep?

# Do NOT include a geom_smooth (which wouldn't make sense here)
penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len)) + 
  geom_point()

Part b

In Part a, you represented a 3D point cloud in 2D. BONUS (just for fun & for better understanding that this is a 3D relationship): Visualize this same data in something more like 3D:

# Don't worry about this code!
# It's specialized to this exercise and we'll rarely use it
library(plotly)
penguins %>% 
  plot_ly(x = ~bill_len, y = ~bill_dep, z = ~flipper_len,
          type = "scatter3d")

Part c

We can think of your plot above as a 3D point cloud. And the estimated formula for the linear regression model is

E[flipper_len | bill_len, bill_dep] = 179.2 + 2.38 bill_len - 5.15 bill_dep

quant_model <- lm(flipper_len ~ bill_len + bill_dep, penguins)
coef(summary(quant_model))

QUESTION:

The model of flipper_len by bill_len and species is represented by 2 parallel lines. How would you draw / represent the model here?!

Part d

Challenge: Interpret the (Intercept) and bill_len coefficients. What physical meaning do they have, i.e. where do they come into the “drawing” of the model? What contextual meaning do they have?





Exercise 11: Challenge Part 2

Next, challenge yourself to anticipate what visualizations / models might be like with 2 categorical predictors: species and sex.

Part a

First, let’s visualize the relationship of flipper_len with species and sex. To do so, start with the side-by-side boxplots of flipper_len by species, and modify them.

penguins %>% 
  ggplot(aes(y = flipper_len, x = species)) + 
  geom_boxplot()

Part b

The estimated formula for the linear regression model is

E[flipper_len | species, sex] = 191.8 + 21.07 speciesGentoo + 8.39 sexmale

cat_model <- lm(flipper_len ~ species + sex, penguins)
summary(cat_model)

QUESTION:

The model of flipper_len by species alone would only produce 2 possible predictions, one for Chinstraps and one for Gentoos. What about the model here?! How many possible predictions would it make? HINT: Checking out the plot again might help.

Part c

Challenge: Interpret the (Intercept) and speciesGentoo coefficients. What physical meaning do they have, i.e. where do they come into the “drawing” of the model? What contextual meaning do they have?





Wrap Up

  • Next Monday (2/16)
    • Quiz 1! Start studying, if you haven’t already.
  • Next Wednesday
    • CP 6 (10 minutes before class)





Solutions

Warm up

Example 1

Solution

answers will vary! in general, we’d expect R-squared to be higher than in either of these models, thus higher than 0.7014.

# Model 1: R-squared = 0.02881
summary(lm(flipper_len ~ bill_len, penguins))

Call:
lm(formula = flipper_len ~ bill_len, data = penguins)

Residuals:
    Min      1Q  Median      3Q     Max 
-30.441 -10.998   3.015   8.942  19.881 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 177.4971    13.6676  12.987   <2e-16 ***
bill_len      0.6712     0.2850   2.355   0.0195 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 11.91 on 187 degrees of freedom
Multiple R-squared:  0.02881,   Adjusted R-squared:  0.02362 
F-statistic: 5.548 on 1 and 187 DF,  p-value: 0.01954
# Model 2: R-squared = 0.7014
summary(lm(flipper_len ~ species, penguins))

Call:
lm(formula = flipper_len ~ species, data = penguins)

Residuals:
    Min      1Q  Median      3Q     Max 
-18.045  -5.045  -1.045   3.955  15.955 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)   196.0448     0.8065  243.07   <2e-16 ***
speciesGentoo  21.0372     1.0039   20.96   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.602 on 187 degrees of freedom
Multiple R-squared:  0.7014,    Adjusted R-squared:  0.6998 
F-statistic: 439.2 on 1 and 187 DF,  p-value: < 2.2e-16

Example 2

Solution

There appear to be 2 groups of points, likely the 2 different species (gentoo and chinstrap). Neither of these groups is well represented by this simple model.

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)

Thoughts? What’s going on here? How does this highlight the limitations of a simple linear regression model?

Example 3

Solution 
# Import data
cps <- read_csv("https://mac-stat.github.io/data/cps_2018.csv") %>% 
  select(-education, -hours) %>% 
  filter(age >= 18, age <= 34) %>% 
  filter(wage < 250000)

# Check it out
head(cps)
# A tibble: 6 × 6
   wage   age marital industry   health    education_level
  <dbl> <dbl> <chr>   <chr>      <chr>     <chr>          
1 75000    33 single  management fair      bachelors      
2 33000    19 single  management very_good bachelors      
3 43000    33 married management good      bachelors      
4 50000    32 single  management excellent HS             
5 14400    28 single  service    excellent HS             
6 33000    31 married management very_good bachelors      
# Build the model
wage_mod <- lm(wage ~ marital, data = cps)

# Summarize the model
coef(summary(wage_mod))
               Estimate Std. Error   t value     Pr(>|t|)
(Intercept)    46145.23    921.062  50.10002 0.000000e+00
maritalsingle -17052.37   1127.177 -15.12839 5.636068e-50

It appears that single people make far less, on average. This is likely explained by other factors such as age and experience, which are ignored by this model. (For example, single people tend to be younger and younger people tend to have less experience thus lower wages.)

Exercises

Exercise 1: Visualizing the relationship

Solution

  1. no wrong answer

  2. There are multiple options!

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point()

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, shape = species)) + 
  geom_point()

Exercise 2: Visualizing the model

Solution

  1. no wrong answer

penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)

Exercise 3: Intuition

Solution
  1. Flipper length is positively associated with bill length. Both lines have positive slopes.
  2. No. The lines are parallel / have the same slopes.
  3. Gentoo tend to have longer flippers. The Gentoo line is above the Chinstrap line.
  4. Intuition.

Exercise 4: Model formula

Solution 
penguin_mod <- lm(flipper_len ~ bill_len + species, data = penguins)
summary(penguin_mod)

Call:
lm(formula = flipper_len ~ bill_len + species, data = penguins)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.4763  -3.2260  -0.1525   3.1581  15.5303 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)   127.7537     6.1175   20.88   <2e-16 ***
bill_len        1.4024     0.1250   11.22   <2e-16 ***
speciesGentoo  22.8480     0.7938   28.78   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.112 on 186 degrees of freedom
Multiple R-squared:  0.8219,    Adjusted R-squared:   0.82 
F-statistic: 429.3 on 2 and 186 DF,  p-value: < 2.2e-16
  1. bill_len + species
  2. E[flipper_len | bill_len, speciesGentoo] = 127.75 + 1.40 * bill_len + 22.85 * speciesGentoo

Exercise 5: Sub-model formulas

Solution

Chinstrap: E[flipper_len | bill_len] = 127.75 + 1.40 bill_len

Gentoo: E[flipper_len | bill_len] = (127.75 + 22.85) + 1.40 bill_len = 150.6 + 1.40 bill_len

Exercise 6: coefficients – physical interpretation

Solution

  1. The intercept coefficient, 127.75, is the intercept of the line for Chinstrap penguins.

  2. The bill_len coefficient, 1.40, is the slope of both lines.

  3. The speciesGentoo coefficient, 22.85, indicates that the intercept of the line for Gentoo is 22.85mm higher than the intercept of the line for Chinstrap. Similarly, since the lines are parallel, the line for Gentoo is 22.85mm higher than the line for Chinstrap at any bill_len.

Exercise 7: coefficients – contextual interpretation

Solution

  1. For Chinstrap penguins with 0mm bills (silly), the expected flipper length is 127.75mm.

  2. For both Chinstrap and Gentoo penguins, the expected flipper length increases by 1.40mm for every additional mm in bill length.

  3. At any bill_len, the expected flipper length for Gentoo penguins is 22.85mm longer than that for Chinstrap penguins.

Exercise 8: Prediction

Solution 
# a
127.75 + 1.40*50 + 22.85*0
[1] 197.75
penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) + 
  geom_point(aes(x = 50, y = 197.75), color = "red", size = 4)

# b
127.75 + 1.40*50 + 22.85*1
[1] 220.6
penguins %>% 
  ggplot(aes(y = flipper_len, x = bill_len, color = species)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) + 
  geom_point(aes(x = 50, y = 220.6), color = "red", size = 4)

# c
predict(penguin_mod,
        newdata = data.frame(bill_len = 50, 
                             species = "Chinstrap"))
      1 
197.872 
predict(penguin_mod,
        newdata = data.frame(bill_len = 50, 
                             species = "Gentoo"))
       1 
220.7201

Exercise 9: R-squared

Solution
  1. species
  2. no wrong answer
  3. It’s higher than the R-squared when we use either predictor alone!

Exercises 10 & 11

Solution

No solutions! We’ll get into this topic in our next activity :)