---
title: "Introduction to multiple regression"
subtitle: "Notes and in-class exercises"
format: 
  html:
    embed-resources: true
    toc: true
---

# Motivation

**EXAMPLE 1**

Let's explore some data on penguins.

First, enter `install.packages("palmerpenguins")` in the *console* (not Rmd).

Then load the `penguins` data.

You can find a **codebook** for these data by typing `?penguins` in your *console* (not Rmd).


```{r warning=FALSE, message=FALSE}
# Load packages
library(readr)
library(ggplot2)
library(dplyr)

# Load data
library(palmerpenguins)
data(penguins)
penguins <- penguins %>% 
    filter(species != "Adelie", bill_length_mm < 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_length_mm` by penguin `sex` and `species`: 

```{r}
summary(lm(flipper_length_mm ~ sex, penguins))$r.squared
summary(lm(flipper_length_mm ~ species, penguins))$r.squared
```

How might we improve our predictions of `flipper_length_mm` using only these 2 predictors?

What do you think our new R-squared would be?






**EXAMPLE 2**

Consider a simple linear regression model of `flipper_length_mm` by `bill_length_mm`:

```{r}
penguins %>% 
    ggplot(aes(y = flipper_length_mm, x = bill_length_mm)) + 
    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](https://cps.ipums.org/cps/resources/codebooks/cpsmar18.pdf).

```{r}
# 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)
```

```{r}
# Check it out
head(cps)
```

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

```{r}
# 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







**Multiple linear regression model formula**

In general, a **multiple linear regression model** of $y$ with multiple predictors $(x_1, x_2, ..., x_p)$ is represented by the following formula:

$$E[y \mid x_1, x_2, ..., x_p] = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p$$


<br><br><br><br>


# 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_length_mm` by just 2 predictors: `bill_length_mm` (quantitative) and `species` (categorical).


## Exercise 1: Visualizing the relationship

We've learned how to visualize the relationship of `flipper_length_mm` by `bill_length_mm` alone:

```{r}
penguins %>% 
    ggplot(aes(y = flipper_length_mm, x = bill_length_mm)) + 
    geom_point()
```

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

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

```{r}
penguins %>%
    ggplot(aes(y = flipper_length_mm, x = bill_length_mm, ___ = ___)) +
    geom_point()
```





## Exercise 2: Visualizing the model

We've also learned that a simple linear regression model of `flipper_length_mm` by `bill_length_mm` alone can be represented by a line:

```{r}
penguins %>% 
    ggplot(aes(y = flipper_length_mm, x = bill_length_mm)) + 
    geom_point() + 
    geom_smooth(method = "lm", se = FALSE)
```

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

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

```{r}
penguins %>%
    ggplot(aes(y = flipper_length_mm, x = bill_length_mm, ___ = ___)) +
    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_length_mm` by `bill_length_mm` and `species` is represented by 2 lines. 

Let's interpret those lines!

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

a. What's the relationship between `flipper_length_mm` and `species`, no matter a penguin's `bill_length_mm`? 


b. What's the relationship between `flipper_length_mm` and `bill_length_mm`, no matter a penguin's `species`?


c. Does the rate of increase in `flipper_length_mm` with `bill_length_mm` differ between the two `species`?









## Exercise 4: Model formula

Of course, there's a formula behind the multiple regression model.

We can obtain this using the usual `lm()` function.

```{r}
# Build the model
penguin_mod <- lm(flipper_length_mm ~ bill_length_mm + species, data = penguins)

# Summarize the model
coef(summary(penguin_mod))
```


a. In the `lm()` function, how did we communicate that we wanted to model `flipper_length_mm` by both `bill_length_mm` and `species`?


b. Complete the following model formula:    
    
    E[flipper_length_mm | bill_length_mm, speciesGentoo] = ___ + ___ * bill_length_mm + ___ * 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_length_mm` and `bill_length_mm` 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: flipper_length_mm = ___ + ___ bill_length_mm

Gentoo:    flipper_length_mm = ___ + ___ bill_length_mm





## 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:        

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

b. The `bill_length_mm` coefficient, 1.40, is the (intercept / slope) of both lines.

c. 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_length_mm`.
    
    



## Exercise 7: coefficients -- contextual interpretation

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

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

b. Interpret 1.40 (slope of both lines). For both Chinstrap and Gentoo penguins, we expect...

c. Interpret 22.85. At any `bill_length_mm`, we expect...
    
    




## Exercise 8: Prediction

Now that we better understand the model, let's use it to predict flipper lengths!

Recall the model summary and visualization:

```{r}
coef(summary(penguin_mod))

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

a. Predict the flipper length of a Chinstrap penguin with a 50mm long bill. Make sure your calculation is consistent with the plot.       

```{r}
127.75 + 1.40*___ + 22.85*___
```    


b. Predict the flipper length of a Gentoo penguin with a 50mm long bill. Make sure your calculation is consistent with the plot.       

```{r}
127.75 + 1.40*___ + 22.85*___
```    

c. Use the `predict()` function to confirm your predictions in parts a and b.

```{r}
# Confirm the calculation in part a
predict(penguin_mod,
        newdata = data.frame(bill_length_mm = ___, species = "___"))

# Confirm the calculation in part b
predict(penguin_mod,
        newdata = data.frame(bill_length_mm = ___, 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:

```{r}
summary(lm(flipper_length_mm ~ bill_length_mm, data = penguins))$r.squared

summary(lm(flipper_length_mm ~ species, data = penguins))$r.squared
```


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

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

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

```{r}
summary(penguin_mod)$r.squared
```









## 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?

> **Response:** Put your response here.

