--- title: "Multiple Linear Regression - Intro" subtitle: "Notes and in-class exercises" format: html: embed-resources: true toc: true --- You can download the .qmd file for this activity [here](../activity_templates/08-mlr-intro.qmd) and open in R-studio. The rendered version is posted in the [course website](https://mutasim221b.github.io/Mac-STAT-155-Fall-25/) (Activities tab). I often experiment with the class activities (and see it in live!) and make updates, but I always post the final version before class starts. To be sure you have the most up-to-date copy, please download it once you’ve settled in before class begins. ```{r setup} #| include: false knitr::opts_chunk$set( collapse = TRUE, warning = FALSE, message = FALSE, error = TRUE, fig.height = 2.75, fig.width = 4.25, fig.env = 'figure', fig.pos = 'h', fig.align = 'center') ``` # Notes ## Learning 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 ## Readings and videos Today is a day to discover ideas, so no readings or videos to go through before class. ## 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 qmd). ```{r} # 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 `sex` and `species`: ```{r} # Model 1: R-squared = 0.1205 summary(lm(flipper_len ~ sex, 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? 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`: ```{r} 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](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: $$E[Y \mid X_1, X_2, ..., X_p] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p$$ # 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: ```{r} penguins %>% ggplot(aes(y = flipper_len, x = bill_len)) + geom_point() ``` a. How might we change the scatterplot points to *also* indicate information about penguin `species`? ```{r} penguins %>% ggplot(aes(y = flipper_len, x = bill_len, color = species)) + geom_point() ``` ```{r} penguins %>% ggplot(aes(y = flipper_len, x = bill_len, shape = species)) + 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: ```{r} penguins %>% ggplot(aes(y = flipper_len, x = bill_len)) + geom_point() + geom_smooth(method = "lm", se = FALSE) ``` a. 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? 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_len, x = bill_len, color = species)) + 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. a. What's the relationship between `flipper_len` and `species`, no matter a penguin's `bill_len`? > **Response:** Put your response here. b. What's the relationship between `flipper_len` and `bill_len`, no matter a penguin's `species`? > **Response:** Put your response here. c. Does the rate of increase in `flipper_len` with `bill_len` differ between the two `species`? > **Response:** Put your response here. ## 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_len ~ bill_len + 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_len` by both `bill_len` and `species`? bill_len + species b. Complete the following model formula: E[flipper_len | bill_len, speciesGentoo] = 127.753693 + 1.402367 * bill_len + 22.848036 * 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 = 1 and speciesGentoo = 0 Chinstrap: flipper_len = 127.753693 + 1.402367* bill_len Gentoo: flipper_len = (127.753693 +22.848036) + 1.402367 * 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: a. The intercept coefficient, 127.75, is the intercept of the line for (Chinstrap / Gentoo) penguins. b. The `bill_len` 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_len`. ## 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_len`, 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_len, x = bill_len, 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_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: ```{r} mod_bill <- lm(flipper_len ~ bill_len, data = penguins) summary(mod_bill) mod_species <- lm(flipper_len ~ species, data = penguins) summary(mod_species) ``` a. If you had to use only *1* of our 2 predictors, which would give the better predictions of `flipper_len`? 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) ``` \ \ \ \ # Solutions {-} ```{r eval = TRUE, echo = FALSE} # Load packages & data library(tidyverse) data(penguins) penguins <- penguins %>% filter(species != "Adelie", bill_len < 57) # 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) ``` ## Exercise 1: Visualizing the relationship a. There are multiple options! ```{r eval = TRUE} 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 a. ```{r eval = TRUE} penguins %>% ggplot(aes(y = flipper_len, x = bill_len, color = species)) + geom_point() + geom_smooth(method = "lm", se = FALSE) ``` ## Exercise 3: Intuition a. Gentoo tend to have longer flippers. b. Flipper length is positively associated with bill length. c. No. the lines are parallel / have the same slopes. ## Exercise 4: Model formula a. `bill_len + species` b. E[flipper_len | bill_len, speciesGentoo] = 127.75 + 1.40 * bill_len + 22.85 * speciesGentoo ## Exercise 5: Sub-model formulas Chinstrap: flipper_len = 127.75 + 1.40 bill_len Gentoo: flipper_len = (127.75 + 22.85) + 1.40 bill_len = 150.6 + 1.40 bill_len ## Exercise 6: coefficients -- physical interpretation a. The intercept coefficient, 127.75, is the intercept of the line for **Chinstrap** penguins. b. The `bill_len` coefficient, 1.40, is the **slope** of both lines. c. 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 b. For both Chinstrap and Gentoo penguins, average flipper lengths increase by 1.40mm for every additional mm in bill length. c. At any `bill_len`, the average flipper length for Gentoo penguins is 22.85mm longer than that for Chinstrap penguins. ## Exercise 8: Prediction ```{r eval = TRUE} # a 127.75 + 1.40*50 + 22.85*0 # b 127.75 + 1.40*50 + 22.85*1 # c predict(penguin_mod, newdata = data.frame(bill_len = 50, species = "Chinstrap")) predict(penguin_mod, newdata = data.frame(bill_len = 50, species = "Gentoo")) ``` ## Exercise 9: R-squared a. species b. no wrong answer c. It's higher than the R-squared when we use either predictor alone!