10: Multiple Linear Regression

IMS, Ch. 8

Benjamin S. Baumer

Smith College

Feb 18, 2026

Recap

Recall the Italian restaurants data

library(tidyverse)
nyc <- read_csv("https://gattonweb.uky.edu/faculty/sheather/book/docs/datasets/nyc.csv")
mod <- lm(Price ~ Food + East, data = nyc)

italian_plot <- ggplot(
    data = nyc, 
    aes(x = Food, y = Price, color = factor(East))
  ) +
  geom_jitter(width = 0.1, alpha = 0.5, size = 2) + 
  scale_x_continuous("Jittered service rating") + 
  scale_y_continuous("Average Price (US$)") +
  scale_color_discrete("East of 5th?")

Parallel slopes model

italian_plot + 
  moderndive::geom_parallel_slopes(se = 0)

Model interpretation

coef(mod)
(Intercept)        Food        East 
 -17.429591    2.874707    1.459285

Interpretation of Slope

  • Among Italian restaurants in NYC in 2001, each additional rating point of Food is associated with a $2.88 increase the expected price of a meal of two, after controlling for location (relative to 5th Avenue).

Interpretation of East

  • On average, restaurants on the East side of 5th Avenue charge $1.46 more (than those on the West side) for food of the same quality.

Your turn: Parallel slopes model

  • How is the quality of the \(Decor\) at these restaurants associated with its price?

  • Build a parallel slopes model by conditioning on the \(East\) variable. Interpret the coefficients of this model.

  • What is the value of being on the East Side of Fifth Avenue?

  • Calculate the expected \(Price\) of a restaurant in the East Village with a \(Decor\) rating of 23.

Multiple Regression with a Second Quantitative Variable

MLR: two numerical explanatory variables

If \(X_2\) is a quantitative variable, then we have

\[ \widehat{y} = b_0 + b_1 \cdot X_1 + b_2 \cdot X_2 \]

Notice that our model is no longer a line, rather it is a plane that lives in three dimensions!

https://en.wikipedia.org/wiki/Analytic_geometry#Lines_and_planes

Italian Restaurants (continued)

  • Consider quality of \(Food\), and also quality of \(Service\)
  • In R, simply add another variable to our model
mod_fs <- lm(Price ~ Food + Service, data = nyc)
coef(mod_fs)
(Intercept)        Food     Service 
 -21.158582    1.495369    1.704101

Your turn

  • Interpret the value of the \(Food\) coefficient

  • Interpret the value of the \(Service\) coefficient

3D plotting with plotly

  • Set up a grid of values in the Food-Service plane
library(modelr)
plane <- nyc |>
  data_grid(
    Food = seq_range(Food, n = 25),
    Service = seq_range(Service, n = 25),
  )
  • Compute the fitted value \(\widehat{Price}\) over the grid
plane <- plane |>
  mutate(Price_hat = predict(mod_fs, newdata = plane))

Build the 3D plot

library(plotly)
data_space_fs <- nyc |>
  plot_ly(x = ~Food, y = ~Service, z = ~Price) |>
  add_markers()

plane_fs <- data_space_fs |>
  add_surface(
    data = plane, 
    x = ~unique(Food), 
    y = ~unique(Service), 
    z = ~matrix(Price_hat, nrow = 25), 
    opacity = 0.7
  )

3D visualization

plane_fs

Your turn: interpretation

  • Interpret the coefficients of this model:
    • What does the coefficient of \(Food\) mean?
    • \(Service\)?
  • How important is \(Service\) relative to \(Food\)?
  • Is it fair to compare the two coefficients?

Your turn: residuals

  • Use broom::augment() to find the expected \(Price\) of a restaurant with a \(Food\) rating of 21 and a \(Service\) rating of 28
  • Calculate the residual for San Pietro. Is it overpriced?
filter(nyc, Restaurant == "San Pietro")
# A tibble: 1 × 7
   Case Restaurant Price  Food Decor Service  East
  <dbl> <chr>      <dbl> <dbl> <dbl>   <dbl> <dbl>
1    44 San Pietro    58    24    21      23     1

Higher dimensions

  • What geometric shape would we have if we added all three explanatory variables to the model?
mod_full <- lm(Price ~ Food + Service + East, data = nyc)

planes <- nyc |>
  data_grid(
    Food = seq_range(Food, n = 25),
    Service = seq_range(Service, n = 25),
    East = seq_range(East, n = 2)
  )

planes <- planes |>
  mutate(Price_hat = predict(mod_full, newdata = planes))

pplanes <- data_space_fs |>
  add_surface(
    data = filter(planes, East == 0), 
    x = ~unique(Food), y = ~unique(Service), 
    z = ~matrix(Price_hat, nrow = 25), 
    opacity = 0.7
  ) |>
  add_surface(
    data = filter(planes, East == 1), 
    x = ~unique(Food), y = ~unique(Service), 
    z = ~matrix(Price_hat, nrow = 25), 
    opacity = 0.7
  )

Big reveal

pplanes