Logistic regression
Lecture 19
November 4, 2026
Recap: Simple linear regression
Numerical outcome and one numerical predictor:
Recap: Simple linear regression
Numerical outcome and one categorical predictor (two levels):
Recap: Multiple linear regression
Numerical outcome, numerical and categorical predictors:
Today: a binary outcome
\[ y = \begin{cases} 1 & &&\text{eg. Yes, Win, True, Heads, Success}\\ 0 & &&\text{eg. No, Lose, False, Tails, Failure}. \end{cases} \]
Example: Is the e-mail spam or not?
\[ \mathbf{x}: \text{word and character counts in an e-mail.} \]
\[ y = \begin{cases} 1 & \text{it's spam}\\ 0 & \text{it's legit} \end{cases} \]
Example: Is it cancer or not?
\[ \mathbf{x}: \text{features in a medical image.} \]
\[ y = \begin{cases} 1 & \text{it's cancer}\\ 0 & \text{it's healthy} \end{cases} \]
Example: Will they default?
\[ \mathbf{x}: \text{financial and demographic info about a loan applicant.} \]
\[ y = \begin{cases} 1 & \text{applicant is at risk of defaulting on loan}\\ 0 & \text{applicant is safe} \end{cases} \]
Example: Did the House seat flip to Democrat or not?
\[ \mathbf{x}: \text{Trump's 2024 vote share in the congressional district.} \]
\[ y = \begin{cases} 1 & \text{the seat flipped to Democrat from 2024 to 2026}\\ 0 & \text{it didn't} \end{cases} \]
How do we model this type of data?
Straight line of best fit is a little silly
Instead: S-curve of best fit
Instead of modeling \(y\) directly, we model the probability that \(y=1\):
- “Given new email, what’s the probability that it’s spam?”
- “Given new image, what’s the probability that it’s cancer?’’
- “Given new loan application, what’s the probability that they default?’’
- “Given an uncalled House seat, what’s the probability that it flips to Democrat?’’
Probability to odds
Odds to log odds
Participate 📱💻
If \(p\) is the probability of success, what is the following called:
\[ \frac{p}{1-p} \]
- Probability of failure
- Odds of failure
- Odds of success
- Log-odds of success
Go to wooclap.com and use the code TCGEJHI.
Visualizing the data
Code
ggplot(
midterm_26,
aes(x = trump_24, y = flip_dem_24_26, fill = flip_dem_24_26)
) +
geom_jitter(
shape = "circle filled",
height = 0.2,
alpha = 0.7,
size = 3,
show.legend = FALSE,
color = "white"
) +
scale_fill_manual(
values = c("0" = "darkgray", "1" = "#0015BC"),
na.value = "#A3B18A"
) +
scale_x_continuous(
limits = c(10, 90),
breaks = seq(0, 100, 10),
labels = label_percent(scale = 1)
) +
labs(
x = "Trump's 2024 vote share (%)",
y = "Flipped to Democrat",
title = "2026 House elections",
caption = "Data from Decision Desk HQ, as of 2026-11-04 at 11:14 am"
) +
facet_wrap(~called_26, nrow = 2, scales = "free_y")
Goal: Predict whether a seat flips to Democrat based on Trump’s 2024 vote share for uncalled seats, using data from called seats.
