Probabilities, odds, odds ratios

Prof. Maria Tackett

Mar 18, 2025

Announcements

  • HW 03 due TODAY at 11:59pm

  • Exploratory data analysis due March 20

    • Next milestone: Project presentations March 31 in lab

  • Statistics experience due April 15

Topics

  • Logistic regression for binary response variable

  • Relationship between odds and probabilities

  • Odds ratios

Computational setup

# load packages
library(tidyverse)
library(tidymodels)
library(knitr)
library(patchwork)
library(Stat2Data) #contains sleep data set

# set default theme in ggplot2
ggplot2::theme_set(ggplot2::theme_bw())

Predicting categorical outcomes

Types of outcome variables

Quantitative outcome variable:

  • Sales price of a house in Duke Forest
  • Model: Expected sales price given the number of bedrooms, lot size, etc.

Categorical outcome variable:

  • Indicator of being high risk of getting coronary heart disease in the next 10 years
  • Model: Probability an adult is high risk of heart disease in the next 10 years given their age, total cholesterol, etc.

Models for categorical outcomes

Logistic regression

2 Outcomes

1: Yes, 0: No

Multinomial logistic regression

3+ Outcomes

1: Democrat, 2: Republican, 3: Independent

Example: Win probability

ESPN Analytics win probability for Duke vs. Louisville (March 15, 2025)

Do teenagers get 7+ hours of sleep?

Students in grades 9 - 12 were surveyed about health risk behaviors including whether they usually get 7 or more hours of sleep.

Sleep7

1: yes

0: no

# A tibble: 446 × 2
     Age Sleep7
   <int>  <int>
 1    16      1
 2    17      0
 3    18      0
 4    17      1
 5    15      0
 6    17      0
 7    17      1
 8    16      1
 9    16      1
10    18      0
# ℹ 436 more rows

Plot the data

ggplot(sleep, aes(x = Age, y = Sleep7)) +
  geom_point() + 
  labs(y = "Getting 7+ hours of sleep")

Let’s fit a linear regression model

Outcome: \(Y\) = 1: yes, 0: no

Let’s use proportions

Outcome: Probability of getting 7+ hours of sleep

What happens if we zoom out?

Outcome: Probability of getting 7+ hours of sleep

🛑 This model produces predictions outside of 0 and 1.

Let’s try another model

✅ This model (called a logistic regression model) only produces predictions between 0 and 1.

The code

ggplot(sleep_age, aes(x = Age, y = prop)) +
  geom_point() + 
  geom_hline(yintercept = c(0,1), lty = 2) + 
  stat_smooth(method ="glm", method.args = list(family = "binomial"), 
              fullrange = TRUE, se = FALSE) +
  labs(y = "P(7+ hours of sleep)") +
  xlim(1, 40) +
  ylim(-0.5, 1.5)

Different types of models

Method Outcome Model
Linear regression Quantitative \(y_i = \beta_0 + \beta_1~ x_i\)
Linear regression (transform Y) Quantitative \(\log(y_i) = \beta_0 + \beta_1~ x_i\)
Logistic regression Binary \(\log\big(\frac{p_i}{1-p_i}\big) = \beta_0 + \beta_1 ~ x_i\)

Linear vs. logistic regression

State whether a linear regression model or logistic regression model is more appropriate for each scenario.

  1. Use age and political party to predict if a randomly selected person will vote in the next election.

  2. Use budget and run time (in minutes) to predict a movie’s total revenue.

  3. Use age and sex to calculate the probability a randomly selected adult will visit Duke Health in the next year.

Probabilities and odds

Data: Concern about rising AI

This data comes from the 2023 Pew Research Center’s American Trends Panel. The survey aims to capture public opinion about a variety of topics including politics, religion, and technology, among others. We will use data from 11201 respondents in Wave 132 of the survey conducted July 31 - August 6, 2023.


The goal of this analysis is to understand the relationship between age, how much someone has heard about artificial intelligence (AI), and concern about the increased use of AI in daily life.


A more complete analysis on this topic can be found in the Pew Research Center article Growing public concern about the role of artificial intelligence in daily life by Alec Tyson and Emma Kikuchi.

Variables

  • ai_concern: Whether a respondent said they are “more concerned than excited” about in the increased use of AI in daily life (1: yes, 0: no)

Source: Pew Research

Variables

  • ai_heard : Response to the question “How much have you heard or read about AI?”

    • A lot
    • A little
    • Nothing at all
    • Refused

  • age_cat: Age category

    • 18-29
    • 30-49
    • 50-64
    • 65+
    • Refused

Data prep

# change variable names and recode categories
pew_data <- pew_data |>
  mutate(ai_concern = if_else(CNCEXC_W132 == 2, 1, 0),
         age_cat = case_when(F_AGECAT == 1 ~ "18-29",
                             F_AGECAT == 2 ~ "30-49",
                             F_AGECAT == 3 ~ "50-64",
                             F_AGECAT == 4 ~ "65+",
                             TRUE ~ "Refused"), 
         ai_heard = case_when(AI_HEARD_W132 == 1 ~ "A lot",
                              AI_HEARD_W132 == 2 ~ "A little",
                              AI_HEARD_W132 == 3 ~ "Nothing at all",
                              TRUE ~ "Refused"
                              ))

# Make factors and  relevel 
pew_data <- pew_data |>
  mutate(ai_concern = factor(ai_concern),
         age_cat = factor(age_cat), 
         ai_heard = factor(ai_heard, levels = c("A lot", "A little", "Nothing at all", "Refused"))
  )

Univariate EDA

Binary response variable

  • \(Y = 1: \text{ yes (success), } 0: \text{ no (failure)}\)

  • \(p\): probability that \(Y=1\), i.e., \(P(Y = 1)\)

  • \(\frac{p}{1-p}\): odds that \(Y = 1\)

  • \(\log\big(\frac{p}{1-p}\big)\): log odds

  • Go from \(p\) to \(\log\big(\frac{p}{1-p}\big)\) using the logit transformation

Example

Suppose there is a 70% chance it will rain tomorrow

  • Probability it will rain is \(\mathbf{p = 0.7}\)
  • Probability it won’t rain is \(\mathbf{1 - p = 0.3}\)
  • Odds it will rain are 7 to 3, 7:3, \(\mathbf{\frac{0.7}{0.3} \approx 2.33}\)

Concerned about AI in daily life?

pew_data |>
  count(ai_concern) |>
  mutate(p = round(n / sum(n), 3))
# A tibble: 2 × 3
  ai_concern     n     p
  <fct>      <int> <dbl>
1 0           5245 0.468
2 1           5956 0.532


\(P(\text{Concerned about AI}) = P(Y = 1) = p = 0.532\)

\(P(\text{Not concerned about AI}) = P(Y = 0) = 1 - p = 0.468\)

\(\text{Odds of being concerned about AI} = \frac{0.532}{0.468} = 1.137\)

From odds to probabilities

odds

\[\text{odds} = \frac{p}{1-p}\]

probability

\[p = \frac{\text{odds}}{1 + \text{odds}}\]

Odds ratios

Concern about AI vs. age

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

Compare the odds for two groups

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

We want to compare concern about increased use of AI in daily life between individuals who are 18-29 years old to those who are 65+ years old

Compare the odds for two groups

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

We’ll use the odds to compare the two groups

\[ \text{odds} = \frac{P(\text{success})}{P(\text{failure})} = \frac{\text{# of successes}}{\text{# of failures}} \]

Compare the odds for two groups

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

  • Odds of being concerned with increased use of AI in daily life for 18-29 year olds: \(\frac{416}{550} = 0.756\)

  • Odds of being concerned with increased use of AI in daily life for those who are 65+ years old: \(\frac{2013}{1376} = 1.463\)

  • Based on this, we see that individuals 65+ years old are more likely to be concerned about the increased use of AI in daily life than 18-29 year olds.

Odds ratio (OR)

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

Let’s summarize the relationship between the two groups. To do so, we’ll use the odds ratio (OR).

\[ OR = \frac{\text{odds}_1}{\text{odds}_2} \]

OR: AI concern by age

Age Not Concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

\[OR = \frac{\text{odds}_{18-29}}{\text{odds}_{65+}} = \frac{0.756}{1.463} = \mathbf{0.517}\]

The odds an 18-29 year old is concerned about increased use of AI in daily life are 0.517 times the odds a 65+ year old is concerned.

More natural interpretation

  • It’s more natural to interpret the odds ratio with a statement with the odds ratio greater than 1.

  • The odds a 65+ year old is concerned about increased use of AI in daily life are 1.934 (1/0.517) times the odds an 18-29 year old is concerned.

Code to make table

pew_data |>
  count(age_cat, ai_concern)
# A tibble: 10 × 3
   age_cat ai_concern     n
   <fct>   <fct>      <int>
 1 18-29   0            550
 2 18-29   1            416
 3 30-49   0           1898
 4 30-49   1           1681
 5 50-64   0           1398
 6 50-64   1           1818
 7 65+     0           1376
 8 65+     1           2013
 9 Refused 0             23
10 Refused 1             28

Code to make table

pew_data |>
  count(age_cat, ai_concern) |>
  pivot_wider(names_from = ai_concern, values_from = n)
# A tibble: 5 × 3
  age_cat   `0`   `1`
  <fct>   <int> <int>
1 18-29     550   416
2 30-49    1898  1681
3 50-64    1398  1818
4 65+      1376  2013
5 Refused    23    28

Code to make table

pew_data |>
  count(age_cat, ai_concern) |>
  pivot_wider(names_from = ai_concern, values_from = n) |>
  kable()
age_cat 0 1
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

Code to make table

pew_data |>
  count(age_cat, ai_concern) |>
  pivot_wider(names_from = ai_concern, values_from = n) |>
  kable(col.names = c("Age", "Not concerned", "Concerned"))
Age Not concerned Concerned
18-29 550 416
30-49 1898 1681
50-64 1398 1818
65+ 1376 2013
Refused 23 28

Application exercise

📋 sta210-sp25.netlify.app/ae/ae-09-prob-odds.html

Click here to add your response to the Google Slide.

Recap

  • Introduced logistic regression for binary response variable

  • Showed the relationship between odds and probabilities

  • Introduced odds ratios