Lecture 20: Logistic Regression Continued

BIOS 600 - Spring 2026

Announcements

Practice problems today and Thursday
Solutions will be posted after Thursday’s class
Please reach out to me/HyungGyu/tutors with any questions on material leading up to the final

Announcements

Final Exam is Tuesday, May 5, 8am-11am in this room (McG 1305).
Final Exam will be similar length to previous exams, and will focus on material since Exam 02.

Course evaluation

Please fill out the course evaluations

Computational setup

library(tidyverse)
library(tidymodels)
library(kableExtra)
library(randomForestSRC) # for pbc data
data(pbc, package = "randomForestSRC")

Quick review

Review

Which R function do we use to run a linear regression? How about for a logistic regression?
What additional piece of information do you need to specify when running a logistic regression, compared to a linear regression?
Suppose you have an outcome variable \(Y\) that takes on one of two values, 0 or 1. What distribution does \(Y_i\) have (one observation)? How about \(Y_1, \ldots, Y_n\) (\(n\) observations)?
What is the logit function? (describe in words & symbols)
(Fill in the blank:) Negative logits correspond to probabilities less than ______.

Logistic regression from last time

Recall the cirrhosis dataset from last class.

Researchers are interested in predicting whether a patient died (status) based on the following variables:

ascites: whether the patient had ascites (abnormal fluid buildup in the abdomen) (1 = yes, 0 = no)
bili: serum bilirubin in mg/dL (higher values may indicate liver damage/disease)
stage: histologic stage of disease (ordinal categorical variable with stages 1, 2, 3, and 4)

Logistic regression model

fit2 <- glm(status ~ ascites + bili + as.factor(stage), 
            data = pbc, 
            family = "binomial")
tidy(fit2) |>
  kable(digits = 3)

term	estimate	std.error	statistic	p.value
(Intercept)	-3.143	1.054	-2.981	0.003
ascites	2.868	1.067	2.689	0.007
bili	0.315	0.064	4.886	0.000
as.factor(stage)2	1.251	1.095	1.142	0.253
as.factor(stage)3	1.716	1.069	1.604	0.109
as.factor(stage)4	2.167	1.075	2.015	0.044

Interpretation

\(\hat\beta_{\text{bili}}\): Each one mg/dL increase in bilirubin is associated with the odds of death being multiplied by exp(.315) = 1.37, holding ascites presence and ascites presence constant.

Another Equivalent Interpretation: A one mg/dL increase in bilirubin is associated with 37% higher odds of death, holding ascites presence and status constant.

Predicted probabilities

There is a one-to-one-relationship between \(p\) and \(logit(p)\). So, if we predict the log-odds, we can back-transform to get back to a predicted probability.
For instance, suppose a patient does not have ascites, has a bilirubin level of 5 mg/dL, and is a stage 2 patient. Their predicted log-odds are:

\[−3.14+0.31×5+1.25=−0.34\]

Thus, their predicted probability of dying is

\[\frac{\text{exp}(-0.34)}{1+\text{exp}(-0.34)} = 0.42\]

In R

# Example patient:
new_patient <- data.frame(
  ascites = 0,
  bili = 5,
  stage = 2
)

# Get predicted log-odds (type = "link")
pred_logodds <- predict(fit2, 
                        newdata = new_patient, 
                        type = "link")

# Convert log-odds to odds
pred_odds <- exp(pred_logodds)

# Convert odds to probability
pred_prob <- pred_odds / (1 + pred_odds)

Output

pred_logodds

         1 
-0.3180994

pred_prob

       1 
0.421139

Alternatively…

We can also calculate the predicted probability directly using predict by setting the type to be "response".

# Get predicted probability (type = "response")
pred_prob2 <- predict(fit2, 
                        newdata = new_patient, 
                        type = "response")
pred_prob2

       1 
0.421139

Application Exercise

Tip

Head to Canvas and download the template for Application Exercise 07. AE 07 will be due this Sunday at 11:59pm.

More practice

term	estimate	std.error	statistic	p.value
(Intercept)	-3.143	1.054	-2.981	0.003
ascites	2.868	1.067	2.689	0.007
bili	0.315	0.064	4.886	0.000
as.factor(stage)2	1.251	1.095	1.142	0.253
as.factor(stage)3	1.716	1.069	1.604	0.109
as.factor(stage)4	2.167	1.075	2.015	0.044

By hand, write out the following. Leave in terms of values given:

What is the predicted log odds of dying for a patient with ascites, a bilirubin level of 3 mg/dL, and stage 3?
Given this value, what is the predicted probability?

Hypothesis tests in logistic regression

Generally, we wish to know whether the OR = 1 or equivalently, whether the logit of \(p\) (a \(\beta\) coefficient) = 0. To test

\[H_0: \beta_k = 0\]

\[H_1: \beta_k \neq 0\]

we can compare the ratio of a parameter estimate to its standard error using the standard normal distribution

(The reason we use Z instead of t is a bit technical, so we won’t go into it in class).

Confidence intervals

CIs for associations on the logit scale are given by

\[\hat \beta_j \pm z^*_{1-\alpha/2} \times \widehat {SE} (\hat\beta_j)\]

These are typically translated into confidence intervals in odds scale by exponentiating the lower and upper limits. That is,:

\[[\text{exp}(\hat \beta_j - z^*_{1-\alpha/2} \times \widehat {SE} (\hat\beta_j)), \text{exp}(\hat \beta_j + z^*_{1-\alpha/2} \times \widehat {SE} (\hat\beta_j))]\]

Connection between hypothesis tests and confidence intervals

If we wish to test \(H_0: \beta_k=\beta_0\), at \(\alpha=.05\), we can test this using a confidence interval
Check whether \(\beta_0\) is contained in a \((1-\alpha)100\%\) confidence interval
If not, reject \(H_0\)
For example, to test \(H_0: \beta_k=0\) at \(\alpha=.05\), check if 0 is contained in a 95% confidence interval
Why?

Recap

Logistic regression review
Interpreting continuous and categorical coefficients in a logistic regression model
Predicting probabilities using a logistic regression model object

Next class

More review for final