Binary Logistic Regression

logistic-regression

odds-ratio

binary-outcome

mcfadden

Modelling a dichotomous outcome; odds ratios, confidence intervals, McFadden R-squared, and calibration

Published

April 17, 2026

Research question

Logistic regression models the probability of a binary outcome as a logit-linear function of predictors. Biomedical example: in patients admitted with acute pancreatitis, does an admission lactate level predict in-hospital mortality after adjusting for age, Ranson’s score, and comorbidities?

Assumptions

Assumption	How to verify in R
Binary outcome (0 / 1)	data check
Independent observations	design
Linearity of continuous predictors on the logit	Box-Tidwell test; splines; `car::crPlots()`
No severe multicollinearity	`car::vif()`
No complete separation	warnings from `glm()`; Firth correction via `logistf` if needed

Hypotheses

For each coefficient: \(H_0: \beta_j = 0\) (equivalently \(OR_j = 1\)) vs. \(H_1: \beta_j \ne 0\).

R code

library(tidyverse); library(rstatix); library(broom); library(performance)
library(gtsummary); library(pROC)
set.seed(42)

# 260 pancreatitis admissions
pan <- tibble(
  age       = rnorm(260, 58, 14),
  ranson    = sample(0:8, 260, replace = TRUE),
  comorb    = sample(0:4, 260, replace = TRUE),
  lactate   = rlnorm(260, log(2.0), 0.5),
  lp        = -5 + 0.02 * age + 0.4 * ranson + 0.2 * comorb + 0.8 * log(lactate)
) |>
  mutate(death = rbinom(260, 1, plogis(lp))) |>
  select(-lp)

fit <- glm(death ~ age + ranson + comorb + lactate, data = pan, family = binomial)

# Coefficients with odds ratios
broom::tidy(fit, conf.int = TRUE, exponentiate = TRUE)

# Pseudo R-squareds
performance::r2(fit)

# Discrimination: ROC AUC
roc_obj <- roc(pan$death, fitted(fit), quiet = TRUE)
auc(roc_obj); ci.auc(roc_obj)

# Calibration
check_model(fit, check = c("pp_check", "binned_residuals"))

tbl_regression(fit, exponentiate = TRUE) |>
  add_global_p() |> add_glance_source_note()

Interpreting the output

Exponentiated coefficients are odds ratios. For lactate, OR \(\approx 2.4\) (95 % CI 1.6-3.5), \(p < .001\): a one-unit increase in log-lactate more than doubles the odds of in-hospital mortality. The McFadden pseudo-\(R^2 \approx 0.23\) indicates good fit. ROC AUC \(= 0.82\) (95 % CI 0.77-0.87) reflects strong discrimination.

Effect size

Odds ratio per predictor (exponentiated coefficient). Interpretive thresholds depend on context; OR = 1 is no effect.
McFadden’s \(R^2\): values 0.2-0.4 are considered excellent fit.
AUC: 0.7-0.8 acceptable, 0.8-0.9 excellent, > 0.9 outstanding.

Reporting (APA 7)

In a multivariable logistic regression, higher admission lactate independently predicted in-hospital mortality in acute pancreatitis (OR = 2.37, 95 % CI 1.58-3.54, p < .001) after adjustment for age, Ranson’s score, and number of comorbidities. The model discriminated well (AUC = 0.82, 95 % CI 0.77-0.87; McFadden’s R^2 = .23).

Common pitfalls

Complete separation: a predictor perfectly predicts the outcome; glm() returns gigantic SEs. Use Firth correction (logistf::logistf()).
Small events-per-variable (EPV) ratio: fewer than 10 events per predictor inflates Type I error. Consider penalised regression.
Reporting only ORs without CIs.
Using the Hosmer-Lemeshow test for calibration; it is known to have low power and arbitrary binning. Prefer performance::check_model() calibration diagnostics.

Parametric vs. non-parametric alternative

Penalised: ridge / lasso logistic regression via glmnet.
Probit regression: glm(..., family = binomial(link = "probit")).
For ordered outcomes, use ordinal logistic regression.
For unordered multi-category outcomes, use multinomial logistic regression.