Binary Logistic Regression

Models the probability of a binary outcome from one or more predictors — e.g., predicting whether an employee quits within a year.

Data requirements

Assumptions

• Independence of observations.
• Linearity of the LOGIT: each continuous predictor relates linearly to log-odds (Box–Tidwell check).
• No severe multicollinearity.
• No extreme influential cases; adequate events-per-variable.
• NO assumptions of normal residuals or homoscedasticity — those belong to OLS.

Hypotheses

The concept

The model fits log(p/(1−p)) = b₀ + b₁X₁ + … — a straight line in log-odds that becomes an S-shaped curve in probability. Exponentiating a coefficient gives the odds ratio: e^b is the multiplicative change in the odds of the outcome per unit of the predictor. OR > 1 raises the odds; OR < 1 lowers them; OR = 1 is no effect.

Fit is judged by the likelihood-ratio χ² against the null model, pseudo-R² (Nagelkerke, McFadden — report which; they are not OLS R²), and classification-independent discrimination via the ROC curve's AUC (.5 = chance, .7+ acceptable, .8+ good). Odds are not probabilities: an OR of 2 does not mean 'twice as likely' unless the base rate is small.

Worked example

Predicting one-year turnover (18% left) from job satisfaction, salary grade, and commute time (N = 420).

Result: model χ²(3) = 48.2, p < .001, Nagelkerke R² = .18, AUC = .74. Satisfaction OR = 0.55 (each satisfaction point nearly halves quit odds); commute OR = 1.03 per minute (p = .01).

How to run it

model <- glm(quit ~ satisfaction + salary_grade + commute,
             data = df, family = binomial)
summary(model)
exp(cbind(OR = coef(model), confint(model)))   # ORs with CIs

# fit and discrimination
anova(model, test = "Chisq")
library(pROC); auc(roc(df$quit, fitted(model)))

import statsmodels.formula.api as smf
import numpy as np
from sklearn.metrics import roc_auc_score

model = smf.logit("quit ~ satisfaction + salary_grade + commute", data=df).fit()
print(model.summary())
print(np.exp(model.params))               # odds ratios
print(roc_auc_score(df["quit"], model.predict()))

Interpreting the output

• Each OR: multiplicative change in odds per unit predictor, other predictors held constant; CI excluding 1 = significant.
• Model χ² and pseudo-R² for overall fit; AUC for discrimination.
• Translate to predicted probabilities for concrete scenarios — stakeholders think in probabilities, not odds.
• Check the events-per-variable ratio before trusting the coefficients.

APA-style reporting

Common mistakes

Related methods