Logistic Regression: Binary Outcomes

When the outcome variable is binary (0/1, yes/no, event/no event), ordinary linear regression produces predicted probabilities that can fall outside [0,1] and assumes a linear relationship between predictors and probability — both are unrealistic. Logistic regression solves this by modeling the log-odds (logit) of the outcome probability as a linear function of predictors. The logistic function (sigmoid curve) maps any real number to a probability between 0 and 1: P(Y=1) = 1 / (1 + e^(−(b₀ + b₁X₁ + ...))). The curve is S-shaped: near 0 for very negative linear predictors, near 1 for very positive values, and 0.5 where the linear predictor equals zero. The log-odds (logit) = ln[P/(1−P)] = b₀ + b₁X₁ + b₂X₂. The log-odds is a linear function of the predictors, but probability is not — the effect of X on probability depends on where on the sigmoid curve you are. A change of 1 unit in X₁ changes the log-odds by b₁, but this corresponds to a large change in probability near P = 0.5 and a small change near P = 0 or P = 1. Logistic regression is fit by maximum likelihood estimation (MLE), not OLS — it finds the parameter values that maximize the probability of observing the data. The log-likelihood is maximized iteratively; standard regression output reports the log-likelihood value and −2 log-likelihood (−2LL) for model comparison.