Logistic Regression for Binary Outcomes: From Odds Ratios to Standardized Risks

This chapter introduces logistic regression as a fundamental statistical method for analyzing binary outcomes in epidemiological research, such as the presence or absence of disease, survival or death, and treatment success or failure. The chapter explains why ordinary linear regression is unsuitable for binary outcomes and how logistic regression addresses this limitation by modeling the log-odds of an event through the logit link function. Students learn the mathematical formulation of the binary logistic regression model, its three components within the generalized linear model (GLM) framework, and the estimation of regression parameters using maximum likelihood estimation (MLE). Particular emphasis is placed on interpreting regression coefficients through adjusted Odds Ratios (ORs), evaluating confidence intervals and statistical significance, and controlling for potential confounders through multivariable modeling. Examples involving smoking and lung cancer, as well as obesity and hypertension, demonstrate how logistic regression can be applied to epidemiological questions using statistical software such as R.

The second major focus is the interpretation and limitations of Odds Ratios, together with the prediction and evaluation of outcome probabilities. Students examine the distinction between odds, risk, Risk Ratios (RRs), and Risk Differences (RDs), recognizing that ORs approximate RRs when outcomes are rare but may substantially overestimate relative risks when outcomes are common. The chapter also discusses the non-collapsibility of ORs, explaining why adjusted and crude estimates may differ even in the absence of confounding. Beyond estimating associations, students learn to calculate predicted probabilities for individuals with specific characteristics and evaluate model performance using discrimination and calibration. Methods discussed include the Area Under the Curve (AUC), calibration plots, the Hosmer–Lemeshow test, calibration intercept and slope, and the Brier Score. These concepts help students distinguish a model's ability to discriminate between individuals with and without an outcome from its ability to produce accurate absolute-risk estimates, which are important for clinical decision-making and public health planning.

The final component focuses on marginal standardization and the estimation of population-level risks from logistic regression models, providing a more interpretable alternative to relying exclusively on adjusted ORs. Students learn to estimate marginal predicted probabilities by assigning exposure values to individuals while retaining their observed covariate distributions, then averaging the predicted probabilities across the population. These standardized risks can be used to calculate adjusted Risk Differences and Risk Ratios, facilitating comparisons between exposed and unexposed populations in terms of absolute and relative risk. The chapter also introduces effect modification through interaction terms, basic model diagnostics, and conditional logistic regression for matched study designs. Through theoretical explanations, worked epidemiological examples, graphical illustrations, and R programming exercises, students develop the ability to construct and interpret logistic regression models, assess predictive performance, and communicate adjusted associations and standardized risks appropriately. Overall, the chapter emphasizes that effective epidemiological analysis requires distinguishing association from absolute risk and selecting measures that align with the research question and public health objectives.