Regression analysis

Regression analysis

Regression is the workhorse of analytics: it quantifies how outcomes relate to inputs and lets you say "holding other things equal, X is associated with this much Y". Its power is also its trap, because a coefficient looks like a causal effect and usually is not. Interpreting it honestly is the skill.

Method

1. Pick the form for the outcome. Linear regression for continuous outcomes, logistic for binary (odds/probability), Poisson-family for counts. Using linear regression on a 0/1 outcome or ignoring the outcome type gives nonsense predictions and misleading coefficients.
2. Read coefficients as "holding others constant". Each coefficient is the association between that predictor and the outcome with the other included variables held fixed. This "controlling for" is only as good as the variables you included; an omitted confounder biases the coefficients you have (see correlation-causation). State the direction, size, and units, not just significance.
3. Judge fit and the residuals, not just R-squared. R-squared says how much variance is explained but nothing about correctness; a high R-squared can hide a mis-specified model, and a low one can still carry a real relationship. Plot the residuals: patterns reveal nonlinearity, heteroscedasticity, and outliers the summary numbers miss.
4. Report uncertainty on every coefficient. Coefficients are estimates with confidence intervals; a "significant" coefficient with a wide interval is weakly known. Give the interval, and remember that with enough data, tiny, unimportant effects become significant (see statistical-inference).
5. Check the assumptions and the specification. Multicollinearity (correlated predictors make individual coefficients unstable and uninterpretable), the right functional form (linear in the right variables, interactions where they matter), and influential outliers. A model that violates its assumptions produces confident, wrong coefficients.
6. Resist the causal leap. A regression coefficient is an association, not the effect of intervening, unless the data came from an experiment or you have carefully handled confounding. "Controlling for X" in a regression is not the same as an experiment; do not tell stakeholders a coefficient is what will happen if they change the input (see correlation-causation, experiment-analysis).

Boundaries

• Interpretive regression (understanding relationships) and predictive modeling (forecasting an outcome) optimize for different things; a model built for insight is judged on honest coefficients and caveats, one for prediction on held-out accuracy (see model-evaluation, ml-problem-framing).
• Regression on observational data supports "associated with", not "causes"; the coefficient's causal interpretation requires a design (experiment, natural experiment) the regression alone does not provide.
• Complex relationships (strong nonlinearity, interactions, high dimensionality) may exceed what a readable regression captures; there the tradeoff is interpretability versus a flexible model you cannot explain as simply.