regresanalyses

Regress analyses, or regression analyses, are statistical methods used to examine the relationship between a dependent variable and one or more independent variables. The goal is to model how changes in the predictors are associated with changes in the outcome, and to quantify the strength and form of those relationships.

The simplest form is simple linear regression, involving one predictor and assuming a linear relationship. More generally, multiple regression handles several predictors. Beyond linear models, regression approaches include logistic regression for binary outcomes, Poisson regression for count data, and survival models for time-to-event data. Nonlinear and nonparametric extensions, such as generalized additive models, broaden the range of possible relationships.

Estimation typically relies on methods like ordinary least squares (OLS) for linear models or maximum likelihood for generalized linear models. Key assumptions for linear regression include linearity, independence of observations, homoscedasticity (constant variance of errors), and normally distributed errors. Violations can be assessed with diagnostics and may require transformations, robust methods, or alternative models. Multicollinearity, outliers, and missing data are common challenges that affect interpretation and precision.

Model performance is usually assessed with metrics such as R-squared and adjusted R-squared, F-statistics, and p-values for terms. Prediction accuracy is evaluated with measures like root mean squared error (RMSE) or mean absolute error (MAE), often using cross-validation to guard against overfitting. Coefficients are interpreted as the expected change in the dependent variable per unit change in a predictor, holding other variables constant; causality cannot be inferred from regression alone without additional assumptions or experimental design.

Applications of regression analyses span economics, engineering, psychology, epidemiology, and many other fields. Extensions include regularization techniques (ridge, Lasso, elastic net) to address overfitting and variable selection, as well as more complex models for longitudinal or hierarchical data.