Regression Analysis Online: Simple and Multiple Linear Regression
What is Regression Analysis?
Regression analysis studies the quantitative relationship between one or more independent variables (X) and a continuous response (Y), fitting an equation Y = b0 + b1X1 + ... + bkXk by least squares to explain influences and predict future outcomes. Fit quality is measured by the coefficient of determination R-squared (the share of total variation explained by the model), while adjusted R-squared penalizes adding variables. Plot a scatter plot matrix of the variables first to inspect relationships and outliers.
Testing the Model and Coefficients
The overall model is tested with an F test for whether all independent variables jointly matter, and each coefficient with a t test for whether that variable has a significant linear relationship with Y after controlling for the others, together with a confidence interval for the coefficient. The tool outputs the regression equation, coefficients, standard errors, t values, p values and standardized coefficients, which let you compare the relative importance of variables on different scales and identify the strongest drivers.
Residual Analysis and Model Diagnostics
Regression assumes residuals that are independent, normal, of constant variance and unrelated to X. The tool outputs residual plots (residuals versus fitted values and versus each variable, plus a normal probability plot) and a Durbin-Watson autocorrelation test. A funnel-shaped residual pattern indicates non-constant variance, curvature indicates a missing nonlinear term or interaction, and a skewed normal plot suggests transforming Y. Investigate outliers with high leverage or strong influence and assess their sensitivity.
Collinearity Diagnostics and Usage Steps
When independent variables are highly correlated, coefficients become unstable and standard errors inflate. The tool computes the variance inflation factor (VIF); VIF above 10 indicates serious collinearity, so drop or merge variables or use ridge or principal component regression. The workflow: paste data, choose the response and predictors, inspect the equation and R-squared, check residual plots and VIF, then use the equation for prediction or explanation. An AI interpretation summarizes key conclusions and model limitations.