In a multiregression model, the dependent variable is assumed to be linearly related to the independent variables, though the relationship can also be adjusted for nonlinear transformations if necessary. The general form of a linear multiregression equation is:
where Y represents the dependent variable, X₁, X₂, ..., Xₖ are the independent variables, β₀ is the intercept, β₁, β₂, ..., βₖ are the coefficients representing the effect of each predictor, and ε is the error term capturing unmodeled variability.
Key components of multiregression include the coefficients, which quantify the change in the dependent variable associated with a one-unit change in the corresponding independent variable while holding other variables constant. The model also provides measures such as R-squared (R²), which indicates the proportion of variance in the dependent variable explained by the independent variables. Adjusted R-squared accounts for the number of predictors in the model, offering a more accurate measure of fit, especially when comparing models with different numbers of variables.
Multiregression relies on assumptions such as linearity, independence of errors, homoscedasticity (constant variance of errors), and normally distributed errors. Violations of these assumptions can affect the validity of the results, necessitating techniques like transformations, robust standard errors, or alternative models. Techniques such as stepwise regression, ridge regression, or Bayesian approaches are often employed to address issues like multicollinearity or overfitting.
The primary advantages of multiregression include its ability to control for confounding variables, enhance predictive accuracy, and provide insights into the relative importance of different predictors. However, it requires careful consideration of variable selection, potential interactions, and model diagnostics to ensure reliable and interpretable results.