backward selection

1. Begin with the full model:

\begin{equation} M_p: y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_p x_p. \end{equation}

1. In step 1, fit all models generated by removing one predictor from the current model, and select the model that exhibits the smallest loss of fit, indicated by the least increase in residual sum of squares (RSS).
2. At step \(k \leq p-1\), from the current model, evaluate the impact of removing each remaining predictor one at a time, and eliminate the one whose removal leads to the least deterioration in fit.
3. Update the model by excluding this predictor and repeat step 3.
4. Terminate the process when the removal of any remaining predictor significantly degrades the model or when a specified criterion (such as AIC, BIC, adjusted \(R^2\), or cross-validation error) is optimized.