Introduction

In the realm of regression analysis, selecting the right model is a crucial decision that directly impacts the accuracy and reliability of our predictions. The process of model selection involves choosing a subset of predictor variables that best explains the relationship with the response variable. This chapter explores two prominent model selection procedures - Stepwise Selection and All-Subset Selection - and how they relate to the essential concept of the bias-variance trade-off.

Model Selection Procedures

1) Stepwise Selection
Stepwise selection is an iterative method that sequentially adds or removes predictor variables from the model based on certain criteria. It can be forward (variables added one by one) or backward (variables removed one by one). The criteria for variable inclusion or exclusion often involve statistical significance, such as p-values or information criteria like AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). The goal is to find a balance between model complexity and goodness of fit.

For instance, let’s consider a real estate dataset aiming to predict house prices. Stepwise selection could start with a simple model and add variables that improve the model’s fit. Variables may be included if their p-values are below a certain threshold, thus indicating statistical significance.

2) All-Subset Selection
All-subset selection involves generating models for all possible combinations of predictor variables and evaluating their performance. This exhaustive method provides insights into various model configurations and identifies the optimal subset. However, as the number of predictors increases, the number of possible models grows exponentially, making it computationally intensive.

Continuing with the real estate example, all-subset selection would examine all possible combinations of variables (e.g., bedrooms, square footage, location) and evaluate which subset yields the best fit based on a chosen criterion.

Bias-Variance Trade-Off

Model selection procedures are closely tied to the bias-variance trade-off. A model with too few variables might underfit the data (high bias), failing to capture its complexities. On the other hand, a model with too many variables might overfit the data (high variance), performing well on training data but poorly on unseen data.

Both stepwise selection and all-subset selection influence this trade-off. Stepwise selection, driven by criteria like p-values, aims to strike a balance between explanatory power and simplicity. All-subset selection, by exploring various model complexities, showcases the trade-off directly by evaluating models of different sizes.