The least squares criterion is a fundamental principle in regression analysis, both simple and multiple. It’s the method used to determine the “best-fit” line (or hyperplane in multiple regression) that represents the relationship between the independent and dependent variables.

Least Squares Criterion Explained:

• Goal: The goal of regression analysis is to create a model that accurately predicts the dependent variable (Y) based on the independent variable(s) (X).
• Residuals: The difference between the actual observed value of Y and the predicted value of Y from the regression model is called a residual (or error).
• Least Squares: The least squares criterion aims to minimize the sum of the squared residuals. In other words, it finds the line (or hyperplane) that minimizes the total squared difference between the observed and predicted values.
• Why Squared Residuals?
  • Squaring the residuals ensures that both positive and negative differences contribute to the overall error.
  • It also gives greater weight to larger errors, which is often desirable.
  • It creates a smooth, differentiable function that is easier to minimize mathematically.

Simple vs. Multiple Regression:

• Simple Regression:
  • In simple linear regression, there is only one independent variable.
  • The least squares criterion is used to find the equation of a straight line (Y = β₀ + β₁X) that best fits the data points.
• Multiple Regression:
  • In multiple linear regression, there are two or more independent variables.
  • The least squares criterion is used to find the equation of a hyperplane (Y = β₀ + β₁X₁ + β₂X₂ + … + βₚXₚ) that best fits the data points in a multi-dimensional space.
• The Criterion Stays the Same:
  • The core principle of the least squares criterion remains the same in both simple and multiple regression.
  • The objective is always to minimize the sum of the squared residuals.
  • The only difference is the complexity of the equation that is being solved. In simple regression the equation describes a line, and in multiple regression it describes a plane, or hyper plane.
• Calculations:
  • The mathematical calculations involved in finding the coefficients (β₀, β₁, β₂, etc.) are more complex in multiple regression due to the increased number of variables.
  • However, statistical software packages handle these calculations efficiently.

In essence:

The least squares criterion provides a consistent and objective method for fitting regression models to data, regardless of the number of independent variables. It’s the bedrock of how regression analysis finds the model that best represents the relationships between variables.