Least Squares Regression

Definition

Least squares regression is a statistical method used to determine the best-fitting line or curve that describes the relationship between a dependent variable and one or more independent variables. It aims to minimize the sum of the squared differences between the observed and predicted values.

5 Must Know Facts For Your Next Test

1. Least squares regression is used to find the line of best fit that minimizes the sum of the squared differences between the observed and predicted values.
2. The method of least squares provides the estimates of the regression coefficients that result in the smallest possible residuals.
3. Least squares regression can be used for both simple linear regression (one independent variable) and multiple linear regression (multiple independent variables).
4. The regression line obtained through least squares regression represents the best linear unbiased estimate (BLUE) of the true relationship between the variables.
5. Least squares regression makes certain assumptions, such as linearity, normality, homoscedasticity, and independence of residuals, which must be checked for valid inference.

Review Questions

• Explain how the method of least squares is used to determine the best-fitting regression line.
  • The method of least squares seeks to find the regression coefficients that minimize the sum of the squared differences between the observed values and the predicted values. This is done by calculating the partial derivatives of the sum of squared residuals with respect to the regression coefficients and setting them equal to zero. The resulting system of equations is then solved to obtain the estimates of the regression coefficients that provide the line of best fit.
• Describe the key assumptions of least squares regression and explain why they are important for valid inference.
  • The key assumptions of least squares regression include linearity (the relationship between the dependent and independent variables is linear), normality (the residuals are normally distributed), homoscedasticity (the residuals have constant variance), and independence of residuals. These assumptions are important because they ensure that the regression coefficients are unbiased, efficient, and have valid standard errors and confidence intervals. Violations of these assumptions can lead to biased or inefficient estimates, and invalid statistical inferences.
• Analyze the role of the coefficient of determination (R-squared) in evaluating the goodness of fit of a least squares regression model.
  • The coefficient of determination, or R-squared, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model. R-squared values range from 0 to 1, with higher values indicating a better fit of the regression line to the data. R-squared can be used to assess the overall fit of the regression model and the strength of the relationship between the variables. However, it is important to consider other diagnostic measures, such as the significance of the regression coefficients and the assumptions of the model, to fully evaluate the validity and usefulness of the least squares regression analysis.

Related terms

Linear Regression:

A statistical method that models the linear relationship between a dependent variable and one or more independent variables.

Residuals:

The differences between the observed values and the predicted values in a regression model.

Coefficient of Determination (R-squared): A statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s).