Model Significance linear Regression

Question

I have a more general question. Do you know about methods to investigate model significance in a sense of testing whether $Y=X\beta+\varepsilon$ is valid, i.e $H_0:\beta=0$ -apart from the typical F-test (for gaussian $\varepsilon$)?

Maybe you can provide some theory and expertise. Thanks in advance.

Answer 1

• t-test: This is a method to test the significance of individual regression coefficients, i.e., H0: βi = 0 for a given i. It is based on the standard error of the coefficient estimate and the t-distribution. It can be used to test whether a certain predictor variable has a linear relationship with the response variable¹.
• F-test: This is a method to test the significance of the overall regression model, i.e., H0: β1 = β2 = ... = βp = 0, where p is the number of predictor variables. It is based on the ratio of the explained variance to the unexplained variance (also known as the mean square error) and the F-distribution. It can be used to test whether at least one predictor variable has a linear relationship with the response variable¹.
• R-squared: This is a measure of how well the regression model fits the data. It is calculated as the proportion of the total variance in the response variable that is explained by the regression model. It ranges from 0 to 1, where 0 means no fit and 1 means perfect fit. It can be used to compare different regression models or to assess the goodness of fit of a single model².
• Adjusted R-squared: This is a modified version of R-squared that takes into account the number of predictor variables in the model. It penalizes models that have too many variables that do not contribute to the explanation of the response variable. It can be used to select the best subset of predictor variables for a regression model².
• AIC and BIC: These are criteria that balance the fit and complexity of a regression model. They are based on the likelihood function and a penalty term that increases with the number of parameters in the model. They can be used to compare different regression models or to select the best subset of predictor variables for a regression model³.

(1) How to Test the Significance of a Regression Slope - Statology. https://www.statology.org/test-significance-regression-slope/.

(2) Simple Linear Regression | An Easy Introduction & Examples - Scribbr. https://www.scribbr.com/statistics/simple-linear-regression/.

(3) Linear regression - Wikipedia. https://en.wikipedia.org/wiki/Linear_regression.

(4) Everything you need to Know about Linear Regression! - Analytics Vidhya. https://www.analyticsvidhya.com/blog/2021/10/everything-you-need-to-know-about-linear-regression/.