How to check if the regression model fits the data well?

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There are a couple of metrics that you can use:
R-squared/Adjusted R-squared: Relative measure of fit. This was explained in a previous answer
F1 Score: Evaluates the null hypothesis that all regression coefficients are equal to zero vs the alternative hypothesis that at least one doesn’t equal zero
RMSE: Absolute measure of fit.

To assess whether a regression model fits the data well, you can consider several evaluation techniques. Here are some commonly used methods:

  1. Residual Analysis: Check the residuals (the differences between observed and predicted values) for randomness and independence. A good model should have residuals that are normally distributed around zero with constant variance.
  2. Coefficient of Determination (R-squared): R-squared measures the proportion of the variance in the dependent variable that is predictable from the independent variables. A higher R-squared value (closer to 1) indicates a better fit, although it’s important to consider other metrics alongside it.
  3. Adjusted R-squared: This is similar to R-squared but adjusts for the number of predictors in the model. It penalizes excessive use of predictors that do not improve the model fit significantly.
  4. F-test: This test evaluates the overall significance of the regression model. It assesses whether the explained variance by the model is significantly higher than the unexplained variance.
  5. Mean Squared Error (MSE) or Root Mean Squared Error (RMSE): These metrics measure the average squared difference between the observed and predicted values. Lower values indicate better fit.
  6. Cross-validation: Use techniques like k-fold cross-validation to assess how the model performs on unseen data. This helps to detect overfitting and provides a more reliable estimate of the model’s performance.
  7. Visualization: Plotting the observed vs. predicted values or residuals can provide insights into how well the model fits the data. Additionally, visualizing the relationship between independent and dependent variables can help identify nonlinearities or heteroscedasticity.

It’s important to use a combination of these techniques to comprehensively evaluate the model’s fit and avoid relying solely on one metric. Different methods may provide complementary insights into the model’s performance.