1. What is the Coefficient of Determination?

The coefficient of determination, denoted R², is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

2. How Does the Calculator Work?

The calculator uses the R² formula:

Where:

• \( SS_{res} \) — Residual sum of squares (sum of squared differences between observed and predicted values)
• \( SS_{tot} \) — Total sum of squares (sum of squared differences between observed values and their mean)

Explanation: R² measures how well the regression predictions approximate the real data points, with values ranging from 0 to 1.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the goodness-of-fit of regression models, helping researchers understand how well their model explains the variability of the response data.

4. Using the Calculator

Tips: Enter the residual sum of squares (SS_res) and total sum of squares (SS_tot). SS_res must be between 0 and SS_tot, and SS_tot must be greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: What does a high R² value indicate?
A: A high R² value (close to 1) indicates that the model explains a large portion of the variance in the dependent variable.

Q2: Can R² be negative?
A: In ordinary least squares regression, R² ranges from 0 to 1. Negative values typically indicate that the model fits worse than a horizontal line.

Q3: What are the limitations of R²?
A: R² always increases as more predictors are added to the model, which can lead to overfitting. It doesn't indicate whether the regression coefficients are statistically significant.

Q4: How is R² different from adjusted R²?
A: Adjusted R² accounts for the number of predictors in the model, penalizing excessive use of variables, while R² does not.

Q5: What is a good R² value?
A: This depends on the field of study. In social sciences, R² of 0.2 might be considered good, while in physical sciences, values above 0.8 are often expected.