1. What is the Coefficient Of Determination?

The Coefficient of Determination (R²) is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It provides an indication of the goodness of fit of a 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 ranges from 0 to 1, where 0 indicates that the model explains none of the variability of the response data around its mean, and 1 indicates that the model explains all the variability.

3. Importance of R² Calculation

Details: R² is crucial for evaluating the performance of regression models, comparing different models, and understanding how well the independent variables explain the variability in the dependent variable.

4. Using the Calculator

Tips: Enter the residual sum of squares (SS_res) and total sum of squares (SS_tot) values. Both values must be positive, and SS_res must be less than or equal to SS_tot.

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 may occur in other contexts but indicate that the model performs worse than simply using the mean.

Q3: What are the limitations of R²?
A: R² can be misleading with nonlinear relationships, doesn't indicate causality, and can be artificially inflated by adding more variables.

Q4: How is R² different from adjusted R²?
A: Adjusted R² accounts for the number of predictors in the model and penalizes excessive variables, providing a more accurate measure for multiple regression.

Q5: When should I use R²?
A: Use R² to evaluate how well your regression model fits the data, but always consider it alongside other metrics and domain knowledge.