This calculator finds the coefficient of determination \( R^2 \) for a simple linear regression model, showing how much of the variation in the response variable can be explained by the predictor variable.
Coefficient of Determination (R2):
Interpretation:
Understanding the Coefficient of Determination (R²)
What is the Coefficient of Determination?
The coefficient of determination, commonly denoted as \( R^2 \), is a statistical metric used in regression analysis to assess the goodness of fit of a model. It quantifies the proportion of the variance in the response variable \( y \) that can be explained by the predictor variable \( x \) in a linear regression model.
The formula for \( R^2 \) is:
\( R^2 = 1 - \frac{SSE}{SST} \)
Where:
- SSE is the Residual Sum of Squares, representing the sum of squared differences between the observed and predicted values.
- SST is the Total Sum of Squares, representing the total variance in the observed data.
An \( R^2 \) value close to 1 indicates a strong model fit, while an \( R^2 \) near 0 suggests that the model does not explain the variability in \( y \) well.
How to Interpret \( R^2 \)
Interpreting the coefficient of determination helps us understand the strength of the relationship between the predictor and response variables:
- High \( R^2 \) (close to 1): A high \( R^2 \) indicates that a large proportion of the variability in the response variable is explained by the predictor variable, suggesting a strong fit.
- Moderate \( R^2 \): A moderate \( R^2 \) (typically between 0.5 and 0.7) suggests that the predictor explains some of the variability in the response variable, but other factors may also influence the response.
- Low \( R^2 \) (close to 0): A low \( R^2 \) means that the predictor variable explains very little of the variability in the response variable, suggesting a weak fit.
In practical terms, if \( R^2 = 0.84 \), you could interpret it as follows: "84% of the variation in the response variable can be explained by the predictor variable." This implies a strong relationship but does not imply causation.
Limitations of \( R^2 \)
While \( R^2 \) is a valuable metric, it has limitations:
- Not Always a Measure of Predictive Power: A high \( R^2 \) does not necessarily mean that the model has strong predictive capability, particularly if overfitting occurs.
- Only Measures Linear Relationships: \( R^2 \) is best suited for linear models; it may not provide an accurate measure of fit for non-linear relationships.
- Does Not Indicate Causation: A high \( R^2 \) shows association, not causation. External factors may influence the observed relationship.