## Coefficient of Determination

The coefficient of determination, denoted as R², is a statistical measure that represents the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression model. In other words, it indicates how well the independent variable(s) can predict the dependent variable’s values.

R² ranges from 0 to 1, with higher values indicating a better fit of the regression model. An R² value of 1 implies that the independent variable(s) perfectly predict the dependent variable’s values, while an R² value of 0 means that the independent variable(s) do not explain any of the variation in the dependent variable.

In the context of a simple linear regression with only one independent variable, R² is equal to the square of the correlation coefficient (r²). This relationship exists because the correlation coefficient measures the strength and direction of the linear relationship between the two variables, and squaring it eliminates the direction (positive or negative) and highlights the strength of the relationship.

However, in multiple regression models with more than one independent variable, R² is calculated differently, taking into account the combined effect of all independent variables on the dependent variable. In such cases, R² is typically calculated using the sum of squares (SS) method.

It’s important to note that while a high R² value may indicate a good fit, it does not guarantee that the model is appropriate for the data or that it will make accurate predictions. Additionally, R² tends to increase as more independent variables are added to the model, even if they are not significantly related to the dependent variable, which can lead to overfitting. To address this issue, researchers often use adjusted R², which takes into account the number of independent variables and the sample size, providing a more accurate measure of the model’s explanatory power.

## Example of the Coefficient of Determination

Let’s consider a simple linear regression example using the same dataset from the previous example, where we want to investigate the relationship between the number of hours studied (X) and exam scores (Y) for ten students.

We already calculated the correlation coefficient (r) in the previous example, which was approximately 0.976.

To find the coefficient of determination (R²) for this simple linear regression, we just need to square the correlation coefficient:

R² = r² = (0.976)² ≈ 0.953

The R² value is approximately 0.953, which means that approximately 95.3% of the variation in exam scores (Y) can be explained by the number of hours studied (X). This high R² value suggests that our linear regression model is a good fit for this data.

Keep in mind that this example uses a simple linear regression model with only one independent variable. If we had multiple independent variables, we would need to calculate R² differently, using the sum of squares (SS) method or a statistical software package.

It’s also important to remember that a high R² value does not guarantee the model’s validity or its ability to make accurate predictions. It merely indicates the proportion of the variance in the dependent variable that can be explained by the independent variable(s). Always consider other factors, such as the appropriateness of the model, the assumptions of the regression analysis, and the possibility of omitted variables or other issues that may affect the model’s validity.