Definition:
The coefficient of determination, usually 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.

Explanation:
The coefficient of determination is a value between 0 and 1, typically expressed as a percentage, that indicates the goodness of fit of a regression model. It measures the proportion of the total variation in the dependent variable that is predictable from the independent variable(s) included in the model.

Interpretation:
An R² value of 0 suggests that the dependent variable cannot be predicted at all using the independent variable(s), whereas an R² value of 1 indicates that the dependent variable can be perfectly predicted from the independent variable(s). Therefore, higher values of R² indicate a better fit of the regression model.

Formula:
The coefficient of determination is calculated as the square of the correlation between the actual values of the dependent variable and the predicted values by the regression model. Mathematically, it can be expressed as:

R² = (Explained variation) / (Total variation)

or

R² = 1 – (Residual variation) / (Total variation)

where:
– Explained variation refers to the sum of the squared differences between the predicted values and the mean of the dependent variable.
– Total variation is the sum of the squared differences between the actual values and the mean of the dependent variable.
– Residual variation represents the sum of the squared differences between the actual values and the predicted values by the regression model.

Limitations:
While the coefficient of determination provides a useful measure of the explanatory power of a regression model, it has certain limitations. R² does not determine whether the independent variable(s) have any causal relationship with the dependent variable and cannot differentiate between correlation and causation. Additionally, R² may increase when additional irrelevant independent variables are added to the model, leading to potential overfitting.

Overall, the coefficient of determination is a valuable tool to assess the reliability and adequacy of a regression model when interpreting the relationship between variables.