What does a higher coefficient of determination (R^2) indicate in a regression model?

A higher coefficient of determination, denoted as R^2, indicates that a larger proportion of the variance in the dependent variable can be explained by the independent variable(s) in the regression model. Essentially, R^2 provides a measure of how well the model fits the data, with values ranging from 0 to 1.

When R^2 approaches 1, it suggests that the model is capturing a significant amount of the variability in the response data, meaning the independent variables in the model have a strong relationship with the dependent variable. This is crucial in regression analysis because it helps determine the effectiveness and predictive power of the model. A high R^2 value thereby infers that the model has a robust explanatory capability regarding the behavior of the dependent variable.

In contrast, a low R^2 value implies that the model fails to capture much of the variance, which can lead to less reliable conclusions. Thus, the assertion that a higher R^2 indicates more variance explained aligns with the primary purpose of regression analysis, which is to evaluate and understand the relationship between variables.