R-Square

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Definition 

The R² value, also known as the coefficient of determination, is a statistical measure used in the context of predictive models or regressions. It represents the proportion of the variance in the dependent variable that is predictable from the independent variables. 

1. Range of R²: The R² value ranges from 0 to 1.

    An R² of 0 indicates that the model explains none of the variability of the response data around its mean.

    An R² of 1 indicates that the model explains all the variability of the response data around its mean.

 2. Interpretation:

    A higher R² value generally indicates a better fit for the model. It means that the model can better predict the dependent variable using the independent variable(s).

    However, a higher R² doesn’t always mean a better model. It doesn’t account for whether the model is overfitting the data or if the model is actually useful for prediction.

 3. Calculation: 

In the context of linear regression, R² is calculated as the square of the correlation coefficient between the observed and predicted values of the dependent variable.

4. Limitations:

    R² alone doesn’t tell you if you’ve chosen the right model or if you’re overfitting.

    It doesn’t indicate if a regression model is adequate. You can have a low R² value for a good model, or a high R² value for a model that does not fit the data at all.

    In models with a large number of predictors, R² can be artificially high, so it’s important to look at other statistics as well.

5. Adjusted R²: 

In multiple regression scenarios, the adjusted R² is often used as it adjusts for the number of predictors in the model. It's a more robust measure as it penalizes for adding predictors that do not improve the model.

 In summary, the R² value is a useful statistic in regression analysis to give a basic idea of the goodness of fit, but it should be interpreted with caution and in conjunction with other statistical measures.

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