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Why does R² exceed 1 or fall below 0? The math behind it!
In statistics, R-squared (R²) is a widely used metric to assess the predictive power of a regression model. The basic concept of R² is to explain how effectively the variable explains the variation of the dependent variable. However, it is confusing that we often see cases where R² exceeds 1 or falls below 0, so it is necessary to dig deeper into the math behind this.
R² is a measure of the goodness of fit of a model and should ideally be between 0 and 1. When this metric does not fit within this range, it usually indicates that there is a problem with the model.
Basic explanation of R²
By definition, R² is the proportion of variation that is explained. When the model fit is very good, R² is close to 1, indicating that the model is able to predict the results of the strain number very well. If the R² is 0, it means that the model cannot explain the variation and the prediction performance is the same as the mean.
Why is R² less than 0
In certain cases, R² may be less than 0. This usually happens when the predicted results are worse than the observed results. This can occur, for example, when the model you use does not fit the data correctly or does not include an intercept term. At this time, R² is negative, which means that the prediction results of the model fitting are not as good as those using the average value of the data.
When R² is less than 0, this indicates that the chosen model may be inappropriate, or even that a simpler prediction from the model, such as using the mean, may be more predictive.
Why R² Exceeds 1
It is somewhat rare for R² to exceed 1, but it can occur in some applications of the model. This is mainly related to the chosen fitting method and the complexity of the model. For example, when incorrect calculations are used or restrictions are applied inappropriately, the R² of a model may turn out to be outside the expected range. This is often the result of choosing the wrong mathematical model or making incorrect assumptions.
Incremental Effect and Model Overfitting
R² tends not to decrease as more variables are included in the model, making it likely that many models will overfit. This is why R² may appear to improve when adding variables, but may not actually increase the actual predictive power. To avoid this phenomenon, it is ideal to use the adjusted R², which adjusts for the number of variables in the model, making the estimate more rigorous.
The adjusted R² takes into account the number of variables and therefore better reflects the true predictive power of the model as variables are added in the future.
How to properly interpret R²
R² can be used to compare the performance of different models, however, relying solely on this single metric to make decisions is not enough. The context of different models, the nature of the data, and other statistical tests should all be considered comprehensively. For example, even if the R² value is high, we still need to correct for possible errors in model assumptions to avoid misleading conclusions.
ConclusionR² is a very valuable tool in model building, but its value must be interpreted with caution. In some cases, this indicator may be outside the normal range, so further consideration is needed on the underlying reasons and data characteristics. How can we correctly use and understand these statistical indicators to build a more accurate model?
