 

R^{2} is the percentage of variation in the response that is explained by the model. It is calculated as 1 minus the ratio of the error sum of squares (which is the variation that is not explained by model) to the total sum of squares (which is the total variation in the model).
Use R^{2} to determine how well the model fits your data. The higher the R^{2} value, the better the model fits your data. R^{2} is always between 0% and 100%.
R^{2} always increases when you add additional predictors to a model. For example, the best fivepredictor model will always have an R^{2} that is at least as high the best fourpredictor model. Therefore, R^{2} is most useful when you compare models of the same size.
Small samples do not provide a precise estimate of the strength of the relationship between the response and predictors. If you need R^{2} to be more precise, you should use a larger sample (typically, 40 or more).
R^{2} is just one measure of how well the model fits the data. Even when a model has a high R^{2}, you should check the residual plots to verify that the model meets the model assumptions.
Adjusted R^{2} is the percentage of the variation in the response that is explained by the model, adjusted for the number of predictors in the model relative to the number of observations. Adjusted R^{2} is calculated as 1 minus the ratio of the mean square error (MSE) to the mean square total (MS Total).
Use adjusted R^{2} when you want to compare models that have different numbers of predictors. R^{2} always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R^{2} value incorporates the number of predictors in the model to help you choose the correct model.
Step  % Potato  Cooling rate  Cooking temp  R^{2}  Adjusted R^{2}  Pvalue 

1  X  52%  51%  0.000  
2  X  X  63%  62%  0.000  
3  X  X  X  65%  62%  0.000 
The first step yields a statistically significant regression model. The second step adds cooling rate to the model. Adjusted R^{2} increases, which indicates that cooling rate improves the model. The third step, which adds cooking temperature to the model, increases the R^{2} but not the adjusted R^{2}. These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.
Predicted R^{2} is calculated with a formula that is equivalent to systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. The value of predicted R^{2} ranges between 0% and 100%.
Use predicted R^{2} to determine how well your model predicts the response for new observations. Models that have larger predicted R^{2} values have better predictive ability.
A predicted R^{2} that is substantially less than R^{2} may indicate that the model is overfit. An overfit model occurs when you add terms for effects that are not important in the population, although they may appear important in the sample data. The model becomes tailored to the sample data and therefore, may not be useful for making predictions about the population.
Predicted R^{2} can also be more useful than adjusted R^{2} for comparing models because it is calculated with observations that are not included in the model calculation.
For example, an analyst at a financial consulting company develops a model to predict future market conditions. The model looks promising because it has an R^{2} of 87%. However, the predicted R^{2} is only to 52%, which indicates that the model may be overfit.
S represents the standard deviation of how far the data values fall from the fitted values. S is measured in the units of the response.
Use S to assess how well the model describes the response. S is measured in the units of the response variable and represents the standard deviation of how far the data values fall from the fitted values. The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.
For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled potato chips per container. You reduce the model to the significant predictors, and S is calculated as 1.79. This result indicates that the standard deviation of the data points around the fitted values is 1.79. If you are comparing models, values that are lower than 1.79 indicate a better fit, and higher values indicate a worse fit.
Mallows' Cp can help you choose between competing multiple regression models. Mallows' Cp compares the full model to models with the best subsets of predictors. It helps you strike an important balance with the number of predictors in the model. A model with too many predictors can be relatively imprecise while a model with too few predictors can produce biased estimates. Using Mallows' Cp to compare regression models is only valid when you start with the same complete set of predictors.
A Mallows' Cp value that is close to the number of predictors plus the constant indicates that the model produces relatively precise and unbiased estimates.
A Mallows' Cp value that is greater than the number of predictors plus the constant indicates that the model is biased and does not fit the data well.