Model summary table for Fit Regression Model and Linear Regression

Interpretation

Use S to assess how well the model describes the response. S is measured in the units of the response variable and represents 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.

Interpretation

Use R² to determine how well the model fits your data. The higher the R² value, the better the model fits your data. R² is always between 0% and 100%.

You can use a fitted line plot to graphically illustrate different R² values. The first plot illustrates a simple regression model that explains 85.5% of the variation in the response. The second plot illustrates a model that explains 22.6% of the variation in the response. The more variation that is explained by the model, the closer the data points fall to the fitted regression line. Theoretically, if a model could explain 100% of the variation, the fitted values would always equal the observed values and all of the data points would fall on the fitted line. However, even if R² is 100%, the model does not necessarily predict new observations well.

Interpretation

Use adjusted R² when you want to compare models that have different numbers of predictors. R² always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R² value incorporates the number of predictors in the model to help you choose the correct model.

The first model yields an R² of more than 50%. The second model adds cooling rate to the model. Adjusted R² increases, which indicates that cooling rate improves the model. The third model, which adds cooking temperature, increases the R² but not the adjusted R². These results indicate that cooking temperature does not improve the model. Based on these results, you consider removing cooking temperature from the model.

Interpretation

Use PRESS to assess your model's predictive ability. Usually, the smaller the PRESS value, the better the model's predictive ability. Minitab uses PRESS to calculate the predicted R², which is usually more intuitive to interpret. Together, these statistics can prevent over-fitting the model. An over-fit 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.

Interpretation

Use predicted R² to determine how well your model predicts the response for new observations. Models that have larger predicted R² values have better predictive ability.

A predicted R² that is substantially less than R² may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the sample data and, therefore, may not be useful for making predictions about the population.

Predicted R² can also be more useful than adjusted R² 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 R² of 87%. However, the predicted R² is 52%, which indicates that the model may be over-fit.

Interpretation

Use AICc and BIC to compare different models. Smaller values are desirable. However, the model with the least value for a set of predictors does not necessarily fit the data well. Also use tests and residual plots to assess how well the model fits the data.

Both AICc and BIC assess the likelihood of the model and then apply a penalty for adding terms to the model. The penalty reduces the tendency to overfit the model to the sample data. This reduction can yield a model that performs better in general.

As a general guideline, when the number of parameters is small relative to the sample size, BIC has a larger penalty for the addition of each parameter than AICc. In these cases, the model that minimizes BIC tends to be smaller than the model that minimizes AICc.

In some common cases, such as screening designs, the number of parameters is usually large relative to the sample size. In these cases, the model that minimizes AICc tends to be smaller than the model that minimizes BIC. For example, for a 13-run definitive screening design, the model that minimizes AICc will tend to be smaller than the model that minimizes BIC among the set of models with 6 or more parameters.

For more information on AICc and BIC, see Burnham and Anderson.¹

Interpretation

Use test S to assess the performance of the model on new data. The lower the value of test S, the closer the predictions of the model are to the actual values in the test data set.

An S value that is substantially less than the test S value may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the sample data and, therefore, may not be useful for making predictions about the population.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled chips per container. You reduce the model to the significant predictors and find that S is 1.79, but test S is 17.63. Because the test S is very different from the S from the training set, you decide that the test S gives a better indication of how the model will perform for new data.

A low test S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

Interpretation

Use test R² to determine how well your model fits new data. Models that have larger test R² values tend to perform better on new data. You can use test R² to compare the performance of different models.

A test R² that is substantially less than R² may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the training data and, therefore, may not be useful for making predictions about the population.

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 R² of 87%. However, the test R² is 52%, which indicates that the model may be over-fit.

A high test R² value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

Interpretation

Use K-fold S to assess the performance of the model on new data. The lower the value of K-fold S, the closer the predictions of the model are to the actual values in the fold when the data in the fold are not part of the estimation of the model.

An S value that is substantially less than the K-fold S value may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the sample data and, therefore, may not be useful for making predictions about the population.

For example, you work for a potato chip company that examines the factors that affect the percentage of crumbled chips per container. You reduce the model to the significant predictors and find that S is 1.79, but K-fold S is 17.63. Because the K-fold S is very different from the S from the training set, you decide that the K-fold S gives a better indication of how the model will perform for new data.

A low K-fold S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

Interpretation

Use K-fold R² to determine how well your model fits new data. Models that have larger K-fold R² values tend to perform better on new data. You can use K-fold R² to compare the performance of different models.

A K-fold R² that is substantially less than R² may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the training data and, therefore, may not be useful for making predictions about the population.

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 R² of 87%. However, the K-fold R² is 52%, which indicates that the model may be over-fit.

A high K-fold R² value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.

Interpretation

Use k-fold stepwise R² to determine the number of terms in a model. Minitab calculates k-fold stepwise R-sq when you perform forward selection with validation with k-fold cross validation. K-fold stepwise R² results from separate forward selections for each fold. Minitab uses the k-fold stepwise R² to determine the best step in forward selection. Once forward selection is complete for each fold, Minitab performs forward selection on the full data set. With the full dataset, Minitab produces regression results for the model at the best step according to the k-fold stepwise R² criterion.

To evaluate the predictive performance of a model with k-fold cross validation, use the k-fold R² statistic instead.

Interpretation

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.