Methods and formulas for stepwise in Fit Regression Model and Linear Regression

Performs variable selection by adding or deleting predictors from the existing model based on the F-test. Stepwise is a combination of forward selection and backward elimination procedures. Stepwise selection does not proceed if the initial model uses all of the degrees of freedom.

Variables to remove

Minitab calculates an F-statistic and p-value for each variable in the model. If the model contains j variables, then F for any variable, x_r , is this formula:

Notation

If the p-value for any variable is greater than the value specified in Alpha to remove, then Minitab removes the variable with the largest p-value from the model, calculates the regression equation, displays the results, and initiates the next step.

Variables to add

If Minitab cannot remove a variable, the procedure attempts to add a variable. Minitab calculates an F-statistic and p-value for each variable that is not in the model. If the model contains j variables, then F for any variable, x_a, is this formula:

Notation

Term	Description
SSE j	SS Error before xa is added to the model
SSE(j + Xa )	SS Error after xa is added to the model
	Degrees of freedom for variable Xa
MSE(j + Xa )	MS Error after xa is added to the model

If the p-value corresponding to the F-statistic for any variable is smaller than the value specified in Alpha to enter, Minitab adds the variable with the smallest p-value to the model, calculates the regression equation, displays the results, then goes to a new step. When no more variables can be entered into or removed from the model, the stepwise procedure ends.

A method for determining which terms to retain in a model. Forward selection adds variables to the model using the same method as the stepwise procedure. Once added, a variable is never removed. The default forward selection procedure ends when none of the candidate variables have a p-value smaller than the value specified in Alpha to enter.

A method for determining which variables to retain in a model. Backward elimination starts with the model that contains all the terms and then removes terms, one at a time, using the same method as the stepwise procedure. No variable can re-enter the model. The default backward elimination procedure ends when none of the variables included in the model have a p-value greater than the value specified in Alpha to remove. Backward elimination does not proceed if the initial model uses all of the degrees of freedom.

The forward selection with validation procedure depends on the validation method.

Test data set

When you use a test data set, the procedure is similar to forward selection. At each step, Minitab adds the term with the smallest p-value to the model. At the end of each step, Minitab calculates the test R² value. At the end of the forward selection procedure, the model with the greatest test R² value is the final model.

K-fold cross-validation

Once the forward selection procedures are complete for each fold, Minitab calculates the overall k-fold stepwise R² values for each step that is in the selection procedure for every fold. The step with the maximum k-fold stepwise R² value becomes the step for the chosen model from a final forward selection procedure.

Last, Minitab performs forward selection on the full dataset. Minitab displays regression results for the model at the step with the maximum overall k-fold stepwise R² value from the k-fold stepwise procedures. The table of model selection details and the graph of the k-fold stepwise R² versus model selection step continue for 8 steps past the step for the regression results.