Why do different regression methods provide different values for R-squared, adjusted R-squared, and S for the same model?

You can get different results for the same model if your data set contains missing values for any predictors.

When you perform Stat > Regression > Regression > Fit Regression Model > Stepwise or Stat > Regression > Regression > Best Subsets, Minitab removes all rows that contain missing values for any predictors that are in the list of predictors. Minitab removes the rows whether or not the predictors are in the model. If you change the lists of predictors, the results can change because of the missing values even though the model is the same.

For example, suppose the data set has the response in C1, the predictors in C2-C4, and one missing value in C4. You perform an analysis and list all of the predictors. Then the row with the missing value is not used to calculate the statistics, even for the model that contains only C2 and C3 as predictors. However, if you redo the analysis and list only C2 and C3 as predictors, the entire data set is used to calculate the statistics. Therefore, R-squared, adjusted R-squared, and S will differ for the same model.