I cannot speak to the behavior of MINITAB but I can explain the difference between the model parameters when the factors are coded or not.
We should at least center our variables when using regression or another linear model technique because it helps reduce the correlation of the estimates. It also provides a meaningful interpretation the intercept: it is the mean response at the origin of the data. Without centering, the intercept often has no real physical meaning. Finally, the scaling aspect means that all the estimates represent half of the change in the response over the entire range of the factor. This way, the estimates a scale invariant and they can be compared, if we are exercising the widest possible range for each factor. (For example, which factors produce the largest effect?) That is not possible without coding. I can make the estimates arbitrarily small or large by a change of scale. (For example, change the units of a dimension from meters to light years and you will see what I mean.)
You can remove the coding after you fit the model. This way you can see the unit change in the original scale. Save it as a column formula, open the formula editor, click the red triangle near the keys at the top of the editor and select Simplify. The centering and scaling in the formula involves only constants so they can be converted without loss.
NOTE: it is very important that you maintain model hierarchy if you intend to convert between the coded scale and the original scale.
Using the original factors and the coded factors give identical model predictions, but we usually expect more of our models.
I attached a script for a demonstration of coded levels from our introductory DOE course. You can play with it and the side panels provide the formulas to convert between the two forms of levels.
I am sorry. I did not answer the part about why it is only possible for hierarchical models.
Actually, it is possible to convert non-hierarchical linear models but it leads to undesirable changes in that case. Terms in the fitted model might not appear in the simplified model. Terms that did not appear in the fitted model might appear in the simplified model. The results are correct, but unsettling. Most analysts do not like this result.
I co-authored an article in an old issue of "The JMPer Cable" about his behavior. It starts on page 9 of Issue 24, Spring 2008.
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