Grafen & Hails: Modern Statistics for the Life Sciences
10.04.08: Ross Farrelly, Macquarie University, Australia
Q: In Chapter 6, page 42 of the Minitab supplement to "Modern Statistics for the Life Sciences" the coefficients are as follows: Constant -0.013, BACBEF 0.8831, TREATMT 1 -1.590, TREATMT 2 -0.726. However in the corresponding SAS and SPSS support documents the coefficients are given as Intercept 2.303341923, BACBEF 0.883070237, TREATMT 1 -3.905961455, TREATMT 2 -3.041813293. Why the difference?
A: Ross has come across, and been understandably puzzled by, the concept of 'aliasing'. Don't worry, its not as bad as it sounds. The background and the term itself are introduced in Section 3.2, and the different behaviours of Minitab, SAS and SPSS are also explained.
Rather than repeat, I'll just refer the reader to that section of that book for an explanation. But here we can use Ross's example to confirm that the packages don't disagree.
Let's ignore BACBEF, as that difference between the packages is just down to rounding conventions. When Minitab gives coefficients for treatments 1 and 2, we need to construct the coefficient for the third treatment by making all three add up to zero. So we add the two coefficients, and change the sign. This gives us 2.316. So, the fitted intercepts for groups 1 to 3, using the appropriate formula (intercept = grand mean plus treatment coefficient), are
1: -0.013 - 1.590 = -1.603
2: -0.013 - 0.726 = -0.739
3: -0.013 + 2.316 = 2.303
With SAS and SPSS the rule for constructing the third coefficient is simpler: just set it to zero. So with those packages we get
1: 2.303 - 3.906 = -1.603
2: 2.303 - 3.042 = -0.739
3: 2.303 + 0 = 2.303
and notice that the fitted intercepts are exactly the same, even though the parameters are different, and it's all down to the question 'what is the rule for calculating the missing treatment coefficient?'.
Now, Ross might have meant 'why don't packages all agree on one method of aliasing, and keep life simple for us'? Users tend to become accustomed to one kind, and not want to shift. But often the user can choose the method of aliasing by setting preferences, so it is mainly in their default aliasing that packages differ. And a final thought: the differences between packages do bring the issue of aliasing to our attention, and it is, really, a useful thing to understand!