# [R] a kinder view of Type III SS

From: Bernard Leemon <bernie.leemon_at_gmail.com>
Date: Thu, 7 Feb 2008 14:42:20 -0700

A young colleague (Matthew Keller) who is an ardent fan of R is teaching me much about R and discussions surrounding its use. He recently showed me some of the sometimes heated discussions about Type I and Type III errors that have taken place over the years on this listserve. I'm presumptive enough to believe I might add a little clarity. I write this from the perspective of someone old enough to have been grateful that the stat programmers (sometimes me coding in Fortran) thought to provide me with model tests I had not asked for when I carried heavy boxes of punched cards across campus to the card reader window only to be told to come back a day or two later for my output. I'm also modern enough to know that anova(model1, model2), where model2 is a proper subset of model1, is all that I need and allows me to ask any question of my data that I want to ask rather than being constrained to those questions that the SAS or SPSS programmer thought I might want to ask. I could end there, and we would probably all agree with what I have said to this point, but I want to push the issue a bit and say: it seems that Type III Sums of Squares are being unfairly maligned among the R cognoscenti. And the practical ramification of this is that it creates a good deal of confusion among those migrating from SAS/SPSS land into R - not that this should ever be a reason to introduce a flawed technique into R, but my argument is that type III sums of squares are not a flawed technique.

In my reading of the prior discussions on this list, my conclusion is that the Type I/Type III issue is a red herring that has generated unnecessary heat. Base R readily provides both types. summary(lm( y ~ x + w + z)) provides estimates and tests consistent with Type III sums of squares (it doesn't provide the SS directly but they are easily derived from the output) and anova(lm(y ~ x + w + z)) provides tests consistent with Type I sums of squares. The names Type I and III are dreadful "gifts" from SAS and others.  I'd prefer "conditional tests" for those provided by summary() because what is estimated and tested are x|w,z w|x,z and z|x,w [read these as "x conditional on w and z being in the model"] and "sequential" for those provided by anova(), being x, w|x, and z|x,w. None of these tests is more or less valid or useful than any of the others. It depends on which questions researchers want to ask of their data.

Things get more interesting when z represents the interaction between x and w, such that z = x * w = xw. Fundamentally everything is the same in terms of the above tests. However, one must be careful to understand what the coefficient and test for x|w,xw and w|x,xw mean. That is, x|w,xw tests the relationship between x and y when and only when w = 0. A very, very common mistake, due to an overgeneralization of traditional anova models, is to refer to x|w,xw as the "main effect." In my list of ten statistical commandments I include: "Thou shalt never utter the phrase main effect"  because it causes so much unnecessary confusion. In this case, x|w,xw is the SIMPLE effect of x when w = 0. This means among other things that if instead we use w' = w - k so as to change the 0 point on the w' scale, we will get a different estimate and test for x|w',xw'. Many correctly argue that the main effect is largely meaningless in the presence of an interaction because it implies there is no common average effect. However, that does not invalidate x|w,xw because it is NOT a "main" (sense "principal" or "chief") effect but only a "simple" effect for a particular level of w. A useful strategy for testing a variety of simple effects is to subtract different constants k from w so as to change the 0 value to focus the test on particular simple effects.

If x and w are both contrast codes (-1 or 1) for the two factors of a 2 x 2 design, then x|w,xw is the simple effect of x when w = 0. While w never equals 0, in a balanced design w does equal 0 on average. In that one very special case, the simple effect of x when w = 0 equals the average of all the simple effects and in that one special case one might call it the "main effect." However, in all other situations it is only the simple effect when w = 0. If we discard the term "main effect", then a lot of unnecessary confusion goes away. Again, if one is interested in the simple effect of x for a particular level of w, then one might want to use, instead of a contrast code, a dummy code where the value of 0 is assigned to the level of w of interest and 1 to the other level.

When factors have multiple levels, it is best to have orthogonal contrast codes to provide 1-df tests of questions of interest. Products of those codes are easily interpreted as the simple difference for one contrast when the other contrast is fixed at some level. Multiple degree of freedom omnibus tests are troublesome but are only of interest if we are fixated on concepts like 'main effect.'

gary mcclelland (aka bernie leemon)