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)
colorado
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