🚀 go-pugleaf

    I have read some of the mail concerning Type I - Type IV sums of squares.
Let me add my two cents worth.
  If the experiment is balanced, then all four types of sums of squares will be
identical, so the difference comes when the experiment is unbalanced and/or
there are missing data.
  In unbalanced cases there are several ways to compute sums of squares, SAS
allows for four different ways of doing it.  If we all wanted to, we could
probably propose other ways to compute sums of squares as well.  One way to
do this is to throw some of the data away to make the experiment balanced,
and then compute sums of squares using balanced data methods.  Clearly, there
are all kinds of ways to throw away data, so there might be hundreds of ways
in which sums of squares could be computed.
  Now the important thing is not how they are computed, but (in terms of the
model parameters) what hypotheses are being tested by these different analyses?
If the experiment has no missing treatment combinations, then the Type III
analysis tests the same hypotheses about the model parameters that would have
been tested if the experiment were balanced.  Thus in almost all cases where
there are no missing treatment combinations, these are going to be the ones
which most experimenters are going to want to use.  One might be able to make
a case for the Type I and Type II analyses if one is in the process of model
building, particularly if there are no CLASS variables, and one is fitting
regression models.
  The Type IV sums of squares are equivalent to the Type III
sums of squares when all treatment combinations are observed at least once.
They differ from the Type III sums of squares when some treatment combinations
are missing.  What hypotheses do they test?  Before I answer this question, let
me note that there are hundreds of ways to compute what could be called Type IV
sums of squares, SAS has one algorithm that it uses.  If you change the values
of the CLASS variables, the Type IV analysis could change.  Thus the Type IV
analysis is not uniquely determined and SAS warns you of this in the printouts.
The hypotheses that are tested by the Type IV analysis can not in general be
predicted by the casual user.  If you use the E4 option on the model statement
in SAS-GLM, the Type IV estimable functions can be obtained, and from these
functions, one can actually determine the hypotheses being tested by the Type
IV analysis.  Rather than looking at the Type IV analysis and trying to figure
out what is being tested, I use ESTIMATE and CONTRAST options to obtain test
statistics for hypotheses that the experimenter is interested in.  I tell my
students that there are two ways to use SAS-GLM.  One is to look at the
answers (the output) and try to figure out what the questions are, or
generate your own questions, and use SAS-GLM to find out the answers.  I hope
everyone out there agrees that the latter choice makes the most sense.
   For more information on these, see Chapters 9-17 in the Analysis of Messy
Data book that I advertised earlier.
  This is about as simple as I can make it, I hope it is useful to someone.

  Dallas E. Johnson, Kansas State University