Newsgroups: bit.listserv.stat-l
Date:         Wed, 30 Sep 1992 10:40:14 EDT
From:         Tim Dorcey <TCD@CORNELLC.BITNET>
Subject:      Re: unusual ANOVA design

On Tue, 29 Sep 1992 16:19:48 -0400 <RJZ@UTXVM> said:
>I am trying to determine the appropriate F denominators for a particular
>factor in the following design:
>
>    Y = Constant + fixed + random + nested(fixed) + fixed by random +
>
>          random by nested(fixed)
>

     The appropriate denominator for testing the "fixed" effect depends
depends upon whether the "nested" effect is considered fixed or random.
If the nested effect is considered a fixed effect, then you can imagine
combining the "fixed" and "nested(fixed)" effects into a single main
effect that has as many levels as the total number of nested levels
(i.e., summed over the different levels of the fixed effect).  The "fixed"
and "nested(fixed)" effects are then simply different contrasts on this
effect, and the most natural denominator for the "fixed" effect is
"fixed by random" i.e., how much the "fixed" effect varies across
the random units.  In order for this test to be valid, you need to
assume that any pairwise difference among the levels of the "fixed"
effect has the same variance, so that e.g., the error term for f1-f2
is the same as the error term for f2-f3.  Of course, this will
always be true if the "fixed" effect has only 2 levels.
     Depending upon the experimental design, it may also be reasonable
to obtain additional error degrees of freedom by pooling "fixed by random"
and "random by nested(fixed)" together into a single error term.  This
would be justified if the contrasts involved in the "nested(fixed)" effect
can be assumed to have the same variance as the contrasts involved in the
"fixed" effect.  To answer this kind of question, a simple mental
experiment is often helpful.  Suppose that "fixed" and "nested(fixed)"
have absolutely no true effect, and you randomly select 2 observations
(from within the same "random" unit).  If these 2 observations were from
the same level of the "fixed" effect would you expected them to be more
similar than if they had different values for the "fixed" effect?
Remember, "fixed" has no true effect; we are asking if there is anything
about the way the experiment was conducted which would tend to make
observations nested within a level of "fixed" effect more similar than
others, even if "fixed" itself has no effect.  If so, then the error terms
should not be pooled; comparisons within a level of "fixed" effect should
have a different error term ("random by nested(fixed)"), than comparisons
involving different levels of "fixed" effect.
      As an aside, I have found this type of mental experiment
especially useful in designs like the one under consideration, except
where "nested" is crossed with "fixed".  I.e., calling the fixed effects
A & B, and the random effect R:
 y = c + A + B + A*B + R*A + R*B + R*A*B
The simple (and more conservative approach) is to simply test each fixed
effect against its interaction with the random effect.  However, it is
often the case that a mental experiment like the one above will suggest
that, while observations under the same level of A are more similar than
those which cross a level of A, observations within a single level of B
are really no more alike than observations which simultaneously differ
on both A & B.  Thus, we are lead to the conclusion that B & A*B should
be tested against the same error term:  R*B + R*A*B = R*B(A), and the
model might be rewritten:
 y = c + A + B + A*B + R*A + R*B(A)
or simply:
 y = c + A + B + A*B + R*A + e
I prefer this approach to memorizing a bunch of weird terminology
(e.g., split plot, split-split plot, randomized complete block, repeated
measures, etc.) that varies from field to field.
    Anyway, to return to the original question, all of the preceding is
based on the assumption that "nested" is fixed.  If "nested" is random,
then observed differences among levels of the "fixed" effect reflect
random variation among both the levels of "nested" and of "random".  An
error term with the correct E(MS) can be obtained by an appropriate
pooling of "fixed by random", "nested(fixed)" and "random by nested(fixed)",
and the Satterthwaite degrees of freedom formula can be employed to
get an approximate p-value for the test.  PROC GLM in version 5 of SAS will
produce a table of E(MS) that can be used to derive an appropriate pooling;
the same procedure in version 6 will construct the "quazi F-ratios"
and calculate the Satterthwaite approximation.  An alternative approach
would be to use maximum likelihood (or REML) estimation as provided
by PROC MIXED in version 6.07 of SAS, or BMDP 3V.  I don't know anything
about the relative merits of these approaches.  The few times I have
tried them, I have seen very similar results between REML estimation
and the E(MS) approach.

Tim Dorcey                        BITNET:   TCD@CORNELLC
Statistical Software Consultant   Internet: TCD@CORNELLC.CIT.CORNELL.EDU
Cornell Information Technologies  Phone:    (607) 255-5715
Cornell University
Ithaca, NY  14853