Help with Linear Mixed Models

[Corrado Corradi-Dell'Acqua]

Dear SPSSers

I am currently trying to re-analyze with SPSS 11.5 data from what was
originally a 2x3 factorial design experiment (both are within-subject
factors employed to 31 subjects). In particular, I am trying to assess
whether an additional factor may affect the original two factors. This
additional factor is subject's "response" during the experiment (either
"yes" or "no"). The problem is that in more than one condition many
subjects responded 100% in the same way, thus leaving empty almost 40%
of the 372 cells of what is now a 2x2x3 design with 31 subjects. Of
course this is a problem with Repetitive Measures GLM in that SPSS
applies a subject-wise deletion of those subjects exhibiting empty cells.

I have been suggested to use Linear Mixed Models, which are held to be
more robust against missing cells. However the tutorials I went through
suggest different approaches: in particular some tutorials suggest to
describe "subjects" as a random factor (this is similar to what is
suggested in other statistical packages such as R), thus carrying out a
model like this:

MIXED
  amp  BY response factor1 factor2 subjects
  /CRITERIA = CIN(95) MXITER(100) MXSTEP(5) SCORING(1)
SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE)
   PCONVERGE(0.000001, ABSOLUTE)
  /FIXED = response factor1 factor2 factor1*response factor2*response
factor1*factor2 factor1*factor2*response  | SSTYPE(3)
  /METHOD = REML
  /RANDOM subjects  | COVTYPE(AR1) .

Other tutorials instead suggest to specify the three factors of the
2x2x3 design as repeated measures:

MIXED
  amp  BY response factor1 factor2
  /CRITERIA = CIN(95) MXITER(100) MXSTEP(5) SCORING(1)
SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE)
   PCONVERGE(0.000001, ABSOLUTE)
  /FIXED = response factor1 factor2 factor1*response factor2*response
factor1*factor2 factor1*factor2*response  | SSTYPE(3)
  /METHOD = REML
  /REPEATED = response*factor1*factor2 | SUBJECT(subjects) COVTYPE(AR1) .

Even though this latter models leads to higher F-rations, both models
show an nice significant interaction between "response" and factor2, but
none show a significant three-way interaction which one would predict
through visual inspection of the plotted data. I am wondering whether
either these models are the correct ones to test what I am planning to
test, or whether my dataset needs to be chewed in another way. In
particular, my design seem pretty heavy for a Mixed Model, in that
attempting to employ a more complex covariance structure leads to an
error message about the Hessian Matrix being not positive definite (no
matter how much I  increase the number of iterations and the step-halvings).

Thanking you in advance for any suggestion you might provide.

Sincerely

Corrado

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