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We performed a randomized controlled trial. I am interested in anxiety levels (dependent), treatment (meds vs. ultra-dance therapy), and personality variables (agreeableness, neuroticism, weeaboo). We measured anxiety levels four times. I am interested in exploring whether participants in one treatment experienced greater reductions in anxiety, and whether the personality variables are associated with reductions in anxiety over time.

My understanding is that subjects are treated as random effects through the repeated measures aspect of the analysis. It seems fairly clear to me that treatment type is a fixed effect. Are the personality variables (scored 0-50 each lets say) best conceptualized as fixed, or random effects?

Here is the syntax I have from SPSS if I put the personality factors as fixed effects:

MIXED ANXIETY BY TREATMENT TIME WITH PERSONALITYXXXX PERSONALITYYYY
  /CRITERIA=CIN(95) MXITER(1000) MXSTEP(100) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, 
    ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE)
  /FIXED=TREATMENT TIME TREATMENT*TIME PERSONALITYXXXX0 PERSONALITYYYY| SSTYPE(3)
  /METHOD=REML
  /PRINT=DESCRIPTIVES  SOLUTION TESTCOV
  /RANDOM=INTERCEPT | SUBJECT(id) COVTYPE(UN)
  /REPEATED=TIME | SUBJECT(id) COVTYPE(DIAG).

Here is the syntax if I put personality factors as random effects

MIXED ANXIETY BY TREATMENT TIME WITH PERSONALITYXCXXX PERSONALITYYYY
  /CRITERIA=CIN(95) MXITER(1000) MXSTEP(100) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, 
    ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE)
  /FIXED=TREATMENT TIME TREATMENT*TIME | SSTYPE(3)
  /METHOD=REML
  /PRINT=DESCRIPTIVES  SOLUTION TESTCOV
  /RANDOM=INTERCEPT PERSONALITYXCXXX PERSONALITYYYY | COVTYPE(UN)
  /REPEATED=TIME | SUBJECT(id) COVTYPE(DIAG).

In SPSS I put "ID" as a subject variable and "TIME" as a repeated variable in the first window. Is this necessary/a good idea? I am concerned these variables may be treated as random more than once, if that is possible.

One issue with having personality variables as a random effect is the output gets quite wacky and I get an error "The final Hessian matrix is not positive definite although all convergence criteria are satisfied."

Thoughts?

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    $\begingroup$ This depends largely on how often you measured personality. I suppose once, so it is a variable on the person (uppermost) level in multilevel language and @Peter Flom's answer applies: fixed. But you could have measured personality four times as well, have you?, and then I would consider it random on level 1. $\endgroup$ – hplieninger Jul 23 '13 at 6:13
  • $\begingroup$ Good point. I measured it four times, but am only interested at time 1, for now. $\endgroup$ – Behacad Jul 23 '13 at 13:29
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I don't know SPSS syntax for these models, but in SAS, the personality variables would be considered fixed effects (that is, part of the MODEL statement) rather than random (that is, part of the RANDOM or REPEATED statements. This is probably true in SPSS as well.

The terminology of "fixed" and "random" is somewhat confusing; some authors (including, e.g., Andrew Gelman) prefer to use the term "multilevel model" instead.

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