[R] Interpretation of hypothesis tests for mixed models

Olof Leimar Olof.Leimar at zoologi.su.se
Sun Dec 15 13:25:03 CET 2002


My question concerns the logic behind hypothesis tests for fixed-effect
terms in models fitted with lme. Suppose the levels of Subj indicate a
grouping structure (k subjects) and Trt is a two-level factor (two
treatments) for which there are several (n) responses y from each
treatment and subject combination. If one suspects a subject by
treatment interaction, either of the following models seem natural
  
> fm1 <- lme(y ~ Trt, random = list(Subj = pdDiag(~ Trt)))
> fm2 <- lme(y ~ trt, random = ~ 1 | Subj/Trt)

These models seem to correspond to the same situation. Both have two
variance components (subject and treatment within subject). However,
they result in different denominator degrees of freedom (denDF) of the
F-statistic for a (fixed-effect) test for treatment. For the case of few
subjects and many observations per subject-treatment combination, denDF
will be much larger for fm1 (denDF = k*2*n-k-1) than for fm2 (denDF =
k-1).

What is the essential difference in the nature of random effects for
situations modelled by fm1 and fm2? On the other hand, if there is no
essential difference between fm1 and fm2, why are the tests different? 

I realize that fm2 corresponds to the classical analysis
> fm3 <- aov(y ~ Trt + Error(Subj/Trt))
but what is the logic behind fm1?

I have looked in Venables & Ripley (2002) and Pinheiro & Bates (2000),
but neither of these excellent books seems to explain how or whether
models like fm1 and fm2 differ.

Olof Leimar, Professor
Department of Zoology
Stockholm University
SE-106 91 Stockholm
Sweden

Olof.Leimar at zoologi.su.se




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