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R-sig-mixed-models Digest, Vol 136, Issue 41

Rune Haubo
On 2 May 2018 at 17:50, Maarten Jung <Maarten.Jung at mailbox.tu-dresden.de> wrote:
I haven't read those articles recently in enough detail that I can comment.
So m_zcp5 and m_zcp7 are identical but I don't see how they are
meaningful in this context. Looking at the random-effect design matrix

image(getME(m_zcp5, "Z")) # identical to image(getME(m_zcp7, "Z"))

you can see that this model estimates a random main effect for
replicate (i.e. (1 | replicate)) and then a random _slope_ for recipe
at each replicate (i.e. recipe in '(recipe || replicate)' is treated
as numeric rather than factor). As far as I can tell this random slope
model is _unrelated_ to models where recipe is treated as a factor
that we have discussed previously: It is a completely different model
and I don't see how it is relevant for this design. (Notice that
'recipe' is equivalent to 'recipe_numeric' in the fixed-effects, but
not so in the random-effects!)
If anything I would say that you should look at all relevant models
and choose the one that represents the best compromise between fit to
data and complexity :-) Likelihood ratio tests can be a helpful guide,
but take care not to formally compare/test models that are not nested.

Here is an example of a set of models and sequences in which they can
be compared with LR tests:

# Random main effect of replicate, no interaction:
fm1 <- lmer(angle ~ recipe + (1 | replicate), data=cake)
# Random interaction recipe:replicate; same variance across recipes;
no main effect:
fm2 <- lmer(angle ~ recipe + (1 | recipe:replicate), data=cake)
# Random interaction with different variances across recipes; no main effect:
fm3 <- lmer(angle ~ recipe +
              (0 + dummy(recipe, "A") | replicate) +
              (0 + dummy(recipe, "B") | replicate) +
              (0 + dummy(recipe, "C") | replicate), data=cake)
# Random main effect and interaction with same variance across recipes:
fm4 <- lmer(angle ~ recipe + (1 | replicate) + (1 | recipe:replicate),
data=cake)
# Random main effect and interaction with different variances across recipes:
fm5 <- lmer(angle ~ recipe + (1 | replicate) +
              (0 + dummy(recipe, "A") | replicate) +
              (0 + dummy(recipe, "B") | replicate) +
               (0 + dummy(recipe, "C") | replicate), data=cake)
# Multivariate structure that contains both main and interaction effects with
# different variances and correlations:
fm6 <- lmer(angle ~ recipe + (0 + recipe | replicate), data=cake)
# Same model, just re-parameterized:
# fm6b <- lmer(angle ~ recipe + (recipe | replicate), data=cake)
# fm6c <- lmer(angle ~ recipe + (1 | replicate) + (0 + recipe |
replicate), data=cake)
# fm6d <- lmer(angle ~ recipe + (1 | replicate) + (recipe |
replicate), data=cake)
# fm6e <- lmer(angle ~ recipe + (1 | recipe:replicate) + (recipe |
replicate), data=cake)
# anova(fm6, fm6b, fm6c, fm6d, fm6e, refit=FALSE) # fm6 = fm6b = fm6c
= fm6d = fm6e

Note that in fm4 and fm5 the random main and interaction effects are
independent, but in fm6 they are not.

No. parameters and log-likelihood/deviance of these models:
as.data.frame(anova(fm1, fm2, fm3, fm4, fm5, fm6,
refit=FALSE))[paste0("fm", 1:6), 1:5]
    Df      AIC      BIC    logLik deviance
fm1  5 1741.019 1759.011 -865.5097 1731.019
fm2  5 1776.967 1794.959 -883.4835 1766.967
fm3  7 1779.571 1804.760 -882.7857 1765.571
fm4  6 1741.067 1762.658 -864.5337 1729.067
fm5  8 1743.437 1772.224 -863.7185 1727.437
fm6 10 1741.003 1776.987 -860.5016 1721.003

The following nesting structure indicate sequences in which models can be
compares with LR tests (arrows indicate model simplification):
fm6 -> fm5 -> fm4 -> fm2
fm6 -> fm5 -> fm4 -> fm1
fm6 -> fm5 -> fm3 -> fm2

Note that fm3 and fm4 are not nested and simply represent different structures
and so no formal LR test is available. The same is true for fm1 and
fm2 as well as
fm1 and fm3.

In addition to these models there are others which are just not as
easily fitted with lmer (to the best of my knowledge) for example a
version of fm5 where the interaction random effect is specified with a
common covariance parameter on top of the 3 variances. Theoretically
there are many options here but obtaining the fits is often not
straight forward and usually no single fit is uniquely better than the
rest.

Cheers
Rune

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Maarten Jung R-sig-mixed-models Digest, Vol 136, Issue 41 May 1 Rune Haubo R-sig-mixed-models Digest, Vol 136, Issue 41 May 1 Maarten Jung R-sig-mixed-models Digest, Vol 136, Issue 41 May 1 Rune Haubo R-sig-mixed-models Digest, Vol 136, Issue 41 May 2 Maarten Jung R-sig-mixed-models Digest, Vol 136, Issue 41 May 2 Maarten Jung R-sig-mixed-models Digest, Vol 136, Issue 41 May 2 Rune Haubo R-sig-mixed-models Digest, Vol 136, Issue 41 May 4 Maarten Jung R-sig-mixed-models Digest, Vol 136, Issue 41 May 5