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

Sorry, I forgot that lmer() (unlike lmer_alt() from the afex package)
does not convert factors to numeric covariates when using the the
double-bar notation!
The model I was talking about would be:

m_zcp5 <- lmer_alt(angle ~ recipe  + (recipe || replicate), cake)
VarCorr(m_zcp5)
 Groups      Name        Std.Dev.
 replicate   (Intercept) 6.2359
 replicate.1 re1.recipe1 1.7034
 replicate.2 re1.recipe2 0.0000
 Residual                5.3775

This model seems to differ (and I don't really understand why) from
m_zcp6 which I think is equivalent to your m_zcp4:
m_zcp6 <- lmer_alt(angle ~ recipe  + (0 + recipe || replicate), cake)
VarCorr(m_zcp6)
 Groups      Name        Std.Dev.
 replicate   re1.recipeA 5.0429
 replicate.1 re1.recipeB 6.6476
 replicate.2 re1.recipeC 7.1727
 Residual                5.4181

anova(m_zcp6, m_zcp5, refit = FALSE)
Data: data
Models:
m_zcp6: angle ~ recipe + ((0 + re1.recipeA | replicate) + (0 + re1.recipeB |
m_zcp6:     replicate) + (0 + re1.recipeC | replicate))
m_zcp5: angle ~ recipe + ((1 | replicate) + (0 + re1.recipe1 | replicate) +
m_zcp5:     (0 + re1.recipe2 | replicate))
       Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
m_zcp6  7 1781.8 1807.0 -883.88   1767.8
m_zcp5  7 1742.0 1767.2 -863.98   1728.0 39.807      0  < 2.2e-16 ***

Yes, m_zcp4 and m_zcp6 are identical.

For m_zcp5 I get:
m_zcp5 <- lmer_alt(angle ~ recipe  + (recipe || replicate), cake)
VarCorr(m_zcp5)
 Groups      Name        Std.Dev.
 replicate   (Intercept) 6.0528e+00
 replicate.1 re1.recipeB 5.8203e-07
 replicate.2 re1.recipeC 2.1303e+00
 Residual                5.4693e+00

and if we change the reference level for recipe we get yet another result:
cake2 <- cake
cake2$recipe <- relevel(cake2$recipe, "C")
m_zcp5b <- lmer_alt(angle ~ recipe  + (recipe || replicate), cake2)
VarCorr(m_zcp5b)
 Groups      Name        Std.Dev.
 replicate   (Intercept) 6.5495e+00
 replicate.1 re1.recipeA 2.5561e+00
 replicate.2 re1.recipeB 1.0259e-07
 Residual                5.4061e+00
This instability indicates that something fishy is going on...

The correlation parameters are needed in the "default" representation:
(recipe | replicate) and (0 + recipe | replicate) are equivalent
because the correlation parameters make the "appropriate adjustments",
but (recipe || replicate) is _not_ the same as (0 + recipe ||
replicate) with afex::lmer_alt. I might take it as far as to say that
(recipe | replicate) is meaningful because it is a re-parameterization
of (0 + recipe | replicate). On the other hand, while the diagonal
variance-covariance matrix parameterized by (0 + recipe || replicate)
is meaningful, a model with (recipe || replicate) using afex::lmer_alt
does _not_ make sense to me (and does not represent a diagonal
variance-covariance matrix).

Do m_zcp5 and Model3b estimate the same random effects in this case?

Well, Model3b makes sense while m_zcp5 does not, but Model3b estimates
more random effects than the others:
Model3b <- lmerTest::lmer(angle ~ recipe + (1 | replicate) + (1 |
recipe:replicate),
                          data=cake)
length(unlist(ranef(Model3b))) # 60
length(unlist(ranef(m_zcp4))) # 45 - same for m_zcp, m_zcp2 and m_zcp6
and Model2

If not, what is the difference between m_zcp5 and Model3b (except for
the fact that the variance depends on the
recipe in m_zcp5) and which one is the more complex model?

There is no unique 'complexity' ordering, for example, Model3b use 2
random-effect variance-covariance parameters to represent 60 random
effects, while m_zcp4 (m_zcp2) use 3 (6) random-effect
variance-covariance parameters to represent 45 random effects. But
usually the relevant 'complexity' scale is the number of parameters,
cf. likelihood ratio tests, AIC, BIC etc. There are corner-cases,
however; if x1 and x2 are continuous then (1 + x1 + x2 | group) and
'(1 + x1 | group) + (1 + x2 | group)' both use 6 random-effect
variance-covariance parameters, but the models represent different
structures and you can argue that the latter formulation is less
complex than the former since it avoids the correlation between x1 and
x2.

Cheers,
Rune

I would be glad if you could elaborate on this and help me and the
others understand these models.

Cheers,
Maarten