Until Rune Haubo pointed out that he cannot see how these models (using the double-bar notation for factorial designs in the way I do in `m1`

) are relevant I was quite sure that they made sense:

`library("lme4")`

`m1 <- afex::lmer_alt(angle ~ recipe + (recipe || replicate), cake)`

He instead suggests that one should use

```
m2 <- lmer(angle ~ recipe +
(0 + dummy(recipe, "A") | replicate) +
(0 + dummy(recipe, "B") | replicate) +
(0 + dummy(recipe, "C") | replicate), cake)
```

or

```
m3 <- lmer(angle ~ recipe + (1 | replicate) +
(0 + dummy(recipe, "A") | replicate) +
(0 + dummy(recipe, "B") | replicate) +
(0 + dummy(recipe, "C") | replicate), cake)
```

which are different from `m1`

and equivalent to

`m4 <- afex::lmer_alt(angle ~ recipe + (0 + recipe || replicate), cake)`

and

`m5 <- afex::lmer_alt(angle ~ recipe + (1|replicate) + (0 + recipe || replicate), cake)`

I’m now quite puzzled because AFAICS you and others (but I don’t remember them) use `m1`

in the `afex`

vignettes and also here.

What is the correct way to specify a `lmer_alt`

model for categorical independent variables?