How to (properly) specify a cross level effect with lme4/nlme

Question

I'm trying to specify properly a cross level effect with lmer() and lme() functions.

I'm working with a modified sleepstudy data version, where I added a level 2 variable: score to each participant:

library(lme4)
data(sleepstudy)
score <- gl(n = 5, k = 10, length = 180)
sleepstudy$score <- as.numeric(score)

My goal is to find the right lme4::lmer() (and a nlme::lme() analogue) syntax that would replicate the following formula:

Level 1:
$Reaction = \\beta_0 + \\beta_1(Days)+ r$

Level 2:
$ \\beta_0= \\gamma_{00} + \\gamma_{01}(Score) + u_0$
$ \\beta_1= \\gamma_{10} + \\gamma_{11}(Score) + u_1$

EDIT: i'm not sure why math equation are not rendered properly, so here goes a img of how it should look like:

For some unknown reason I'm having a hard time translating normal equations to lme4/nlme syntax.

PS. I would use a read (if there is any) where regression equations are translated to syntax. So far I didn't find any.

Answer 1

By trial and error I found the proper solution for the given example:

lmer(Reaction ~ score*Days  + (Days|Subject), data = sleepstudy)

I can leave this Q/A as is or delete if that's considered best practice.