Have I correctly specified my model in glmer ?

#1
The following multilevel logistic model with one explanatory variable at level 1 (individual level) and one explanatory variable at level 2 (group level) :

\(\text{logit}(p_{ij})=\pi_{0j}+\pi_{1j}x_{ij}\ldots (1)\)
\(\pi_{0j}=\gamma_{00}+\gamma_{01}z_j+u_{0j}\ldots (2)\)
\(\pi_{1j}=\gamma_{10}+\gamma_{11}z_j+u_{1j}\ldots (3)\)

where , the group-level residuals \(u_{0j}\) and \(u_{1j}\) are assumed to have a multivariate normal distribution with expectation zero . The variance of the residual errors \(u_{0j}\) is specified as \(\sigma^2_0\) , and the variance of the residual errors \(u_{1j}\) is specified as \(\sigma^2_1\) .

Substituting equation (2) and (3) in equation (1) yields ,

\(\text{logit}(p_{ij})=\gamma_{00}+\gamma_{10}x_{ij}+\gamma_{01}z_j+\gamma_{11}x_{ij}z_j+u_{0j}+u_{1j}x_{ij}\ldots (4)\)

Now I want to specify the formula in glmer . So I wrote

Code:
m1 <- glmer(Y~X*Z+(X|group),data=dat,family=binomial)
where "group" is the grouping variable that tells R who's in which .

Is this correct formula to fit the model in equation (4) ? I have scoured Google and I am still confused about the formula . It will be very helpful if anyone could explain the notation .


Many thanks! Regards .