Simulating hierarchical data (nested structure), containing categorical variable

Question

I am simulating dyadic data, which is cluster size is fixed to two.

I have continuous IV (e.g.,education score) and categorical variable (e.g., gender)

these two variable is level-1 variable.

target ICC is 0.3 (ICC=between cluster variable / (within+ between))

so, when I creat education, it was fine. what I did is,

genderating between-cluster column, and within-cluster column for member 1, and within-cluster column for member 2.

so, education for member 1= ICC weight * between cluster + icc weight * within 1 member 2= ICC weight *between cluster + iccc weight * within 2

cn <- 100 #number of pairs
mu <-c(0,0,0,0)
vcovmatrix <- matrix(c(1,0,0,0,
                       0,1,0,1,
                       0,0,1,0,
                       0,1,0,1),
                    nrow = 4,
                    byrow = TRUE,
                   dimnames = list(c("withn_1","btw_1","withn_2","btw_2"),
                           c("withn_1","btw_1","withn_2","btw_2")))


dat<-mvrnorm(cn, # sample size
            mu=mu, # Mu
            Sigma = vcovmatrix) # Covariance matrix

dat<-as.data.frame(dat)
dat$pid<-1:cn # pair ID (group ID)

dat%>%
  dplyr::select(c(withn_1,btw_1,withn_2,btw_2))%>%
  cor()


## Let's say that ICC is 0.3, This means that between:within ratio is 3:7. 
## 3/(7+3)=0.3

withn_weight=sqrt(0.3)
btw_weight=sqrt(0.7)


dat <- dat %>%
  mutate(
    edu_1=withn_weight*withn_1+btw_weight*btw_1,
    edu_2=withn_weight*withn_2+btw_weight*btw_2
  )

However, I cannot easily envisioning how I can creat gender.

gender have two levels.

as I did in the education,

I can create between-cluster effect in gender, and within_1 and within_2

and gender_1 <-ICCweightgenderbetween + Iccweightgenderwithin1 gender_2<-ICCweightgnederbetween+ICCweightgenderwithin2

but gender is categorical variable, so how I can create my gender variable?