I’m trying to figure out the interpretation of a group level correlation named as follows in the model summary: `cor(Intercept,year)`

Could you please explain this simple words? For example, how should I simulate my data to see a positive or negative correlation.

**Simulated data**

6 groups, each next has higher mean “y” with higher SD, association between “y” and year is positive within each group

```
set.seed(1)
n = 7
df2 = bind_rows(
tibble(y = rnorm(n, 10, 2), group = rep(times = n, "A")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)),
tibble(y = rnorm(n, 20, 6), group = rep(times = n, "B")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)),
tibble(y = rnorm(n, 30, 10), group = rep(times = n, "C")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)),
tibble(y = rnorm(n, 40, 14), group = rep(times = n, "D")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)),
tibble(y = rnorm(n, 50, 16), group = rep(times = n, "E")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)),
tibble(y = rnorm(n, 60, 20), group = rep(times = n, "F")) %>% arrange(y) %>% mutate(year = seq(1, n, 1)))
```

**Model**

`m= brm(y ~ year + (year |d| group), df2)

m`

```
> Family: gaussian
> Links: mu = identity; sigma = identity
> Formula: y ~ year + (year | d | group)
> Data: df2 (Number of observations: 42)
> Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
> total post-warmup samples = 4000
>
> Group-Level Effects:
> ~group (Number of levels: 6)
> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
> sd(Intercept) 17.53 6.63 8.74 34.65 1.00 1371 1891
> sd(year) 3.05 1.41 1.35 6.73 1.00 1115 1843
> cor(Intercept,year) 0.25 0.39 -0.58 0.84 1.00 1586 2178
>
> Population-Level Effects:
> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
> Intercept 20.12 6.95 6.39 34.02 1.00 1441 1835
> year 3.93 1.28 1.27 6.50 1.00 1446 1298
>
> Family Specific Parameters:
> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
> sigma 3.99 0.56 3.07 5.25 1.00 2108 2274
>
> Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
> and Tail_ESS are effective sample size measures, and Rhat is the potential
> scale reduction factor on split chains (at convergence, Rhat = 1).
```

**Group level effects**

```
conditions = tibble(group = unique(df2$group),
row.names = unique(df2$group))
conditional_effects(m_default2, effects = "year", re_formula = NULL, conditions = conditions)
```