Hi, my question will be a more general question regarding the model formula. I have been searching for many web sources just to try to get any information on this issue. I am working on a **two-level multilevel model**.

### Problem Background

In level one, I have one response variable with one predictor variable. Therefore, I have **one intercept** and **one slope coefficient** at this level.

In level two, I have **one model that predicts the level-one intercept** and **another model that predicts the level-one slope**. In my study, **I will be using different predictors in both level-two models**. Nearly all tutorials in the internet and the books I have read assume the same level-two predictors are being used to predict the level-one intercept and level-one slope.

#### Level One Model

y_{ij} = \beta_{0j} + \beta_{1j}\cdot x_{ij} + \varepsilon_{ij}

#### Level Two Model

\beta_{0j} = \gamma_{00} + \gamma_{01}\cdot z_j + u_{0j}

\beta_{1j} = \gamma_{10} + \gamma_{11} \cdot k_j + u_{1j}

### Question

How should I specify my model in r? (Packages in use: `lmerTest`

, `blmer`

, and `brms`

. They all use the same model formulation)

### What I Know

Null Model: This is simple. I think I have done it correctly.

```
y ~ (1 | j)
```

Random Intercept Model: I am pretty sure this is correct too. `x`

will be a fixed effect predictor and this only allows the incept to be varying across the different groups.

```
y ~ x + (1 | j)
```

### What I Don’t Know

How do I make a formula for random intercept and random slope and beyond? I know that when you have the same level-two predictors for both the intercept and the slope, it is

```
y ~ x + (z | j)
```

But to my understanding, this assumes `z`

to be the level-two predictor of both Level-one intercept and slope. How do I formulate my model when the level-two predictors for level-one intercept and slope are different? I hope my question makes sense to everybody.

### Note:

Please let me know what I should do to make the question clearer to you. I have recently been experimenting with `brms`

package for a thorny model estimation issue. Thank you.