*Bounty: 50*

*Bounty: 50*

Suppose that a dependent level variable $y$ is measured at a unit level (level 1) that is nested within units of type $A$ (level $2$), and that units of type $A$ are nested within levels of type $B$ (level $3$).

Suppose that I fit the following formula:

```
y ~ "FIXED EFFECTS [my syntax]" + (1 + x | B/A)
```

where $x$ is some predictor at level $1$.

My understanding is that the mathematical representation of such a formula is the following. Is it correct?

In what follows, $y_{b,a,i}$ is the output of the $i$th data point in unit $a$ of $A$ nested in unit $b$ of $B$. This data point has a corresponding predictor $x_{b,a,i}$.

$y_{b,a,i} = text{“fixed effects”} + u_{b} + u_{b,1,a} + (beta_{b} + beta_{b,1,a})x$

where

$u_{b} sim N(0, sigma_{B})$

$u_{b,1,a} sim N(0, sigma)$

$beta_{b} sim N(0, rho_{B})$

$beta_{b,a} sim N(0, rho)$

That is, $sigma_{B}$ is a standard deviation term that varies across level $3$. On the other hand, given any $b$, a unit in level $3$, and $a$, a unit contained in level $2$, then the standard deviation term for $a$ is $sigma$. That is, $sigma$ is constant for any level $2$ units.

Is this correct (I based this reasoning by inferring from a related presentation on page 136 of Linear Mixed Models: A Practical Guide Using Statistical Software))? If this is correct, then is there any way to make $sigma$ be dependent on which unit of level $A$ the data point belongs to.