#StackBounty: #r #lme4-nlme #random-effects-model The mathematical representation of a nested random effect term

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.

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