# Notation of Variance of Residuals in Multilevel Modeling

I am having some trouble to understand the notation of variance of residuals in multilevel modeling . In this paper "Sufficient Sample Sizes for Multilevel Modeling" , in p.87 below equation (3) , they mentioned

the variance of residual errors $u_{0j}$ and $u_{1j}$ is specified as $\sigma_{u0}^2$ and $\sigma_{u1}^2$ .

And in p.89 in the first para , they mentioned

Busing (1993) shows that the effects for the for the intercept variance $\sigma_{00}$ and the slope variance $\sigma_{11}$ are similar ; hence we chose to set the value of $\sigma_{11}$ equal to $\sigma_{00}$ .

$\bullet$ Does $\sigma_{00}$ denote the variance of residual errors $u_{0j}$ , so that $\sigma_{u0}^2 = \sigma_{00}$?

$\bullet$ Similarly , does $\sigma_{11}$ denote the variance of residual errors $u_{1j}$ , so that $\sigma_{u1}^2 = \sigma_{11}$?

If so , since it is also mentioned in p.89 in the first para that :

The residual variance $\sigma_{u0}^2$ follows from the ICC and $\sigma_{e}^2$ , given Equation 6.

Then for the $\sigma_{e}^2=0.5$ and ICC$=0.1$ , from Equation (6) , $$\rho=\frac{\sigma_{u0}^2}{\sigma_{u0}^2+\sigma_{e}^2} \Rightarrow 0.1=\frac{\sigma_{u0}^2}{\sigma_{u0}^2+0.5} \Rightarrow \sigma_{u0}^2=\frac{1}{18}$$

$\bullet$ Hence from the second highlighted para , will I take the value of $\sigma_{u0}^2 = \sigma_{00}=\sigma_{u1}^2 = \sigma_{11}=\frac{1}{18}$ ?

• I think you are all right. But there are more rigorous ICC definitions for mixed models with both random intercept and slope, though not used in the paper. One of them is defined by Goldstein et al.. After equation (2), ICC depends on the values of predictors. – Randel Jul 27 '15 at 19:52

Here is an elaboration of my comment above.

Let's rewrite the multilevel model into a single model,

$$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\boldsymbol z_{ij}^{'}\boldsymbol u_j+e_{ij},$$

where $j$ denotes the cluster, $\boldsymbol u_j=(u_{0j},u_{1j})$. The variance of the residual errors $e_{ij}$ is specified as $\sigma^2_e$ and the variances of the residual errors $u_{0j}$ and $u_{1j}$ are specified as $\sigma^2_{u0}$ and $\sigma^2_{u1}$. Note that there is a typo at the end of the paragraph below equation (3). In the first paragraph on Page 89, the authors followed the notation in Busing (1993) and referred $\sigma_{00}$ and $\sigma_{11}$ to $\sigma^2_{u0}$ and $\sigma^2_{u1}$.

Their simulation procedure for $\sigma^2_{u0}$ and $\sigma^2_{u1}$ is:

1. Calculate $\sigma^2_{u0}$ based on $\sigma^2_e=0.5$, the intra-class correlation (ICC = 0.1, 0.2, 0.3), and the ICC definition in equation (6) below $$\rho=\frac{\sigma_{u0}^2}{\sigma_{u0}^2+\sigma_{e}^2}.$$
2. Set $\sigma^2_{u1}=\sigma^2_{u0}$ by following Busing (1993).

Note that the ICC definition above is for random-intercept-only model. There are several more rigorous ICC definitions for mixed models with both random intercept and slope. One of them is defined by Goldstein et al. (2002). As shown in the second formula after equation (2) in their paper, ICC depends on predictors in the model.