# Notation of Variance of Residuals in Multilevel Modeling

#### Cynderella

##### New Member
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 $$\sigma_{00}$$ ."

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

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) ,
$$\rho=\frac{\sigma_{u0}^2}{\sigma_{u0}^2+\sigma_{e}^2} \Rightarrow 0.1=\frac{\sigma_{u0}^2}{\sigma_{u0}^2+0.5} \Rightarrow \sigma_{u0}^2=\frac{1}{18}$$

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

Many Thanks! Regards .

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#### hlsmith

##### Not a robit
Good luck, notation on multilevel models is definitively more tricky (sophisticated) than most other models. I will look forward to what others post.

#### Jake

It's hard to say with absolute certainty without reading the rest of the paper for clarification. Ideally one would like the author to make this clear. But I suspect everything you say is right except that $$\sigma_{00}$$ is probably the square root of $$\sigma^2_{u0}$$ (and similarly for the other two sigmas).

#### Cynderella

##### New Member
I suspect everything you say is right except that $$\sigma_{00}$$ is probably the square root of $$\sigma^2_{u0}$$ (and similarly for the other two sigmas).
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 $$\sigma_{00}$$ .
indicates $$\sigma_{00}$$ is the intercept "variance" .