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 .

Last edited:

hlsmith

Less is more. Stay pure. Stay poor.
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" .