Notation of Variance of Residuals in Multilevel Modeling

#1
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

Not a robit
#2
Good luck, notation on multilevel models is definitively more tricky (sophisticated) than most other models. I will look forward to what others post.
 

Jake

Cookie Scientist
#3
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).
 
#4
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).
Thanks for reply .

But
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" .

Thanks & regards .