# ICC reduced to 0 in a multi-level model. Should this be suspicious?

I calculated a sequence of multi-level models with stata (xtmixed). I have a sample of 800 kids in 46 classes and I want to explain their individual level of prejudice by the proportion of fully prejudiced respondents in the class. This is based on a three categorical index, the level of prejudice is measured by a principal component. In the empty model the ICC is 0.12. In a model where I control for the usual suspects (age, gender, school type, etc), it is reduced to 0.08. In a third model where I include the "proportion of fully prejudiced students in the class" variable it has a significant** positive effect, just as I excepct it to. The ICC is reduced to 0.0000 (23 0-s in a row). My colleagues tell me that this is not usual, they've never seen anything like this before. In order to test for validity I calculated a linear regression with cluster and robust options. To me it seemed reassuring: Model with control variables had R square of 7%, model with the "proportion of fully prejudiced students in the class" variable had 15% of explained variance. This suggested me that the variable "proportion of fully prejudiced students in the class" has an additional explanatory power of 8%. And as the ICC reduced from 8% to 0 in the ML model, this seemed logical to me. Have you ever seen an ICC of 0.0000000000000000? Is it OK? Thanks

## 1 Answer

This seems pretty unsurprising to me.

In your model, the ICC can be interpreted as the proportion of the variance not explained by covariates that is due to variation between classes.

If instead of using "proportion of fully prejudiced respondents in the class", you used "average level of prejudice within each class" as a covariate, this would certainly reduce the ICC to zero, as this covariate explains all the variation between classes by definition.

So if "proportion of fully prejudiced respondents in the class" is strongly correlated with "average level of prejudice within each class", as you might well expect, it's not surprising that it also reduces the ICC to zero.