ICC low, but still team variables in individual model?

Question

I have data on two levels: team and individual. However, since the ICC on team level is low, I’ll not use a multilevel model but just a regular regression on individual level. After all, the impact of the team variables and the estimation of clustered standard error is small and has not a significant impact on the final results.

However, is it still valid to use some team variables (for each individual) in your regular regression model? E.g. gender diversity within team? Especially with regard to the endogeneity problem / error term is not independent..

(I know that there is in general a discussion whether ICC is a valid analysis to decide on multilevel models [e.g. Nezlek, 2008]. But let's just ignore that for now….)

Answer 1

My instinct is similar to the article you cited: if there is some sort of nesting in your data, such as players nested within a team, then some sort of multilevel model is justified. In fact I would encourage it, even if the ICC is low.

I said "some sort of multilevel model" because economists typically encourage the use of fixed effects instead of random effects, e.g. add a fixed effect for team. For example, Bowblis and Hyer (2013) used a fixed effect model for an analysis of nursing homes, nested within states (and further, they have multiple observations on each nursing home). That is, they added a fixed effect for each NF. That's also a valid way to accommodate the clustering (and they give the typical argument for accommodating that via fixed effect in their paper).

But, if you must, then it is valid to include the effect of team variables in an OLS regression without any other treatment of the team. For example, if you're doing a random effect model, then you're estimating something like this, where i indexes teams and j indexes individual players:

$$ Y_{ij} = \beta_0 + \beta_1 Gender Diversity + U_i + \epsilon_{ij} $$

If you chose to eliminate $U_i$, which can be a fixed OR random effect, then all you're doing is ... eliminating that fixed or random effect, which disappears into your error term. You'll still be estimating

$$ Y_{ij} = \beta_0 + \beta_1 Gender Diversity + \epsilon_{ij} $$