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One commonly given reason to use mixed-effects model rather than a fixed-effects model is “the assumption of independence of errors does not hold in a multilevel structure, this will cause the standard errors of regression coefficients to be wrongly estimated”. Is this what is also referred to as “non-sphericity”?

But if groupID variables are included as fixed-effects, haven't we already solved for dependence of error within each observation of a particular group?

If the answer requires further context, consider the following case: I have publisher-campaign binomial data with dependent variable isClick (0/1) and independent variables publisherID, campaignID. 80% publisher-campaign combinations are not observed, 15% have 0-1000 data-points, and 5% have 1000-million data-points. The click rate is of order 10^-4. If I train a logistic regression model with publisherIDs and campaignIDs as fixed-effects, what error independence assumption is violated by the model?

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Fixed effects do not necessarily solve the dependence of errors within each particular group. The helpful article "A Practioner's Guide to Cluster-Robust Inference" by A. Colin Cameron and Douglas L. Miller addresses this point in particular.

In your situation, suppose you have some omitted variable that influences observations within a given group. E.g., maybe the campaigns have different effects based on what browser the user has, but for some reason you don't observe this. In that case, you'll get correlation of errors amongst users with different types of browsers, something that will violate your regression assumptions. Something similar could occur with things like different effects of the campaigns based on time of day, how long the campaign has been running, etc.

As I discuss in my answer here it is usually a good idea to at least try the cluster robust estimator. If your errors are indeed uncorrelated within your groups, then your variance will change little. As they say, the innocent (uncorrelated) have nothing to hide.

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