Should a nested random effect be included if it has only one level for ONLY ONE fixed effect level?

Question

I have one fixed effect (group) and one nested random effect (nested). There are 20 levels of group, and all but one of them has 4 levels of nested. There is significant variance in nested that I feel should be included in the ANOVA model, but because a single level of group has only a single corresponding nested level, this seems technically inappropriate.

Moreover, when I run a mixed model anova while including nested (either with lme4 in R or proc MIXED in SAS), there is always some message about "Convergence criteria met but final hessian is not positive definite" (SAS warning). I found an informative SAS paper that mentioned that this error is often caused by mis-specified models (see https://support.sas.com/resources/papers/proceedings12/332-2012.pdf). However, I also saw another question on this site that seemed pertinent and whose answer implied that it was a valid approach (Is it inappropriate to do a One-way independent ANOVA when one of the levels has only one participant?).

From what I could interpret from a different question on this site, if a random effect has only one level then the mixed model's optimizer will really only be estimating the random effect based on rounding error (Mixed model runs well in R whereas a random effect has only one level). In my opinion this seems fine in my case since 1) I just need to account for the effects of nested on group, 2) if the estimate for nested in the single troublesome group level is based on rounding error then it probably won't dramatically change the estimate for that level's fixed effect, and 3) the benefit to accounting for nested within all other group levels seems worth the tradeoff.

Note that I cannot exclude the troublesome group level from the analysis.

QUESTION: Should I fit a mixed model that includes nested, and if so, with how much caution should I interpret the results?

Answer 1

This sounds like a case of partial clustering, which frequently occurs in clinical psychology treatment evaluation studies, in which individuals exposed to the treatment are clustered within therapists, but individuals in the control condition are not (i.e., they are all in one group). There has been some literature on this matter and well-specified guidelines on how to proceed when using linear mixed models. The general advice is to allow the group with only one level of clustering to be the reference group, and then to allow random slopes on the coefficients for the other levels but not a random intercept. This is critical to maintaining accurate type I error rates if the clusters are thought to be random (e.g., therapists randomly selected from a population of therapists). Some papers discussing this issue include Sterba (2017) and Baldwin et al. (2011).

In your case, this would involve setting the group with one level to be the reference level and then placing random slopes on the coefficients for the 19 other levels. You should ensure that the random effects covariance matrix is specified to be compound symmetric (which it is with mixed ANOVA). This is more easily done with PROC MIXED.