Interpreting random effects in zero-inflated models

Question

For context, I have a longitudinal study measuring counts of bacterial sequences in human stool collected during a dietary intervention.

Initially, I was going model the change in each bacterium (sequence) over time using a Negative Binomial generalized mixed model (lme4::glmer.nb) with a random intercept for subject. However, there is considerable between-person variability in the microbiome, and I have plenty of cases where, for example, there is a time-trend in 11 subjects that have a certain bacterium, but then 4 subjects with counts of 0 across the study period. I have 6-10 samples per subject, so it seems very likely that these subjects simply do not have that bacterium.

I have modeled the data using a zero-inflated Negative Binomial mixed model (glmmTMB::glmmTMB) with a random intercept for subject for both the conditional part of the model and the zero-inflated part (and only an intercept for the fixed effects of the zero-inflated part). Specifically:

glmmTMB(sequence1 ~ time + (1|subject_id), 
        ziformula = ~ 1 + (1|subject_id),
        family = "nbinom2",
        data = data)

Including the random effect for zero-inflation substantially improves the model fit.

Does this zero-inflated model effectively remove the subjects that the bacterium was never detected in when estimating the Negative Binomial component? How does this type of model differ from simply dropping subjects where the bacterium was never detected and running a non-zero-inflated model?

Answer 1

Does this zero-inflated model effectively remove the subjects that the bacterium was never detected in when estimating the Negative Binomial component?

No, the model you wrote takes account of their being correlations for each subject in both the zero part and the conditional part. In other words, the propensity to either have or not have the bacterium detected is more similar within the same subject than different subjects; and for those subject that do have the bacterium detected the numbers counted are likely to be more similar within the same subject than different subjects. In both cases the variation of the random intercepts captures these correlations.

How does this type of model differ from simply dropping subjects where the bacterium was never detected and running a non-zero-inflated model?

In that case you are simply modelling the counts in those subjects who have the bacterium detected and ignoring the question of possible immunity or undetectable infection levels (or whatever the reason for zero counts).

Also, don't forget that the negative binomial distribution has support for zeros. We fit a zero-inflated model where there are excess zeros. So removing zeros altogether is likely to be a bad idea for this reason alone.