glmmTMB truncated models with zero inflation

Question

everyone. I am fitting a glmm model using the R library glmmTMB for predicting a count response variable with excess-zeros and overdispersion (nbinom2> Poisson).

Additionally, I am insterested in exploring the zero-inflation component of my sample data, therefore a zero-inflation vs hurdlequestion arises. Even though both approaches are two-component models with finite mixture, two base assumptions between them arises

• hurdle models are those in which the zero vs non-zero component of the count are mechanistically splitted (i.e. you assume mechanism producing the zero part are different from the non-zero part), which mean assuming all your sample zeros are true or structural zeros. Then, a logit model for the zero vs no-zero plus a count model for counts > 0 are presented together.
• zero-inflatedmodels, on the other hand, while also splitting the zero vs non-zero components, allow the count model to produce zero observations (i.e., true zeros), while the zero-inflation component becomes a point-mass extra-zero generator (i.e., false or excess zeros). Those zero counts would represent false abscenses e.g. a sampling error or a mistake in the measurement (maybe you were not paying enough attention...)

Choosing between the two becomes a crucial analytical design conditioning the interpretation of the results, conditional to the study design and tested hypotheses.

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I should note that I am adding random effects in my model, and I believe glmmTMBis the only package that can handle all the aspects at once. If that's not the case, please let me know.

When implementing the hurdle/zi in glmmTMB, the argument ziformula = enables the zero-inflation model, while family = truncated_nbinom2() defines the hurdle model

However, I feel puzzled since attempting to fit a hurdle truncated_nbinom2 model yield the following error message:

'y' contains zeros (or values close to zero). Zeros are compatible with a truncated distribution only when zero-inflation is added

Then my model can be defined as follows:

m0 <- glmmTMB(y ~ var1 + ... + varN + (1 | id),   
          offset = log(offset), 
          data = df,                  
          family = truncated_nbinom2(),
          ziformula= ~ .)

which is producing a nice model that meets my requirements. In fact, it is the only model that i have been able to affectively fit.

I have seen these kind of models in other examples, and it is obvious glmmTMB has specific implementations of zero-inflation with truncated families, but I have never read in the literature about hurdle + zi models as they are always presented as mutually exclussive, focusing on the criteria for selecting and comparing between them

What am I missing here??

1. Is this approach statistically valid?
2. How is the result interpreted? I guess the zero-inflation component is the same as any other zi model. But my conditional model, as a truncated family, should no be able to account for true zero counts and therefore the count model might be biased

I'd love if someone could shed some light into this issue

P.S: i also found some questions regarding prediction with these kind of models, truncated with zero-inflation here

EDIT

As per request, I include a very silly repex that shows the issue

# generate data with 75% zero
y <- tibble(observed = rep(c(1,0,0,0), 25))

# fit the model w/o zero-inflation, then turn to ~1 to fix and run it
mod <- glmmTMB(observed ~ 1, 
            family = truncated_nbinom2(),
            data = y, 
            ziformula = ~ 0)

summary(mod)

Answer 1

I think there is some confusion about what the two zero-inflated models actually are. Your two dot point definitions in the question are good. However, there appears to be some confusion in the interpretation and implementation.

ZIP

A zero-inflated model (let's just say Poisson, for example's sake) is a mixture of a Bernoulli and a Poisson. The probability of zero is the probability that the Bernoulli produces a zero, plus the probability that the Bernoulli doesn't produce a zero but the count process does.

You would implement this in glmmTMB like glmmTMB(formula = y ~ count_predictors, ..., family = poisson(), ziformula = ~ zi_predictors).

ZAP

A zero-altered (or Hurdle) model is a two-part distribution, or you could think of it as a mixture of a Bernoulli and a zero-truncated Poisson. The probability of a zero is just the probability that the Bernoulli produces the zero.

You would implement this in glmmTMB like glmmTMB(formula = y ~ count_predictors, ..., family = truncated_poisson(), ziformula = ~ zi_predictors).

Truncated distribution

Note: truncated_poisson is just a truncated distribution. It does not allow zeroes, which is why you were getting an error.