# Do zero-inflated models induce selection bias?

Zero-inflated models (e.g., ZI poisson, ZI negative binomial, hurdle) assume two processes for the generation of the observed outcome variable: a process for deciding whether the outcome is zero or not, and a process, for those for whom the outcome is not definitely zero, for assigning a value to the outcome. In these models, each process has its own regression model and (implied) disturbance. Presumably, in most cases, the disturbances are correlated (i.e., the unmeasured causes of one process also cause the other process).

In the nonzero part of the model, we condition on not receiving a zero. For example, in the hurdle model, the nonzero part can be estimated directly by simply excluding those who received a zero and estimating a zero-truncated model for everyone else. Those who did receive a zero do not contribute to the likelihood for the nonzero process model estimation.

Conditioning on receiving a zero would seem to me to be conditioning on a collider, opening up a noncausal path from the predictors to the nonzero outcome through the shared disturbance, which to me would seem to bias the relationship between the predictors and the observed nonzero outcome.

Does this bias indeed occur, or is my logic or understanding faulty? Are there ways to mitigate this bias other than trying to eliminate the shared disturbance through control variables or instrumental variables?

• What "disturbance" is being shared? I don't see any "disturbance", just a probability distribution whose parameters we are modeling, possibly as a function of some variables and other parameters. The probability distribution has a particular functional form, for the ZIP having two parameters (let us say $p$ and $\lambda$), and we're modeling $p$ and $\lambda$ (in the case of the ZIP.) – jbowman Mar 13 at 23:10
• "In the nonzero part of the model, we condition on not receiving a zero." - this is definitely not true of the regular ZI poisson and ZINB models. I'm not familiar with the hurdle models, but I suspect it is true of those. – Weiwen Ng Mar 13 at 23:16
• @jbowman my understanding is that generalized linear models have a disturbance, which is the difference between the observed Y and the E[Y|X]. The conditional distribution of Y|X is the distribution of the disturbances. The Y* presentation of GLMs makes this clear. The hurdle model can be estimated as two GLMs (see stats.stackexchange.com/a/320965/116195). It's reasonable to think the disturbances would be correlated. – Noah Mar 13 at 23:31
• @WeiwenNg I mean that the interpretation of the coefficients in the poisson part of the ZIP is the effect of an increase in X by 1 unit on $\lambda$ for those in the class for which nonzero values can be observed (i.e., those not in the structural zero class). This seems to imply conditioning on membership in the class that has the capability of not receiving a zero. – Noah Mar 13 at 23:34
• To clarify the "disturbance" issue, if I have a Poisson variate with mean $\lambda$, you'd refer to the observed value of the variate as a disturbance, rather than as an observation from the distribution? And the $0$ or $1$ observed for a Bernoulli($p$) variate as well? – jbowman Mar 14 at 2:20