# Interpretation of zero-truncated Poisson regression coefficients

For non-truncated Poisson regression, a count random variable $$Y$$ is assumed to follow a Poisson distribution
$$\mathbb{P} \left\lbrack Y = k \right\rbrack = e^{-\mu} \frac{\mu^k}{k!}, \; k = 0, 1, 2\ldots$$
with parameter $$\mu > 0$$. The expectation value of $$Y$$ is easily shown to be equal to $$\mu$$. If we want to model $$Y$$ with respect to some predictors $$x \in \mathbb{R}^p$$, we use a log link function and write
$$\label{eq:link} \log \left( \mathbb{E} \left\lbrack Y \right\rbrack \right) = \log \mu = x^T \beta$$
with a vector $$\beta \in \mathbb{R}^p$$ of regression coefficients to be esimtated. Once one has computed the coefficient estimates $$\hat{\beta}$$, it is possible to interpret the value of one single regression coefficient estimate $$\hat{\beta}_j$$ as follows, by virtue of the previous equation: if the value of the predictor $$x_j$$ increases by one, the other predictors kept fixed, then the expected count $$\mu$$ increases by a factor $$e^{\beta_j}$$.

In the context of zero-truncated Poisson regression, we consider a random variable $$Z$$ following the probability distribution
$$\mathbb{P} \left\lbrack Z = k \right\rbrack = \frac{\lambda^k}{\left(e^\lambda -1\right) k!}, \; k = 1, 2, 3\ldots$$
with $$\lambda > 0$$. When modeling $$Z$$ with respect to a set of predictors $$x$$, a log link is again chosen such that
$$\log \lambda = x^T \beta$$
(see, for instance, the R family object ztpoisson for lme4::glm described at https://rdrr.io/rforge/countreg/man/ztpoisson.html). Here, the expectation value of $$Z$$ is not equal to the distribution parameter $$\lambda$$, but amounts to
$$\mathbb{E} \left\lbrack Z \right\rbrack = \frac{\lambda}{1 - e^{-\lambda}} = \frac{e^{x^T \beta}}{1-e^{-e^{x^T \beta}}}.$$
This makes quantifying the effect of regression coefficient estimates $$\hat{\beta_j}$$ not as straightforward as for non-truncated Poisson regression.

I am wondering: Is there any intuitive way of quantifying the effect of a zero-truncated Poisson regression coefficient estimate analogous to non-truncated Poisson regression?

Sometimes the usual multiplicative effect on $$\lambda$$ has a natural interpretation. Namely, if the expectation of the underlying untruncated counts is of interest.
Moreover, it is clear that the multiplicative effect of a change in $$x_j$$ is at most $$\text{e}^{\beta_j}$$. For an observation with high $$\lambda$$ (i.e., sufficiently far away from the truncation point 0) the effect will become closer to $$\text{e}^{\beta_j}$$. In contrast, for an observation with low $$\lambda$$ (i.e., close to or even below the truncation point 0) the effect will be almost zero. But this, of course, depends on the entire regressor vector $$x$$ and not just the change in one of the regressors.