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?


Unfortunately not. There is (to the best of my knowledge) no easy ceteris paribus interpretation of the coefficients' effect on the expectation of the zero-truncated response.

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.