# Zero-inflated Poisson Regression for Continuous data

I have a continuous variable that I'm trying to model but have a number of issues. The variable is continuous, positive, right skewed and has a large zero-inflation. Whilst the formulation of the score means that they are technically continuous, in practice they are measured (and rounded), results are typically more categorical in nature. In addition, the zero inflation is part of the process and is a well documented phenomenon.

I have seen some conflicting advice for how to model this. I've tried modelling by categorising the variable into groups and model using a zero-inflated Poisson regression. But I'm told that categorising is the wrong thing to do, and to model it as continuously. I've thought about modelling using a hurdle model (Gamma/Lognormal with binomial for zeroes) however the assumption there is that zero isn't part of the data generation process and that the zeroes are a separate process which seems to violate the above.

Any advice about the best way to model this data, with any references to support it would be great.

Important takeaways for analysts:

1. Categorizing a continuous predictor will always result in a loss of information
2. "A lot of zeroes" is not 0 inflation, regardless of skew. I can find a negative binomial parametrization that can generate an arbitrary proportion of zeroes and an arbitrarily high maximum. 0 inflation is a theoretical process, like getting input from a broken Geiger counter: you have to know what's going on in the data
3. The usual way of handling real zero inflation is with a mixture model
4. Be especially cautious of truncation... that is when the instrument is not sensitive to detect values below a lower limit of detection and the result comes out to 0. An EM algorithm or Bayesian approach is needed
5. Zero-inflated Poisson is the most frequently cited zero-inflated model. It uses a mixture model for the 0s, and the Poisson GLM for the "non-zero part" (a misnomer because some of the positive-mean values may be 0)
6. You can have a zero-inflated "anything" model by using an EM fitter to iteratively predict the 0s that are 0-inflated and the effects for the non-0-inflated part
7. A Poisson GLM is completely reasonable for a continuous response provided: a) the log of the mean response is related to a linear combination of regressors and b) the variance is equal to the mean. But I gather this almost certainly not what you would have done, I think you were distracted by the oft cited 0 inflated Poisson model.
• Thanks for this. The variable I'm modelling is a composite score based on a clinical questionnaire and the overinflation of zeroes are a well documented phenomenon, which is fine. I'm also looking into the zero-inflated negative binomial as well to model any overdispersion in the data. I'm mainly confused to whether categorising the score is the correct thing to do as I can't find any references to it, modelling it as a continuous variable in a hurdle model seems like the wrong thing to do... Getting conflicting advice... Feb 17, 2020 at 15:53
• @Thomas Is the clinical questionnaire a validated instrument? In other words, has the sum score been demonstrated to have the appropriate properties? If so, then a simple a simple comparison of mean response via ANOVA or linear regression is a justified and powerful approach to analysis. In any case, I think this answers all your questions about zero-inf Poisson models, and I think we agree it's not the model for your data. Feb 17, 2020 at 16:06
• Yes the clinical questionnaire is validated and has long been used in the field. Just to be clear, you're saying I shouldn't be using the ZI-Poisson or the ZI-NegBin models? If so, how would you suggest modelling this variable in my setting? Feb 17, 2020 at 16:36
• @Thomas reread my previous comment Feb 17, 2020 at 16:44
• Are you able to expand a little please? I'm a little confused. Your help is much appreciated! Feb 17, 2020 at 17:00