Table 1 Models for non-count data adapted for count data and their limitations
From: Forecasting emergency department arrivals using INGARCH models
Model | Advantages | Disadvantages |
|---|---|---|
Normal linear regression \(y=x\beta +\epsilon\) \(\epsilon \sim N\left(0,{\sigma }^{2}\right)\) | Normal distribution approximates the Poisson distribution if the mean is higher than 20 | No possible inference on single outcomes The model allows for a negative outcome The prediction is not coherent, i.e., the forecast is not an integer-valued outcome |
Log-linear model \(\mathrm{log}\left(y\right)=x\beta +\epsilon\) \(\epsilon \sim N\left(0,{\sigma }^{2}\right)\) | The variable y is modelled as a log-normal variable | The zeros in the data have to be deleted to estimate this model, which leads to endogenous sample selection problems The prediction is not coherent, i.e., the forecast is not an integer-valued outcome There is a restriction on the conditional variance, i.e., it must be quadratic in the conditional expectation.* |
Log-linear model with constant c to deal with zeros \(\mathrm{log}\left(y+c\right)=x\beta +\epsilon\) \(\epsilon |x\sim N\left(0,{\sigma }^{2}\right)\) | The model can be estimated even if there are zero elements in the dataset | The log(y) is not linear in x, which introduces bias in the estimation of the model The prediction is not coherent, i.e., the forecast is not an integer-valued outcome |
Non-linear model \(y=\mathrm{exp}\left(\mathrm{x\beta }\right)+\upepsilon\) \(\epsilon \sim N\left(0,{\sigma }^{2}\right)\) | There is no problem in dealing with zero values | The model allows for a negative outcome The prediction is not coherent, i.e., the forecast is not an integer-valued outcome |
Ordered probit and logit state equation: \({y}^{*}=x\beta +\epsilon\) Observation equation: \(y=0\;\text{if}\;{y}^{*}<{\alpha }_{0}\) \(y=1\;\text{if}\;{\alpha }_{0}\le {y}^{*}<{\alpha }_{1}\) \(y=2\;\text{if}\;{\alpha }_{1}\le {y}^{*}<{\alpha }_{2}\) \(\vdots\) | The integer-valued structure of the data is considered The prediction can be coherent, i.e., if we wanted to forecast the future median value, it would be an integer-valued outcome | The underlying count process is not reflected The forecast is limited to values already observed in the data Complexity is excessive when the number of counts is high |