# Towards Improving Transparency of Count Data Regression Models for Health Impacts of Air Pollution
## 2 Illustration of False Inference and Overfitting Due to pmf Misspecification  
The consequences of misspecifying the pmf in count data regression analysis can be seen in our own analysis of the relationship between air quality and childhood asthma in Houston, Texas, during the summers of 2003-2011. Concentrations of aeroallergens (mold and pollen) and anthropogenic contaminants (butane, nitrous oxide, ozone, sulfur dioxide, and particulates) were initially included in the model as covariates. The number of children arriving per day at particular hospital emergency departments for asthma was the response variable. We initially assumed the Poisson distribution for the pmf. With this pmf, a strong association between the response variable and the mold concentration was found, with \(p\)-value \(<10^{-15}\). However, the most appropriate CMM and pmf among those being considered may be identified as that which yields the lowest Akaike information criteria (AIC) value [6]  
\[AIC=-2\cdot ln(\mathcal{L})+2\cdot k \tag{5}\]  
where \(\mathcal{L}\) is the likelihood function value for the selected pmf, and \(k\) is the number of parameters that may be adjusted to increase \(\mathcal{L}\). The second term is thus a way of penalizing the inclusion of parameters, as including an additional adjustable parameter will always increase \(\mathcal{L}\), even if the parameter is not truly representative of actual statistical relationships. Variations of the AIC may also be used. We use the original AIC here because it is commonly  
Figure 1: Emergency department childhood asthma arrivals in response to mold during summers of 2003–2011 in Houston, Texas. available in software packages. The chooseDist() function of the R gamlss package [7] runs through dozens of pmfs for building likelihood functions, adjusts parameters to maximize each, and then identifies the one with the lowest AIC. By using this process, dozens of pmfs were found, which yielded a lower AIC than did the Poisson distribution. An alternative pmf, the zero-inflated Poisson, which allows for a higher number of zeros than would be expected for the Poisson and thus, in turn, has a substantially broader spread than the Poisson would show for our dataset, was found to yield the lowest AIC among the dozens of available pmfs. The resulting \(p\)-value for the mold covariate was now 0.051, a \(p\)-value increase of many orders of magnitude compared to that provided by the Poisson pmf, leading the mold covariate to be accepted as statistically significant only under far less strict criteria. Figure 1 shows a plot of the best-fit line through the data based on the Poisson pmf (gray) and the zero-inflated Poisson pmf (green). Due to the non-linearity of the CMM and other factors, one would be hard-pressed to say whether either of the lines fits the data well, let alone which fits the data better to justify the use of one CMM or pmf over the other. Indeed, as we will see in the following discussion of the generation and analysis of synthetic data, radically different pmfs may yield essentially identical CMMs, completely eliminating the usefulness of plots, such as in Figure 1, in determining which pmf is superior. To show that the impact of pmf misspecification on \(p\)-values is not unique to peculiarities of the somewhat small air quality and childhood asthma dataset we ourselves are working with, we developed a synthetic dataset that readers are free to view, re-generate with parameters of their choice, and re-test through the link provided in the data availability statement below. Figure 2 shows how we generated the synthetic dataset and how the reader could use the code to generate their own. The three blocks forming the left column of the schematic are all the reader would need to select to build the synthetic dataset.