# Towards Improving Transparency of Count Data Regression Models for Health Impacts of Air Pollution
###### Abstract  
In studies on the health impacts of air pollution, regression analysis continues to advance far beyond classical linear regression, which many scientists may have become familiar with in an introductory statistics course. With each new level of complexity, regression analysis may become less transparent, even to the analyst working with the data. This may be especially true in count data regression models, where the response variable (typically given the symbol y) is count data (i.e., takes on values of 0, 1, 2,... ). In such models, the normal distribution (the familiar bell-shaped curve) for the residuals (i.e., the differences between the observed values and the values predicted by the regression model) no longer applies. Unless care is taken to correctly specify just how those residuals are distributed, the tendency to accept untrue hypotheses may be greatly increased. The aim of this paper is to present a simple histogram of predicted and observed count values (POCH), which, while rarely found in the environmental literature but presented in authoritative statistical texts, can dramatically reduce the risk of accepting untrue hypotheses. POCH can also increase the transparency of count data regression models to analysts themselves and to the scientific community in general. count data; correlation; regression models +
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where \(res_{i}=y_{i}-\hat{y}_{i}\), the residual for the \(i-\text{th}\) observed value, and \(\sigma\) is the standard deviation of the residuals. The closer \(res_{i}\) is to 0, the higher the value of \(f(res_{i})\). If the residuals are also identically distributed (i.e., come from the same, vast, imaginary pool of residuals) and independently distributed (i.e., one residual is not useful in predicting the value of another), then the pdfs may be multiplied together to form the normal-based likelihood function:  
\[ \begin{split}&\mathcal{L}_{normal}=\prod_{i=1}^{n}\frac{1}{ \sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{res_{i}}{\sigma}\right)^{2 }\right)\\ &=\prod_{i=1}^{n}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2 }\left(\frac{y_{i}-\left(\hat{y}_{0}+\hat{y}_{1}x_{1i}+\hat{y}_{2}x_{2i}+\ldots+ \hat{y}_{n}x_{ni}\right)}{\sigma}\right)^{2}\right)\end{split}  \tag{3}\]  
The best estimates of CMM parameters may be found by adjusting them until they maximize this normal-based likelihood function. Several properties of this likelihood function allow classical regression analysis to be transparent, both to the analyst working with the data and to the general audience reviewing the published results. Maximizing the likelihood function corresponds to minimizing the sum of the squares of the residuals, and thus a plot of the resulting conditional mean shows it passing more-or-less through the middle of the scattering of observed values. One senses that shifting or rotating that best-fit line would not improve the fit. Also, the relatively simple R2, which varies from 0 to 1, and is a measure of the portion of the variation in the response variable accounted for by the conditional mean model, is visually represented in the plot. In classical linear regression, the standard deviation appearing in the likelihood function can be estimated directly from the residuals to give a fairly reasonable representation of the spread of the data, even if the residuals are not exactly normally distributed. This, in turn, allows for \(p\)-values that tend to be relatively trustworthy. There is still a risk that a covariate that is not truly associated with the response variable will have a low \(p\)-value due to mere chance. This risk increases as the number of covariates under consideration for inclusion in the CMM increases. Overfitting of the CMM (i.e., the inclusion of covariates or other complexities that represent merely random effects rather than actual associations) may then occur. However, the dataset can be divided into two subsets--training data (to build the CMM) and testing data.