# Towards Improving Transparency of Count Data Regression Models for Health Impacts of Air Pollution
## 3 The Predicted-And-Observed Count Histogram  
Figures 3 and 4 are predicted-and-observed count histograms (POCH), similar to what is presented but not formally named in authoritative count data regression analysis texts [1; 2]. Black dots and other markers are where the tops of the more traditional vertical histogram bars would be to represent the number of times that the response variable takes  
on the count value. The black dots represent the number of occurrences of the observed response variable values, while the green and gray markers represent the number of occurrences predicted by the models. In Figure 3, for example, the black dot at the count of 0 indicates that there were 0 childhood asthma emergency department arrivals on 57 of the summer days, while the gray square indicates that the Poisson pmf anticipates 0 arrivals to occur on only 40 of the summer days. Figure 3 shows that while the Poisson and zero-inflated Poisson had similar performance in predicting the number of days for which three or more arrivals occur, the zero-inflated Poisson pmf was a substantial improvement overall for the lower arrival numbers. In Figure 4, the model having two covariates and using the negative binomial pmf was clearly a better fit than was the three-covariate model with the Poisson pmf. Though not shown in either POCH, the analyst may generate predicted values from the pmf that best fits the observed histogram directly, i.e., without a CMM, note the resulting AIC value, and thus have a baseline AIC value from which to develop the CMM.  
In both Figures 3 and 4, the POCH shows Poisson pmfs (as opposed to the zero-inflated and negative binomial pmfs) have a more narrow distribution than does the actual data. It thus clearly warns that \(p\)-values with the Poisson pmf for these particular datasets will be falsely low. Such charts immediately provide transparency of the complicated count data regression analysis to the analyst working with the data and to the broader audience.  
The POCH is easily generated for even the most complex count data regression analysis models, including ones that incorporate smoothing splines, autoregressive parameters, etc., as in generalized additive models and models in which the count data is binary, such as in case-crossover studies. The POCH merely requires a predicted response variable and a representation of the distribution of residuals, and so can be developed even for a quasi-likelihood method [9].  
A POCH helps assess the correctness of the pmf not only in regards to spread, but also in regard to skewness, zero-inflation, hurdles, and other potentially important features. A POCH will not entirely address every violation of statistical assumptions. For example, one still needs to check for autocorrelation among residuals. However, where the POCH does not directly address them, it may provide a solid starting point. For example, testing for autocorrelation in count models requires standardizing the residuals before plotting the autocorrelation function [2; 10]. The POCH can help identify the correct pmf for the stan  
Figure 3: Predicted-and-Observed Count Histogram for modeling of emergency department arrivals with mold as the covariate, for summers from 2003–2011 in Houston, Texas.  
dardization. Once the final model is selected, perhaps including autoregressive parameters, the POCH should be re-generated to re-confirm the appropriateness of the model.  
In our literature review of the impact of air quality on health, we found no histogram such as a POCH, or any explicit evidence that the most appropriate pmf was used. This absence even among excellent articles [11; 12; 13; 14; 15; 16; 17; 18] suggests a systemic issue extending beyond individual authors. We recommend that publishers require a POCH for articles involving count data regression models.