## 3 Results
### Model Diagnostic Results  
The dispersion statistic from the Poisson model was greater than one (1.68), eliminating the model from the selection process due to overdispersion. Quasi-Poisson, negative binomial, and zero-inflated negative binomial (ZINB) models were then created to address overdispersion and the high number of zeroes in the data. The examination of the deviance residual indicated that the quasi-Poisson model was not the best fit for the data, and the model was eliminated. The quasi-Poisson model displayed residuals in the negative range of values, which should not occur with count data since negative values are not present. A review of the negative binomial model revealed similar issues that were present in the quasi-Poisson model. Residuals for the negative binomial model were also in the negative range of values. The ZINB model had a band of residuals close to zero, but that is expected, given how the model adjusts for excess zeroes. Furthermore, the residual plot points did not range into negative values the same way the quasi-Poisson and negative binomial models did. The fitted vs. observed plot also indicated ZINB to be the better model, as the predicted and observed values were closest to the fit line. The AIC for the ZINB model (193.89) was \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Species Count**} & **HPAI\({}^{+}\) Counties** & \multirow{2}{*}{**Median**} & **HPAI\({}^{+}\) Counties \textgreater{}** & \multirow{2}{*}{\(\chi^{2}\) (_p-Value_)} \\  & & **Count** & & & **Median** & \\ \hline Canada goose counts greater & & & & & \\ than or equal to median (full & 2,369,666 & 310,198 & 3947 & 9 (32\%) & 4.29 (0.04) \\ \multirow{2}{*}{Canada goose counts greater & & & & & \\ than or equal to median 2022 & & 1,293,380 & 152,166 & 2363 & 11 (39\%) & 1.39 (0.24) \\ \multirow{2}{*}{Canada goose counts greater & & & & & \\ than or equal to median 2023 & & 1,076,316 & 158,032 & 1600 & 11 (39\%) & 1.39 (0.24) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Descriptive statistics and Pearson \(\chi^{2}\) test. Figure 3: Frequency distribution of HPAI outbreaks in all Iowa counties during the reference period, 2022—2023. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Year**} & **Canada** & **Mallard** & \multirow{2}{*}{**Wood Ducks**} & **Blue-Winged** & **Green-** \\  & **Geese** & **Ducks** & & **Teals** & **Winged Teals** & **Snow Geese** \\ \hline
2022 & 1,293,380 & 334,310 & 28,872 & 66,320 & 67,224 & 568,426 \\
2023 & 1,076,316 & 297,374 & 28,649 & 57,007 & 68,878 & 433,246 \\ Total & 2,369,666 & 631,697 & 57,521 & 123,327 & 136,102 & 1,001,672 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Waterfowl species counts during the study reference period. lower than the AIC for the negative binomial model (212.01). BIC for the ZINB model was (263.96) was higher compared to the negative binomial model (248.34). A Likelihood ratio test was then employed to evaluate which model performed better. The results indicated the ZINB model was a better fit for the data than the negative binomial model, as the ZINB model yielded a significant _p_-value. (Likelihood ratio \(p\) = < 0.01). The combination of small discrepancies between BIC values, better plot performance, Likelihood ratio test results supporting ZINB, and 72% of the outcome data containing zeroes led us to select the ZINB model over the negative binomial model. Before evaluating the variables that resulted in the best model fit, the full model was summarized. The summarization revealed that the full model produced NaNs (not a number). A VIF (variance inflation factor) test to evaluate collinearity was conducted, but the test did not provide evidence of collinearity. In an attempt to identify which variables were contributing to NaNs, each variable was excluded, one by one, from subsequent models. This revealed that including the green and blue teal variables resulted in unstable ZINB models.