## 3 Results
### Generalized Additive Model for Nonlinearity  
The GAM results have a few implications (Table 5). First, it further reduces the AICc by 2.04 from 440.90 to 438.86, which suggests that adding non-linear smoothing terms improved the fitness of the model. Note that the pseudo-R\({}^{2}\) reported here is not comparable to those from negative binomial linear models. It is offered to show how much the model improves over the null model. Second, many predictors still show linear relationships. Of particular interest is the temperature. Although the general trend fitting in the \"General Pattern and Correlation\" section visually shows a non-linear relationship between ambient temperature and the number of questing ticks, such nonlinearity does not hold well in the GAM. This is because GAM penalizes adding complex smoothing terms, which is similar to adding more predictors to a linear regression. When the temperature is modeled as non-linear, the improvement to the model fitness does not offset the cost of adding more variables. Therefore, temperature turns out to be a linear predictor (Figure 7). By contrast, the relative humidity has a strong non-linear relationship. An EDF value of six means there are six knots or turning points in the fitted relationship (Figure 7). Again, this is different from the general trend in the \"General Pattern and Correlation\" section. However, the complex curve was estimated by the GAM and the smoothed term could significantly improve model fitness even with higher degree of freedom. Note that the y axis in Figure 7 is transformed by the negative binomial function, therefore, different from the y axis in Figures 5 and 6.  
The generalized additive model (GAM) allowed us to assess the nonlinearity of the environmental factors. Overall, the relative humidity has a strong non-linear relationship with the modeled quantity of questing ticks, as well as the percentage of the dwarf shrubs land cover. Other statistically significant factors, including temperature and other types of vegetation, reveal a linear relationship, which is rather unexpected but could be explained  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Variable** & **EDF** & \(\mathbf{\chi^{2}}\) & \(\mathbf{p}\)**-Value** \\ \hline Temperature & 1 & 64.786 & \textless{}0.001 *** \\ Relative Humidity & 6.05 & 29.834 & \textless{}0.001 *** \\ Vegetation Land Cover Percentage & & & \\ _Mixed Hardwood_ & 1 & 12.029 & \textless{}0.001 *** \\ _Dwarf Shrub_ & 4.002 & 22.226 & \textless{}0.001 *** \\ _Grassland_ & 1 & 7.038 & 0.008 ** \\ _Pine Forest_ & 1 & 5.973 & 0.015 * \\ \hline \hline \end{tabular}  
* EDF = effective degree of freedom. An EDF of 1 means the model fits a linear relationship between the independent and dependent variables. A higher EDF implies more complex non-linear relationships. \(\chi^{2}\) is the test statistic for the significance of the smoothing terms. \(AIC=438.86\), _Pseudo \(-R^{2}=0.52\)_. Significance Level: 0 ***+- 0.001 ***+- 0.01 ***+- 0.05.  
\end{table}
Table 5: Generalized Additive Model Estimation.  
by the distribution of the sample data. The linearity of ambient temperature, for instance, mainly results from the bimodal distribution of lower temperature in the spring and higher temperature in the fall.