Data Notes

Note: The data used in this document are imported from retrospective_traffic_data_20200401-c.xlsx, a file obtained from Stantec technical consultant.

PVM counts are from the state of Florida. 1995-2015 annual panther counts are from Figure 4 in McBride & McBride (2015). 2015 was the last year of the annual panther counts and 1995 was chosen as a starting year due to Texas Cougar introductions.

Data Visualization

Time series plots

Scatterplots

Scatterplot with SLR fitted line

Fit model and look at estimates

#Not accounting for serial autocorrelation
ll_lm_out <- lm(ln_PVM ~ ln_annual_count, data=pop_PVM_df)
summary(ll_lm_out)
## 
## Call:
## lm(formula = ln_PVM ~ ln_annual_count, data = pop_PVM_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.91112 -0.13790 -0.02463  0.18178  0.61237 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)
## (Intercept)      -5.5604     0.6629  -8.388 1.24e-07
## ln_annual_count   1.7314     0.1496  11.571 9.05e-10
## 
## Residual standard error: 0.3246 on 18 degrees of freedom
##   (3 observations deleted due to missingness)
## Multiple R-squared:  0.8815, Adjusted R-squared:  0.8749 
## F-statistic: 133.9 on 1 and 18 DF,  p-value: 9.048e-10

Backtransform results for interpretation

2^ll_lm_out$coef[2] #3.32
## ln_annual_count 
##         3.32055
2^(confint(ll_lm_out)[2,]) #2.67, 4.13
##    2.5 %   97.5 % 
## 2.670389 4.129008
with(dplyr::filter(pop_PVM_df, year < 2016), cor(ln_PVM, ln_annual_count)) #0.9388
## [1] 0.9388771

Check residuals to assess assumptions

Check for autocorrelation in residuals

Summary of SRL results

A doubling in the annual count is associated with a 3.3 times increase in median reported PVM (95% confidence interval from about 2.6 to 4.1 times.)

The p-value associated with a statistical hypothesis test associated with the null hypothesis of a slope of zero is less than 0.00001, under the assumed model. The observed correlation coefficient for the log transformed variables is 0.94 (R-squared from SLR is 0.88); the strong, positive association is clear from the above figures. The linear relationship on the log scale is clearly positive and strong. Temporal autocorrelation is not accounted for because autocorrelation in the residuals does not look concerning (see above acf and partial acf plots). Violations in other assumptions not deemed serious enough to affect conclusions, despite the response of small counts. The log transformation is reasonable in this case given the fact there are no PVM counts of zero.

Poisson log-linear regression

Fit model and look at estimates

#No accounting for serial correlation
glm_out <- glm(PVM ~ ln_annual_count, data=pop_PVM_df, family="poisson")
summary(glm_out)
## 
## Call:
## glm(formula = PVM ~ ln_annual_count, family = "poisson", data = pop_PVM_df)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -1.51495  -0.47904  -0.05553   0.52427   1.05746  
## 
## Coefficients:
##                 Estimate Std. Error z value Pr(>|z|)
## (Intercept)      -6.2411     1.0804  -5.777 7.62e-09
## ln_annual_count   1.8871     0.2299   8.209 2.22e-16
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 108.4034  on 19  degrees of freedom
## Residual deviance:   9.9605  on 18  degrees of freedom
##   (3 observations deleted due to missingness)
## AIC: 92.648
## 
## Number of Fisher Scoring iterations: 4

Backtransform for interpretation

2^(coef(glm_out)[2]) #3.69
## ln_annual_count 
##        3.698862
2^(confint(glm_out)[2,]) #2.74, 5.12
##    2.5 %   97.5 % 
## 2.740514 5.121466

Summary of Poisson log-linear results

Results are very similar and consistent with the SLR model, as expected. There is some evidence of under-dispersion in comparing the residual deviance to the residual df (less variance in the residuals than expected under the Poisson model). Temporal autocorrelation is not modeled, as there is little evidence in the residuals from the SLR model and there is already underdispersion - p-value may be larger and confidence interval wider than if Poisson model was a better fit.

A doubling of panther annual count is associated with an estimated 3.7 times increase in mean PVM (95% confidence interval from about 2.7 to 5.1).

Based on both models, a doubling of panther annual count is associated with an estimated 3 fold increase in PVM, with an associated 95% confidence interval from about 2.5 to 5 times.