Figure 1. Cooper’s Hawk, Accipiter cooperii. Image source: Flickr

Figure 2. Sharp-shinned Hawk, Accipiter striatus. Image source: Pixels

Figure 3. Red-tailed Hawk, Buteo jamaicensis. Image source: SVCnews

Introduction

In order of increasing size, sharp-shinned, Cooper’s, and red-tailed hawks are common raptors in North America, and all three are commonly known as sparrowhawks or chickenhawks, for their propensity to feed on smaller bird (Oldham, 2019). All of them, and especially the red-tailed hawk, however, also feed on rodents and other small mammals, large insects, lizards and other reptiles and amphibians, and in the case of red-tailed hawk, even carrion (Animal Spot, 2014; Animalia, 2020). Though not adept fishers, red tailed hawks have also been known to eat crustaceans and fish. It has long been suggested that rostrum biometrics, such as the length of the culmen - the upper ridge of the bill from the flesh to the tip - for example, may be a result of prey-specialisation adaptation(Huzley, 1942; Schoener, 1965). As these birds are predatory, and I had reliable data of culmen length on 892 observations, I wanted to see if I could culmen, as an aspect of gape, had any infuence on weight of a bird. Knowing that these birds are in different size classes, from the quick and acrobatic sharp-shinned hawk to the large, soaring red-tailed hawk, I inspected the grouped species and weight to determine their interation with culmen length. This has a practical application, when it comes to determining the ecology of a novel species, as well as a potential metric for either obscure species identification in the field, or confirmation of identification upon examination of a dataset.

Research Question

Is there a relationship between the culmen length, measured along the top of the rostrum, and species-specific weight, amongst hawks sampled at Lake MacBride, Iowa?

Data Description

The data for this report was gathered over 908 observations of hawk, all identified to species and many to sex and age, between the months of September to December over the course of 1992 through 2003, at Lake MacBride near Iowa City, Iowa. Special thanks to students and faculty at Cornell College in Mount Vernon, Iowa, for their work in capturing, measuring, and compiling data for these birds, and Professor Bob Black for making the dataset available. Biometrics on the three species where n>1; sharp-shinned hawk, Cooper’s hawk, and red-tailed hawk were taken on 6-10 variables. The variables examined in this report are: body weight of the hawk, in grams; length of the culmen, in mm; and species. The samples are purported to be random, but I have no way to confirm this.

Summary Figures

For a preliminary examination of the data, culmen length and hawk weight have been rendered in a scatterplot.
# Plot culmen by weight
ggplot(Hawk, aes(x=Culmen, y=Weight))+
  labs(title="Relationship of Culmen Length and Weight for Hawks at Lake MacBride.",
       x="Culmen Length (mm)", y="Weight (g)")+
  geom_jitter()+geom_smooth(method=lm)
## `geom_smooth()` using formula 'y ~ x'
Figure 4. Scatterplot of relationship between Culmen Length and Hawk Weight. There is extreme clumping about three or four locations along a linear trajectory, which is probably the result of difference species' body morphologies. There may be an instance of sexual dimorphism or age-class distinction that causes the additional segregated plot locality.

Figure 4. Scatterplot of relationship between Culmen Length and Hawk Weight. There is extreme clumping about three or four locations along a linear trajectory, which is probably the result of difference species' body morphologies. There may be an instance of sexual dimorphism or age-class distinction that causes the additional segregated plot locality. 

# # # note: code intentionally left visible
A second scatterplot was rendered, introducing species to the equation.
# Render scatterplot of Culmen length by Weight across species w/ best fit line
ggplot(Hawk, aes(x=Culmen, y=Weight, color=Species))+
  labs(title="Relationship of Culmen Length and Weight by Species of Hawk",
       x="Culmen Length (mm)", y="Weight (g)")+
  geom_jitter()+geom_smooth(method=lm)
## `geom_smooth()` using formula 'y ~ x'
Figure 5. Scatterplot of relationship between Culmen Length and Hawk Weight, grouped by Species. When plotted by species, the distinct body morphologies becomes immedietly apparent. The species-specific interaction of culmen length and weight appears to have a linear relationship, but requires thorough examination.

Figure 5. Scatterplot of relationship between Culmen Length and Hawk Weight, grouped by Species. When plotted by species, the distinct body morphologies becomes immedietly apparent. The species-specific interaction of culmen length and weight appears to have a linear relationship, but requires thorough examination.

Checking Assumptions

The assumptions for linear regression are many: does it vary across category? We can see from the scatterplot that it does. Are the X and Y variables continuous? Yes. Are they expirimentally independant of each other? Yes. Now, for the following charts: Is the relationship linear within the examined range? Are the reiduals normally distributed? Is the variance homoscedastic? The linear plot follows.
# Testing assumptions of linear regression
lm(Weight ~ Culmen * Species, data =Hawk)
## 
## Call:
## lm(formula = Weight ~ Culmen * Species, data = Hawk)
## 
## Coefficients:
##                 (Intercept)                       Culmen  
##                    -365.846                       44.721  
##           SpeciesRed Tailed         SpeciesSharp Shinned  
##                     106.420                       96.162  
##    Culmen:SpeciesRed Tailed  Culmen:SpeciesSharp Shinned  
##                       5.482                       -8.294
Linear_Regression_Model<-lm(Weight ~ Culmen * Species, data =Hawk)
 plot(Linear_Regression_Model)
Figure 6, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook's Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook's Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook's Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of linear regression.

Figure 6, A-D. Residual and Cook's Distance Plots for testing assumptions of linear regression.

The assumptions being tested did not have great results. The Figure 6A shows real heteroscedasticity, with more closely clumped variables giving way to more loosely clumped variables along the regression line. Figure 6B (Normal Q-Q) shows a poor linear fit at either extreme of the examined range. Lastly, Cook’s Distance plot, Figure 6D (Residuals vs Leverage), has an outlier beyond the accepted distance of 0.5. As a result, the numeric values have been transformed, and we will now check the same assumptions in a linear regression with a power model in log transformation.
# Testing assumptions of log-transformed linear regression
lm(log(Weight) ~ log(Culmen) * Species, data =Hawk)
## 
## Call:
## lm(formula = log(Weight) ~ log(Culmen) * Species, data = Hawk)
## 
## Coefficients:
##                      (Intercept)                       log(Culmen)  
##                           1.4929                            1.5649  
##                SpeciesRed Tailed              SpeciesSharp Shinned  
##                           0.8995                           -1.6117  
##    log(Culmen):SpeciesRed Tailed  log(Culmen):SpeciesSharp Shinned  
##                          -0.1716                            0.5121
Log_Linear_Regression_Model<-lm(log(Weight) ~ log(Culmen) * Species, data =Hawk)
 plot(Log_Linear_Regression_Model)
Figure 7, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook's Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook's Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook's Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook&#39;s Distance Plots for testing assumptions of log-transformed linear regression.

Figure 7, A-D. Residual and Cook's Distance Plots for testing assumptions of log-transformed linear regression.

The transformation is not perfect, but it works well. The data in Figure 7A is now much more homoscedastic. While clumped horizontally, the regression distances are much more constant. The relationship is linear along the regression line (Fig. 7B) to a greater extent, but we still have some variables at the low end of fitted values that have excessive residual distance. Lastly, the Cooks distance (Fig. 7D) no longer returns any outliers. Having met the assumptions, we can proceed to run the test.

Statistical Test

Summary statistics for the log-transformed linear regression (power model) of log(weight) by way of log(culmen length) and species.
# run statistical analysis
summary(Log_Linear_Regression_Model)
## 
## Call:
## lm(formula = log(Weight) ~ log(Culmen) * Species, data = Hawk)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.25666 -0.07120  0.00288  0.09073  0.70964 
## 
## Coefficients:
##                                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                        1.4929     0.5153   2.897  0.00386 ** 
## log(Culmen)                        1.5649     0.1802   8.685  < 2e-16 ***
## SpeciesRed Tailed                  0.8995     0.6315   1.425  0.15465    
## SpeciesSharp Shinned              -1.6117     0.5585  -2.886  0.00400 ** 
## log(Culmen):SpeciesRed Tailed     -0.1716     0.2116  -0.811  0.41753    
## log(Culmen):SpeciesSharp Shinned   0.5121     0.2007   2.551  0.01090 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2048 on 886 degrees of freedom
## Multiple R-squared:  0.9548, Adjusted R-squared:  0.9545 
## F-statistic:  3741 on 5 and 886 DF,  p-value: < 2.2e-16
The log-transformed linear regression resulted with a very high F-statistic for the given degrees of freedom, and a very low P-value (F= 3741; df= 5, 886; P< 2.2x10-16). We can safely reject the null hypothesis that there is no relationship. There is in fact a significant relationship, with regards to the logistic culmen length and the weight by way of species. Additionally, the adjusted (standardised) coefficient of determination is quite high (R2=0.955), indicating that the predictor variables and response variable fit well along the regression line.

Discussion

Ultimately, we wanted to take a look at what the culmen, the upper ridge of the rostrum, might say about the hawk’s specific capacity to acquire and consume prey. In order to ensure they were bird-specific and not species-specific metrics, we included the interaction of culmen and species, to illuminate the influence that culmen, as grouped by species, has on the individual’s weight. The results indicated a strong linear relationship when log transformed, which makes sense, as the data was positive (right-) skewed, due to the natural laws of lifespan and growth. Once log transformed, we still found some indivduals contributing large residuals to the model at either end, and these are likely the result of compounding factors such as disease, territory, or as barely visible in the Figure 5, indicitive of a possible mis-identification. Most raptors have variable eye color, morphs, and ecotypes, which can result in misidentification without expert experience, and student researchers are not infallible.
Finally, because these birds were collected near a lake, there may have been a higher than normal incidence of squirrel predation and carrion take, due to the proximity to human activity, and fish directly from the water. This may be specific ecotypes that are better suited to the lake and surrounding environs than would otherwise be found over a broader study range, resulting in less of a prey-specific rostrum development.

References

Animal Spot. 2014. Sharp Shinned Hawk Facts, Habitat, Diet, Call, Images. https://www.animalspot.net/sharp-shinned-hawk.html.
Animalia. 2020. Cooper’s Hawk - Facts, Diet, Habitat & Pictures on Animalia.bio. http://animalia.bio/coopers-hawk.
Flickr user: Gary Fairhead. 2011. Coopers Hawk Portrait. https://www.flickr.com/photos/15576619@N07/13682890834/.
Huxley J. 1942. Evolution The Modern Synthesis. http://archive.org/details/in.ernet.dli.2015.280031.
Lack D. 2008. Ecological Aspects of Species-formation in Passerine Birds. Ibis. 86(3):260–286. doi:10.1111/j.1474-919X.1944.tb04092.x.
Oldham C. 2019. Red Tailed Hawk - Description, Habitat, Image, Diet, and Interesting Facts. Animals Network. https://animals.net/red-tailed-hawk/.
Pixels; Sharp-shinned Hawk by James Zipp. 2019. https://pixels.com/featured/2-sharp-shinned-hawk-james-zipp.html.
Schoener TW. 1965. The Evolution of Bill Size Differences Among Sympatric Congeneric Species of Birds. Evolution. 19(2):189–213. doi:https://doi.org/10.1111/j.1558-5646.1965.tb01707.x.