Counting Hawks

1. Counting Hawks Sylvia R. Mori PSW 2005

2. (Accipiter striatus) (Accipiter cooperii) A small, jay-size hawk (avg. 10-14" long), with dark gray back, a rusty-barred breast, a slender square-tipped tail. A medium crow-size hawk (avg.14-20" long) with a dark gray back, a rusty-barred breast, dark cap,and a long, rounded tail.

3. Immature Sharp-shinned Hawk Immature Cooper's (Accipiter striatus) (Accipiter cooperii)

4. Immature Northern goshawk (Accipiter gentilis )

5. Number of sightings Relative Abundance Cooper’s hawk Sharp-shinned hawk Northern goshawk

6. The Golden Gate Raptor Observatory (GGRO) is three staff members and more than 250 community volunteers, all dedicated to studying the autumn hawk migration The GGRO is a program of the Golden Gate National Parks Conservancy in cooperation with the National Park Service http://www.ggro.org/

7. Every autumn, thousands of migrating birds of prey appear over the Golden Gate near San Francisco, California

8. Questions • How is the Hawk population doing? Are they increasing, decreasing or the same? Statistical technique: Trend analysis • What are the proportions of misidentification for Copper’s hawks and Sharp-shinned hawks by age class and sex? Statistical technique: Estimation of misclassification probabilities

9. Counting Hawks 18 years of count data (1986-2003) • Measured variables for trend analysis: • Number of birds counted by a group of trained observers (2-20), • Year of the observation (Y), • Age class (adult, juvenal or unknown) (A) , • Time of the year in Julian days (J), • Time of the observation in hours on day of the observation (T), • Duration of the observation in minutes (D), • Number of observers’ class (1 if <7; 2 if ≥ 7) (O), • Visibility class (1if <10 miles; 2 if ≥ 10 miles) (V). Number of birds counted is the response of interest

10. Trend analysis Statistical model The parameters in the functions are estimated via Maximum Likelihood Estimation (MLE) Example: ^=estimate

11. Example

12. Julian Day of maximumabundance calculation Fix all the explanatory variables except for the Julian day J, therefore the above equality is equivalent to Using the derivative of zo with respect to J, the maximum is given by: The Julian day 298 corresponds to October 25

13. Julian Day of Maximum abundance and variance The delta method The delta method expands a (non-linear) function of a random variable about its mean with a one-step Taylor approximation, and then takes the variance. f(x) ≈ f(mu) + (x-mu)f'(mu) so that Var(f(x)) = Var(x)*[f'mu)]2 where f() is differentiable and f'() = df/dx. Expanded to vector-valued functions of random vectors, Var(f(X)) = f'(mu) Var(X) [f'(mu)]T and that in fact is the basis for deriving the asymptotic variance of maximum likelihood estimators. X is a 1 x p column vector; Var(X) is its p x p variance–covariance matrix; f() is a vector function returning a 1 x n column vector; and f'() is its n x p matrix of first (partial) derivatives. T is the transpose operator. Var(f(X)) is the resulting n x n variance–covariance matrix of f(X).

14. Vector: X=(a,b) Using the Delta Method, the estimated variance of the Julian day of maximum abundance is approximated by:

15. A statistical computing procedure estimates the parameters of the statistical model via MLE. This procedure can also estimate the variance of the estimated Julian day of maximum abundance and its 95% confidence interval:

16. Julian Day of maximum abundance and confidence intervals Unknown age Species 1 Species 2 Juvenal Adult Sept. 30