Turkey Hunting Survey Project Dongchu Sun

1. Turkey Hunting Survey Project Dongchu Sun University of Missouri-Columbia April 26, 2002

2. Jointly with • Chong He (UMC) • Roger Woodard, Ohio State U. • Steven L. Sheriff, MDC • John Molitor, U of S. California • Jacob Oleson, UMC • Larry Vangilder, MDC

3. Where…… • Where we were? • Where we are? • Where we go?

6. Where we were? • Preparation: 1995--1997 • He, Sun, Sheriff……. • Pilot Study: 1997--1998 • Woodard • Phase 1: 1998—2000 • Woodard, Molitor……

7. Where we are? Ph.D Dissertations • R. Woodard , 1999 Bayesian Hierarchical Models for Hunting Success Rates • J. Molitor, 2000, Bayesian Analysis for various order restricted problems

8. Articles in Refereed Journals • He & Sun (1998). EES, 223-236. • Woodard, Sun, He & Sheriff (1999). JABES, 456-472. • He & Sun (2000). Biometrics, 360-367.

9. Data • 1988, 1990, 1992, 1994, 1996 , 1998 Spring Turkey Hunting Survey • Hunting success rates during the entired spring season in 1994 ---- He and Sun (1998). • Here we use 1996 data for illustration.

10. 1996 Turkey Hunting Survey • Regulations in spring of 1996: • Two weeks • Buy license anywhere in Missouri • May hunt anywhere in Missouri • May kill one turkey per trip per week • A simple random survey collected • # of trips in each county each week • Birds taken in each county each week

11. Goals • Estimates: --- Hunting success rates per trip --- hunting pressure --- harvest • Accurate for State: --- permit buyers: 95,800 --- large sample sizes: 7,000 --- returned: 5005 (71.5%)

12. Likelihood (model the data)

13. Naïve frequency estimate of p_ij y_ij / n_ij

14. Problems • Post-stratification --- cann’t draw samples at count level • Small samples for some counties --- county 72 (4 trips in week 1) --- county 73 (3 trips in week 1) --- county 35 (0 trips in both weeks)

15. Method 1: A beta-gamma prior

17. Remarks on the Binomial- beta-gamma model • Good properties --- better than the naïve frequency estimators --- quite robust in terms of choices of (a_i, b_i) and (c_i,d_i). • Problems --- contiguity between regions ? --- hard to include covariates

18. Method 2:Hierarchical linear mixed model

19. Stage 2 prior for Method 2

20. Stage 3 prior for Method 2

21. Method 2

22. Computation via MCMC • Calculation of posterior via integration is infeasible • Iterative method • Produce random sample from joint posterior distribution

23. Model Fitting • Hyperparameters: • Estimates

25. Robustness in changing hyperparameters

26. Model Comparison • Want to examine if Method 2 can be simplified.

27. AR or CAR • He and Sun (2000), AR and CAR are equally good for estimating p_ij

28. Improving estimates by adding covariates? Woodard, Sun, He, and Sherif (1999) considered

31. Ramdom Trips n_ij • Currently we have checkstation data • k_ij : # killed turkeys in week I and county j • Woodard (1999) modeled random trips • Unde the same prior for p_ij, the estimates of p_ij does not seem ``spatially’’ correlated as these when n_ij are treated as fixed.

32. Estimating k_ij • If we can estimate k_ij, could remove check stations. • Woodard, He and Sun (2001) assumed random trips n_ij,

34. Where we go? • Can we simultaneously estimate total # of trips, harvest, hunting success rates at county level? • How about pre-stratification? • Properties of hiecarchical lines mixed models? --- Molitor, Sun and He (2000)……

35. Comments • Hierarchical Bayes models are superior to simple frequency estimates. • Normal linear mixed priors are superior to beta-gamma priors. • Spatial correlation among counties does exist.

36. More Comments • Include covariates such as forest coverage may not be helpful. • Assuming random trips will improve the estimates? • It can be used in a general small area estimation context.