1. Ka-fu Wong © 2003 Lab 1-1 Dr. Ka-fu WONG ECON1003 Analysis of Economic Data

2. Ka-fu Wong © 2003 Lab 1-2 Counting the cards We are interested in knowing the number of cards in box. We only need to do to obtain a reasonable estimate of cards in the box – allowing for errors of counting or estimation.

3. Ka-fu Wong © 2003 Lab 1-3 Two examples Example #1: The box contains only a small number of cards. Example #2: The box contains a lot of cards that will take days to count.

4. Ka-fu Wong © 2003 Lab 1-4 Capture/Re-capture GOAL: 1.Illustrate that how to estimate the population size when the cost of counting all individuals is prohibitive. 2.Illustrate how easy and intuitive statistics could be. Statistics need not be completely deep, murky, and mysterious. Our common sense can help us to negotiate our way through the course. In-class Lab

5. Ka-fu Wong © 2003 Lab 1-5 History and examples of capture / recapture method Capture-recapture methods were originally developed in the wildlife biology to monitor the census of bird, fish, and inset populations (counting all individuals is prohibitive). Recently, these methods have been utilized considerably in the areas of disease and event monitoring. http://www.pitt.edu/~yuc2/cr/history.htm

6. Ka-fu Wong © 2003 Lab 1-6 The fish example Estimating the number of fish in a lake or pond. C fish is caught, tagged, and returned to the lake. Later on, R fish are caught and checked for tags. Say T of them have tags. The numbers C, R, and T are used to estimate the fish population.

7. Ka-fu Wong © 2003 Lab 1-7 Cards in a box The objective is to estimate the number of fish (represented by black cards) in a box. Capture one handful of fish (black cards). Count them and call it C. Mark the fish by replacing the black cards with red cards. Put them back into the box. Capture another handful of fish (cards). Count the total number of fish or cards (R) and the number of marked fish or red cards (T). Based on this information, How to obtain a reasonable estimate of the number of fish or cards in the box?

8. Ka-fu Wong © 2003 Lab 1-8 Cards in a box We know that C/N ≈ T/R Hence, a simple estimate is CR/T C= the number of fish or cards captured in the first round. R= the total number of fish or cards captured in the second round. T= the number of marked fish or red cards captured in the second round.

9. Ka-fu Wong © 2003 Lab 1-9 Simulations to see the properties of this proposed estimator How good is the proposed estimator? To see the properties of this proposed estimator, I have use MATLAB to simulation our Capture- recapture experiment with different numbers of capture (C) and different numbers of recapture (R), relative to the total number of fish in the pond. Throughout, N=500 and 1000 simulations

10. Ka-fu Wong © 2003 Lab 1-10 Simulation design – via MATLAB Individual simulation experiment: Create 500 “black” fish, labelled 1 to 500. Capture a random sample of C fish, mark them by converting their label to zero (i.e., red fish). Capture another random sample of R fish. Count the number of marked fish in the sample. Call it T. Compute the estimate as CR/T. If T=0, we are in trouble. Such experiments with T=0 are dropped. Repeat this experiment 1000 times. Hence, we have 1000 estimates. Compute the mean and standard deviation of these 1000 estimates.

11. Ka-fu Wong © 2003 Lab 1-11 Properties of our estimator Increasing C and R NCRSMeanStd 50040 971640.76401.57 50060 1000579.22321.54 50080 1000533.61154.67 500100 1000522.85104.29 500120 1000513.8277.41 500140 1000507.0460.98 500250 1000500.6422.93 500 1000500.000.00 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with at least one marked fish in recapture.

12. Ka-fu Wong © 2003 Lab 1-12 Properties of our estimator Constant C and increasing R NCRSMeanStd 500120401000507.8675.07 500120601000513.4079.55 500120801000508.1973.56 5001201001000511.2474.55 500120 1000510.9375.41 5001201401000511.2175.63 5001202501000510.4974.04 5001205001000507.4777.32 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with at least one marked fish in recapture.

13. Ka-fu Wong © 2003 Lab 1-13 Properties of our estimator Increasing C and constant R NCRSMeanStd 50040120961646.59405.72 500601201000582.17327.97 500801201000533.28142.23 5001001201000512.2895.40 500120 1000508.7878.75 5001401201000507.5060.61 5002501201000500.8622.38 500 1201000500.000.00 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with at least one marked fish in recapture.

14. Ka-fu Wong © 2003 Lab 1-14 Conclusion from the simulations The proposed estimator generally overestimate the number of fish in pond, i.e., estimate is larger than the true number of fish in pond. That is, there is a bias. Holding R constant, increasing the number of capture (C) helps: Bias is reduced, i.e., Mean is closer to the true population The estimator is more precise, i.e., standard deviation of the estimator is smaller. Holding C constant, increasing the number of recapture (R) does not help: Bias is more or less unchanged. The precision of the estimator is more or less unchanged.

15. Ka-fu Wong © 2003 Lab 1-15 Additional issues Our proposed estimator is good enough but it can be better. Alternative estimators have been developed to reduce or eliminate the bias of estimating N. For instance, Seber (1982, p.60) suggests an estimator of N (C+1)(R+1)/(T+1) – 1 (Note that our proposed formula is CR/T.) Seber, G. (1982): The Estimation of Animal Abundance and Related Parameters, second edition, Charles.

16. Ka-fu Wong © 2003 Lab 1-16 Simulations to see the properties of this modified estimator How good is the modified estimator? To see the properties of this modified estimator, we repeat the above simulation exercise with this new formula. (C+1)(R+1)/(T+1) – 1

17. Ka-fu Wong © 2003 Lab 1-17 Properties of modified estimator Increasing C and R NCRSMeanStd 50040 1000488.60271.05 50060 1000504.39202.16 50080 1000498.88121.47 500100 1000501.7291.20 500120 1000498.1072.01 500140 1000501.1458.44 500250 1000498.6021.72 500 1000500.000.00 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with non-zero marked fish in recapture.

18. Ka-fu Wong © 2003 Lab 1-18 Properties of modified estimator Constant C and increasing R NCRSMeanStd 500120401000498.5567.38 500120601000500.0571.54 500120801000495.5869.22 5001201001000497.0171.14 500120 1000498.4571.05 5001201401000495.1767.46 5001202501000500.4175.29 5001205001000496.7374.27 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with non-zero marked fish in recapture.

19. Ka-fu Wong © 2003 Lab 1-19 Properties of modified estimator Increasing C and constant R NCRSMeanStd 500401201000491.84291.00 500601201000499.33216.81 500801201000496.51117.05 5001001201000493.5087.53 500120 1000503.2473.65 5001401201000498.5956.30 5002501201000499.7622.58 500 1201000500.000.00 N = Total number of fish in the pond. C = number of captured fish. R = number of re-captured fish. S = number of simulation with non-zero marked fish in recapture.

20. Ka-fu Wong © 2003 Lab 1-20 Conclusion from the simulations The modified estimator performs better than the original estimator. There is no apparent bias. The estimator is more precise. Holding R constant, increasing the number of capture (C) helps: The estimator is more precise, i.e., standard deviation of the estimator is smaller. Holding C constant, increasing the number of recapture (R) does not help: The precision of the estimator is more or less unchanged.

21. Ka-fu Wong © 2003 Lab 1-21 What to take away today Statistics could be easy and intuitive. Statistics need not be completely deep, murky, and mysterious. Our common sense can help us to negotiate our way through the course. Syllabus will be distributed and discussed on Thursday 11 February 2003.

22. Ka-fu Wong © 2003 Lab 1-22 - END - In-class Lab Capture / recapture