Points on this side, only 5% chance from distribution A.

1. Critical value at alpha=0.05 Points on this side, only 5% chance from distribution A. Frequency A Duckweed lambdas Area = 5% A could be control (just duckweed) treatment B could be duckweed + high algae treatment

2. Power Analysis Theory (graphical) Conservation examples

3. If null hypothesis true, A and B are identical Probability that any value of B is significantly different than A = 5% A B Probability that any value of B will be not significantly different from A = 95%

5. If null hypothesis true, A and B are identical Probability that any value of B is significantly different than A = 5% = likelihood of type 1 error A B Probability that any value of B will be not significantly different from A = 95%

6. If null hypothesis false, two distributions are different Probability that any value of B is significantly different than A = 1- beta = power A B Probability that any value of B will be not significantly different from A = beta = likelihood of type 2 error

7. Effect size A B

8. 1. Power increases as effect size increases Power Effect size A B Beta = likelihood of type 2 error

9. 2. Power increases as alpha increases Power A B Beta = likelihood of type 2 error

10. 3. Power increases as sample size increases Low n A B

11. 3. Power increases as sample size increases High n A B

12. Example 1: Wildlife monitoring programs www.mp2-pwrc.usgs.gov/powcase

13. To detect a 3% decline with 90% power: • How many sites should you monitor? • How intensively should you monitor sites? • How many years should you monitor sites?

14. Number of sites needed to detect 3% decline with 90% power Intensity Number of sites needed

15. Example 2: Fox hunting in the UK

16. Hunt banned (one year only) in 2001 because of foot-and-mouth disease. • Can examine whether the fox population increased in areas where it used to be hunted (in this year). • Baker et al. found no effect (p=0.474, alpha=0.05, n=157), but Aebischer et al. raised questions about power. Baker et al. 2002. Nature 419: 34 Aebischer et al. 2003. Nature 423: 400

17. 157 plots where the fox population monitored. If hunting had no effect, expect 50% of plots to show increases, 50% to show decreases. If hunting had an effect, expect 63% of plots to show increase (as 63% of UK surface area hunted). Effect size if hunting affected fox populations: 13% (63%-50%)

18. 157 plots where the fox population monitored. If hunting had no effect, expect 50% of plots to show increases, 50% to show decreases. If hunting had an effect, expect 63% of plots to show increase (as 63% of UK surface area hunted). Effect size if hunting affected fox populations: 13% (63%-50%) Power = 0.95 !