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## Statistical Power

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**Ho : Treatments A and B the same**HA: Treatments A and B different**Critical value at alpha=0.05**Points on this side, only 5% chance from distribution A. Frequency A Area = 5% A could be control treatment B could be manipulated treatment**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%**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%**What you say:**Reality**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**Effect size**A B Effect size = difference in means SD**1. Power increases as effect size increases**Power Effect size A B Beta = likelihood of type 2 error**2. Power increases as alpha increases**Power A B Beta = likelihood of type 2 error**3. Power increases as sample size increases**High n A B**Alpha**Effect size Power Sample size**Types of power analysis:**A priori: Useful for setting up a large experiment with some pilot data Posteriori: Useful for deciding how powerful your conclusion is (definitely? Or possibly). In manuscript writing, peer reviews, etc.**Example : Fox hunting in the UK**(posteriori)**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**157 plots where the fox population monitored.**Alpha = 0.05 Effect size if hunting affected fox populations: 13%**157 plots where the fox population monitored.**Alpha = 0.05 Effect size if hunting affected fox populations: 13% Power = 0.95 !**Class exercise:**Means and SD of parasite load (p>0.05): Daphnia magna 5.9 ± 2 (n = 3) Daphnia pulex 4.9 ± 2 (n = 3) (1) Did the researcher have “enough” power (>0.80)? (2) Suggest a better sample size. (3) Why is n=3 rarely adequate as a sample size?**How many samples?**PCBs in salmon from Burrard inlet and AlaskaIn an initial survey (3 individuals each), we find the following information (mean, standard deviation) Burrard – 120.5 ± 75.9 ppb Alaska – 75.2 ± 71.9 ppb The two error bars overlap, but that’s still a big difference and we only took 3 samples The difference could be “hidden” the sizes of the errors This would be reduced by increased samples, but how many should we take?**How many samples?**Our difference between (q) is ~40, therefore if our confidence limits (SE) were <20ppb, we should have adifference between populations, Burrard – 120.5 ± 75.9 ppb Alaska – 75.2 ± 71.9 ppb How many samples do we therefore need??**Re-arrange the equation…**So we should take 56 samples to be reasonably sure of a significant difference