In SAS March 11, Mariya Cheryomina

 power (π) + β = 1  β = (1- π) = probability of accepting false Ho (ie. reject true Ha) - probability of Type II error – false positive  Power (π) = (1- β) = probability of detecting a difference when a difference does exist - probability of accepting true Ha (ie. reject false Ho) – how sensitive your test is to the existing difference between the compared samples

 Generally, the minimal sufficient (acceptable) value of power is 0.80  π ≥ 0.80

 1) Before gathering data  To determine the minimal sample size needed to have desired power in statistical testing (to detect a particular effect size)  2) After gathering data  To determine the magnitude of power that your statistical test will have given the sample parameters (n and s) and the magnitude of the effect that you want to detect

 Sample size (n)  Standard deviation (s)  Alpha level (α )  Size of effect/difference that you want to detect  Type of statistical test performed

 One-sample t-test  One-way Anova

proc power; onesamplemeans (type of statistical test you want the power to be calculated for) mean = ____ (difference you are interested in detecting) ntotal = ____ (sample size) stddev = ____ power = ____ ; run; *One of the four variables must be left blank – this is what you want SAS to calculate

You are asked to determine whether the use of a new soil type leads to a significantly different average height of young pine trees on plantation (compared to the historical/hypotherical mean height of 110cm recorded with the old soil type). You want to conduct a one-sample t test with a 2-sided α = You select150 trees (n). You decide that the minimal difference in height worth addressing is 8cm (effect size) α = 0.05 s = 40 n = 150 “mean” = 8 (effect size) What will the power of your statistical test be? Ho:  = 110cm Ha:  ≤ 102cm OR  ≥ 118cm

SAS text: proc power; onesamplemeans mean = 8 ntotal = 150 stddev = 40 power =.; run; *Unless you indicate otherwise, SAS will automatically assume that α = 0.05

There is a there is a probability of 0.68 that the t test will produce a significant result indicating a difference in mean tree height of at least 8cm

 Sample size  Standard deviation

You are interested in finding out how the changes in samples size (n), standard deviation (s) and minimal effect size of interest (“mean”) will effect the power of your one-sample t- test: proc power; onesamplemeans mean = 5 10 ntotal = 150 stddev = power =.; plot x=n min=100 max=200; run;

Effect size Increasing sample size does not infinitely increase power

 Example 2: You want to compare the average diversity of Canada’s native bee species in four types of habitat: 1) Urban 2) Agricultural : Monoculture plantations (with pesticide use) 3) Agricultural : Organic farms (no pesticide use) 4) National parks Ho: μ 1 = μ 2 = μ 3 = μ 4 Ha: one of μ is different

 What is the minimal number of sites of each habitat type that needs to be surveyed (ie. minimal sample size of each group) to detect whether a significant difference in bee species diversity exists between any of the four habitat types? (using a one-way Anova test with α = 0.05)  The desired power of 0.9.

*Analysis of power usually involves a number of simplifying assumptions Assumptions: Average number of bee species surveyed per site in Canada is ~35 species with a standard deviation of ~ 10 species Based on available research, you predict the following average bee species diversity for each habitat type: 1) Urban - 25 species 2) Monoculture plantations - 30 species 3) Organic farms– 40 species 4) National parks – 45 species *Assume that all groups have the same stdev (s) All numbers used in example were invented for the purpose of the exercise

proc power ; onewayanova groupmeans = 25| 30 | 40 | 45 stddev = 10 alpha = 0.05 npergroup =. power =.9; run;

SAS finds the group sample size that gives a power (actual power) closest to the power you desire (ie. to the nominal power)

 You will need to survey at least 7 locations with each of the four habitat types (ie. 7 urban sites, etc.) to detect the desired (significant) difference between the mean diversities of bee species found at each of the habitat types

Now you are trying to detect a smaller difference between sample means (at a significant level) Observe the new minimal sample size for each group

 Sample size (↑) X approaches  when n approaches N  Standard deviation (↓) Difference between samples is less likely to cur simply due to random sampling effects  (α ) (↑) Higher α leads to lower β which results in higher power (ie. the more willing you are to reject Ho, the less likely you are to accept false Ho, which leads to a higher probability of detecting a truly existing significant difference)  Minimal effect size (↑) (difference that you want to detect) A large difference between samples is less likely to occur due to random variability between samples than a small difference is

 When you are expecting a large effect size, but are not fully confident that the true effect is as large, use a larger sample size (ie. one that the analysis of power suggests for detecting a smaller effect )

 Analysis of power for Anova (in SAS):  Analysis of power for one-sample and two-sample t-tests (in SAS):