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Notes on Power and Sample Size D. Keith Williams PhD Department of Biostatistics

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Area = 0.16 1.00

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Area = 0.47 2.00

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Area = 0.81 3.00

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3.87 Area = 0.955

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Goals n Remove the ‘mystery’ of power and sample size n Introduce the main ideas n See how a ‘statistician’ views the topic n Provide information on how to do it yourself, or at least get started (Its no big deal)

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Buzzwords n Alpha (α) = P(Type I error) = P(Conclude experimental groups are different when they really are the same) n Beta ( ) = P(Type II error) = P(Conclude the experimental groups are the same when they really are different) n Power = 1 - = P(Conclude experimental groups are different when they really are!)

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Thoughts n When planning an experiment, one should determine a sample size that results in a statistical test powerful enough to declare significance for a reasonable difference in the means…if that difference truly exists in the population

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Thoughts n Generally speaking...in order to calculate power/sample size, one needs a ‘guess’ about the pattern of the population means and an estimate of their variance n Otherwise the statistician feels that they have the role of dreaming up what the population means and variances are…YIKES!

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Thoughts n α n 1- n Variance n Population means n N: n Represent the five items involved in power and sample size. n One needs to recognize that that you must input 4 of these items to get the fifth.

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One or the other…. n Input ⇒ (α, 1- , variance,population means) ⇒ gives N n Input ⇒ (α, variance,population means, N) ⇒ gives 1- n One usually ends up iterating between the above to arrive at a sample size that has a desirable level of power.

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How to Help Your Statistician Help You! n Usually a study has several questions to be answered…and a statistical test that goes with each. n Prioritize which of these are most important and arguably the ones power should be based on. n Organize your best bet on the population means and their variances…or some scenarios that are clinically important that you wish to detect (if they truly exist in the population).

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How to Help Your Statistician Help You! n Determine what the resources of the study are…how many subjects can you afford. Communicate this up front. n Try to do some preliminary power calculations on your own.

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The Non Centrality Parameter Two Group t-test

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Scenario 1 n Alpha =0.05, sigma=2 n |mu1 – mu2| = 2, that is, a two unit diff in means for a population n Propose n1 = 10 and n2 = 10

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Rejection region for two tailed t- test alpha=0.05, df = 18

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Noncentrality value =2.236, Critical value = |2.101| Table B.5, Values between 2.0 and 3.0, alpha = 0.05, df = 18 Power between 0.47 and 0.81, SAS calculation 0.56195

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Now one has a couple of choices n Decide that a 2 unit difference in the means is reasonable and you can afford 30 subjects in each group

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Rejection region for two tailed t- test alpha=0.05, df = 58

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Noncentrality value =3.87, Critical value = |2.00| Table B.5, Values between 3.0 and 4.0, alpha = 0.05, df = 58 (60) Power between 0.84 and 0.98, SAS calculation 0.97044

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Now one has a couple of choices n Decide that a 3 unit difference in the means is reasonable and you can only afford 10 subjects in each group

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Noncentrality value =3.35, Critical value = |2.101| Table B.5, Values between 3.0 and 4.0, alpha = 0.05, df = 18 Power between 0.81 and 0.97, SAS calculation 0.88621

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Now Lets Turn it Around Sample Sizes for Given Power Values n Δ = max(mu) – min(mu) n Determine k = Δ/s.d. (‘effect size’) n Use table B.12 n Different levels of alpha = 0.2, 0.1. 0.05, and 0.01 n 1 – β = 0.7, 0.8, 0.9, and 0.95 n r : number of treatments, 2, 3, …., 10

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From Our Earlier Anova Setting.. n mu1 = 20, mu2 = 15, mu3 = 15, mu4 = 12 n Δ = 20 – 12 = 8, sigma = 4 n K = Δ/sigma = 8/4 = 2, r = 4 Powern per groupTotal N 70624 80728 90936 951040

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