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EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 1 §3. Sample Size Estimation A key to study design are sample size or “power” calculations. Required of ever grant proposal In this section: (1) we begin with theory behind power calculations and demonstrate how simple formulae for power and sample sizes are derived. (2) Next, show unified treatment of power for RD, OR, RR based on this theory. (3) Then, describe how varying the question being asked can have substantial effect on the required sample sizes. (4) Brief explanation of the information needed for power calculations for matched pair studies. (5) Some demonstrations on how to use and interpret software for power calculations. Goals – To be able to understand what affects power, how to define the problem, and how to get the computer to give you the answer you need. EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 2 §3.1 Power In General Sample Size Estimation: Terminology Review Null hypothesis (Ho): specified value for a parameter (OR, RR, RD, IRR, IRD, for example) Alternative hypothesis (Ha): specified alternative value for a parameter Type I error = Pr(Reject Ho | Ho is true) = α Type II error = Pr(fail to reject Ho | Ha is true) = β Pr(fail to reject Ho | Ho is false) Power = Pr(reject Ho | Ha is true) = 1- β 1-α = ? (“Pr” signifies probability over repetitions of the study) (References: Woodward, chap 8; Rothman and Greenland, pp. 184-8)EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 3 Notes: (1) α-level is not a p-value. P-value is a quantity computed from and varying with data. α is fixed and is specified without seeing the data. (2) p-value is not the Pr(Ho vs Ha). Is loosely defined: Pr(observed result or more extreme than observed|Ho true). (3) p-value is not Pr(data|Ho). That is the likelihood. Likelihood is usually much smaller than the p-value, because p-value includes not only Pr(data|Ho) but also the Pr(all other more extreme data configurations Ho). (4) Absence of evidence is not evidence of absence. Failing to reject Ho ≠ accept Ho as true. (5) Studies with low power to produce results with appropriately narrow confidence intervals (as defined by the purpose of the study) are not “negative studies” – they are “indeterminate”. An initial description of what we are doing will help. EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 4 Type I error (α) -- H0 is true but you will reject H0 in favor of Ha. Suppose that 2 is your threshold (critical value) for rejecting H0. So, you have only a very small chance of observing a value to the right of 2, and a large chance of observing something to the left of 2, if H0 is true. . . . 0 2 3 4 H0 HaEP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 5 Type II error(β)– If Ha is true, then you have a chance of observing a value to the left of 2, below the critical value, but it is not great. You have a much larger chance of observing a value to the right of 2. How big a chance you have of observing a value at 2 or to the right of 2, if Ha is true depends upon how far Ha is away from H0. If Ha is far away, then power is bigger, and type II error is smaller. Now what happens when sample size increases (or when variance decreases). The distributions become narrower. (This is the distribution of the mean, for example). Holding everything else constant, what does that do to my power to detect a difference? At 2, I have little chance of falsely rejecting H0. This would be a very high critical value for rejecting H0. But if Ha is true, you have an almost certain chance of observing a value at least 2, meaning that power is almost 1.0 and Type II error is almost 0. EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 6 0 .2 .4 .6 .8 0 1 2 3 4EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 7 I can pick a vertical line (2, for example) to correspond to a type I error. This is usually the case. Then I can posit what Ha is (3 or 4), and if the sample size tells me how broad the distributions of the effect size is under H0 and Ha, then I can estimate what Type II error and power will be. Alternatively, I can specify Type I error, and power (and thus Type II error) and estimate just how close Ha can be to H0 to achieve this level of power. We now change the paradigm only slightly. Every estimate has a distribution. The estimate can be of a sample mean, or a measure of association or effect size in a sample. We now think in terms of a distribution of OR, RR, or RD. We think of the distribution of OR, RR, or RD if the null hypothesis were true vs the distribution if the alternative were true. EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 8 Type I error Type II error From: Methods in Observational Epidemiology by J.L. Kelsey, A.S. Whittemore, Alfred S. Evans and W. Douglas Thompson, 1996, New York, Oxford University Press, p. 328.EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 9 Power calculations are based on the sampling distribution of the difference (means, proportions) of the groups being compared. d = value of "difference" [Risk difference, log OR, difference in means, etc.] when null is true (d = 0) dc = value of difference that is just significantly different from d at significance level α critical value d* = value of difference when null is false, i.e, when Ha is true. Some key numbers to remember on SS calcs (For purposes of this presentation) Quantity Interpretation Value Zα/2 Type I error 1.96 of 0.05 Zβ Type II error 0.2 (80% power) +0.84 0.1 (90% power) +1.28 EP 521 Spring, 2007 Vol I, Part 5 Copyright © 2006, Trustees of the University of Pennsylvania 10 ()22ZZαβ+ Used in SS calcs Type I =0.05;Type II=0.2 7.85 Type I=0.05; Type II=0.1 10.5 Some texts refer to Zβ as Z 1-β and Zα/2 as Z 1-α/2 and thus have slightly different formulae. The key to all these sample size formulae is to look at the two distributions: the difference under Ho vs the


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Penn EPID 521 - Sample Size Estimation

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