Slide Number 1Slide Number 2Sample size and PowerSlide Number 4Sample size estimation: Why?Sample size estimationFirst objective: measure with a precisionSlide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Stata Slide Number 16Slide Number 17Sources of variance information: Study design and sample sizeSample Size Under SRS Without Replacement Slide Number 21Alternative Specification (in two-stages): Slide Number 23Derivation (alternative two-stage formula):Sample Size Based on Coefficient of Variation Slide Number 26Caution about using coefficient of variation (CV)Cost considerations for sample size Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Stata implementationStata implementationSlide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45An exampleSlide Number 47Slide Number 48Stata’s add-on programs for sample size estimationSlide Number 50Additional topics to be coveredCopyright 2009, The Johns Hopkins University and Saifuddin Ahmed. 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Your use of this material constitutes acceptance of that license and the conditions of use of materials on this site.Methods in Sample Surveys 140.640 3rd Quarter, 2009Sample Size and Power EstimationSaifuddin Ahmed, PHDBiostatistics DepartmentSchool of Hygiene and Public HealthJohns Hopkins UniversitySample size and Power“When statisticians are not making their lives producing confidence intervals and p-values, they are often producing power calculations”Newson, 2001”In planning of a sample survey, a stage is always reached at which a decision must be made about the size of the sample. The decision is important. Too large a sample implies a waste of resources, and too small a sample diminishes the utility of the results.“Cochran, 1977Sample size estimation: Why?• Provides validity of the clinical trials/intervention studies – in fact any research study, even presidential election polls• Assures that the intended study will have a desired power for correctly detecting a (clinically meaningful) difference of the study entity under study if such a difference truly existsSample size estimation• ONLY two objectives:– Measure with a precision:• Precision analysis– Assure that the difference is correctly detected• Power analysisFirst objective: measure with a precision• Whenever we propose to estimate population parameters, such as, population mean, proportion, or total, we need to estimate with a specified level of precision• We like to specify a sample size that is sufficiently large to ensure a high probability that errors of estimation can be limited within desired limitsStated mathematically:• we want a sample size to ensure that we can estimate a value , say, p from a sample which corresponds to the population parameter, P.• Since we may not guarantee that p will be exact to P, we allow some error• Error is limited to certain extent, that is this error should not exceed some specified limit, say d.• We may express this as:p - P = ±d, i.e., the difference between the estimated p and true P is not greater than d (allowable error: margin-of-error)• But do we have any confidence that we can get a p, that is not far away from the error of ±d?• In other words, we want some confidence limits, say 95%, to our error estimate d.That is 1-α= 95% It is a common practice: α-error = 5%prob {-d p - P d} 1 -≤≤≥αIn probability terms, that is,In English, we want our estimated proportion p to vary between p-d to p+d, and we like to place our confidence that this will occur with a 1-αprobability.From our basic statistical course, we know that we can construct a confidence interval for p by:p ± z1-α/2 *se(p)where zα denotes a value on the abscissa of a standard normal distribution (from an assumption that the sample elements are normally distributed) and se(p) = σp is the standard error.Hence, we relate p ± d in probabilities such that: pzpdpσα2/1−±=±np)p(1ZZd −=−−=2/12/1αασIf we square both sides,np)p(1 d2/12/1−==−−αασZZ2d)p(12Znp)p(1d22Znnp)-p(12Zd22/12/12/1p−=−==−−−ααα932.922)^10(.6.0*4.0*)96.1(2≈==nFor the above example:Note that, the sample size requirement is highest when p=0.5. It is a common practice to take p=0.5 when no information is available about p for a conservative estimation of sample size. 40038516.3842)^05(.5.0*5.0*)96.1(2≈===nAs an example, p = 0.5, d = 0. 05(5% margin-of-error), and α-error = 0.05:. di 1.96^2*.5*(1-.5)/(.05^2)384.16. di (invnorm(.05/2))^2*.5*(1-.5)/(.05^2)384.14588Stata. sampsi .5 .55, p(.5) onesampleEstimated sample size for one-sample comparison of proportionto hypothesized valueTest Ho: p = 0.5000, where p is the proportion in the populationAssumptions:alpha = 0.0500 (two-sided)power = 0.5000alternative p = 0.5500Estimated required sample size:n = 385Sample Size Estimation for Relative Differences If d is relative difference, pdptdppptn2)1(22)*()1(2−=−= Consider that 10% change is relative to p=.40 in the above example. Then, d = 0.4*0.10=0.04, that is, p varies between 0.36 to 0.44. Now, Note, d is very sensitive for sample size calculation. 57724.5762)^4.0*10(.6.0*4.0*)96.1(n2≈==Sample Size for Continuous Data ntd=222σ Change the varianceSources of variance information:• Published studies – (Concerns: geographical, contextual, time issues – external validity)• Previous studies• Pilot studiesStudy design and sample size• Sample size estimation depends on the study design – as variance of an estimate depends on the study design• The variance formula we just used is based on “simple random sampling” (SRS)• In practice, SRS strategy is rarely used • Be aware of the study designSample Size Under SRS Without ReplacementWe know that under SRSWOR,