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Surveys Sample Size By R. Heberto Ghezzo Ph.D. R. Heberto Ghezzo Ph.D. Meakins-Christie laboratories McGill University - Montreal - Canada

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Objective of the Study Estimation Comparison Prevalence Odds- Ratio [Relative Risk if Cohort] Prevalence Odds-ratios [Relative Risk if Cohort]

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Estimation Confidence level - 90 %; 95 %; 99 % Acceptable width of interval - 1 %, 5 %, 10 %, 20 %

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Comparison Error type 1 - alpha ; 0.01 Smallest difference worth detecting - delta Error type 2 - beta ; 0.05 ; 0.01

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Error type 1 - alpha Error in claiming a difference when there is none. Alpha percent of normal people are thus classified into “abnormal”

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Error type 2 - beta Error of not finding a difference when the difference is greater than the threshold or value of delta. Depends on the definition of the threshold i.e. the difference worth detecting, delta.

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Which size? In surveys the errors are generally the same i.e. alpha = beta The level depends on the importance of the issue. Critical studies use beta=0.01

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Estimation of a Prevalence n = z 2 1-a/2 p(1 - p) / d 2 n = z 2 1-a/2 (1 - p) / e 2 p a = error type 1 - alpha d = absolute width of conf.interval e = relative width of conf.interval

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Estimation of an Odds-Ratio n = z 2 1-a/2 {1/p1(1-p1) + 1/p2(1-p2)} / ln 2 (1-e) a = error type 1 - alpha e = relative width of conf.interval p1 = proportion exposed in cases p2 = proportion exposed in controls. OR = p1(1-p2)/(1-p1)p2

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Estimation of a Relative Risk n = z 2 1-a/2 {(1-p1)/p1 + (1-p2)/p2} / ln 2 (1-e) a = error type 1 - alpha e = relative width of conf.interval p1 = proportion exposed in cases p2 = proportion exposed in controls. RR = p1/p2

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Comparing 2 prevalence n = {z 1-a/2 2p(1-p) + z 1-b p1(1-p1)+p2(1-p2)} 2 /(p1-p2) 2 p = (p1 + p2)/2 If p < 0.05 N = (z 1-a/2 + z 1-b ) 2 / [ (arcsin p2 - arcsin p1) 2 ] b = beta = 1-Power

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Testing Odds Ratio > 1.0 n = {z 1-a/2 2p2(1-p2) + z 1-b p1(1-p1)+p2(1-p2)} 2 /(p1-p2) 2 b = beta = 1-Power p1 = prevalence of exposure in cases p2 = prevalence of exposure in controls

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Total Sample Size If design is stratified and tests/estimations will be done at each strata. The sample size applies to each strata. Otherwise all within strata comparisons or estimations will have larger errors or confidence intervals.

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True Size I These formulae are theoretical. No real variable is truly normal. The estimator of variability has its own variability. There is no guarantee that the precision postulated will be achieved.

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True Size II The estimator of variability comes from a different study. If the variability of the proposed study is larger the precision will deteriorate. Always use a beta error smaller than really needed and adjust the sample size upwards to a round number.

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Non Response The sample size refers to the number of complete responses needed. Non response must be estimated and taken into account to arrive to the final size

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Imputation To impute is to fake a value that does not exist Only to complete observations for a multivariate technique

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