Statistical power of the classical twin design wasrevisited. The approximate sampling variances of a least-squares estimate of the heritability in a univariate analysis and estimate of the genetic corre-lation coefficient in a bivariate analysis were derivedanalytically for the ACE model. Statistical power todetect additive genetic variation under the ACEmodel was derived analytically for least-squares,goodness-of-fit and maximum likelihood-based teststatistics. The noncentrality parameter for the likeli-hood ratio test statistic is shown to be a simplefunction of the MZ and DZ intraclass correlation coef-ficients and the proportion of MZ and DZ twin pairsin the sample. All theoretical results were validatedusing simulation. The derived expressions can beused to calculate power of the classical twin designin a simple and rapid manner.Power calculations for twin designs are useful whendesigning experiments to estimate variance compo-nents and to test hypotheses regarding the nature of phenotypic similarity of twins. Power calculationscan be performed using asymptotic theory (Lynch & Walsh, 1998; Martin et al., 1978) or computersimulation studies based upon, for example, likeli-hood theory (Neale et al., 1994; Neale & Maes,2004; Posthuma & Boomsma, 2000). Martin et al.(1978) provided a comprehensive theoretical analysisof the power of the classical twin design usingweighted least-squares to estimate variance compo-nents and a goodness-of-fit test to reject ‘false’models. Nowadays, maximum likelihood is com-monly used to estimate variance component fromtwins or twin families using versatile computer programs such as Mx (Neale et al., 2002).Surprisingly, most of the literature on the estimationof parameters from twin designs is of the ‘black-box’category, in that no explicit equations are given forthe sampling variances of the parameter estimates andfor statistical power. In this study we derive simpleequations to calculate the power of twin designsunder the common ACE model, constrasting least-squares with maximum likelihood and goodness-of-fittests. Equations are presented for the sampling vari-ance of the estimate of the heritability, the proportionof variance due to common environmental effects,and the estimate of the genetic correlation coefficientin a bivariate analysis. The noncentrality parameterfor a maximum likelihood-ratio test for genetic vari-ance is given as a simple function of the population parameters. All predictions are verified using com-puter simulation.Assumptions and NotationThroughout, we assume the commonly used ACEmodel, for which the phenotypic variance is partitioned in an additive genetic (A), common environmental (C) and residual environmental (E) component. The proportions of phenotypic variance due to these random effects are h2, c2and e2,respectively. Predictions are first made using leastsquares (LS), from the properties of mean squares,which are the underlying sufficient statistics in theclassical twin design. Subsequently, derivations arederived for (residual) maximum likelihood.Parameters are scaled so that total phenotypic variance is 1.0. The total variance (var(y)) is then partitioned as, var(y) = h2+c2+e2= 1. For a bivariateanalysis, a derivation is given for the sampling variance of the estimate of the genetic correlationcoefficient, using least squares. Other parameterisa-tions and analysis methods (e.g., Jinks & Fulker,1970) were not investigated because they are not used in practice.TheoryUnivariate modelsLeast squaresConsider the between-pair (B) and within-pair (W)observed mean squares (MS) in the standard ANOVATable for n pairs, where the pairs can be either dizy-gotic (DZ) or monozygotic (MZ)df MS E(MS)between pairs n–1 B 2σb2+ σw2within pair n W σw2505Twin Research Volume 7 Number 5 pp. 505–512Power of the Classical Twin Design RevisitedPeter M. VisscherSchool of Biological Sciences, University of Edinburgh, United KingdomReceived 13 May, 2004; accepted 14 June 2004.Address for correspondence: Peter M. Visscher, School of BiologicalSciences, University of Edinburgh, West Mains Road, Edinburgh EH93JT, Scotland, UK. E-mail: [email protected]= (1–tMZ2) / [(1–tDZ2) + (1–tMZ2)] [2]Except for the trivial case when h2= 0, this ratio issmaller than 1/2. Hence, if the cost of phenotyping islimiting and many twin pairs are available for pheno-typing, then an optimum design would have more DZthan MZ twin pairs if the data are analyzed usingleast squares. For example, for tMZ= 0.5 and tDZ=0.25, n/m = 1.25, that is, approximately 56% DZ and44% MZ pairs. If tDZ= 1/2tMZ(AE model), and the cor-relation is small, then n/m ≈ 1 + 1/2h4. Unless theheritability is very large (>> 0.50), this suggests thatthe optimum design is close to a 1:1 ratio of DZ andMZ pairs.Power and sample size. For large samples, the quantityλ = (h2/SE(h^2)) is the expected mean test statistic of anormal test. Its square is approximately equal to thenoncentrality parameter (NCP) of a chi-square teststatistic. The NCP per total number of pairs (N) is,from Equation [1],NCPLS/N = (tMZ–tDZ)2/ [(1–tMZ2)2/pMZ+ (1–tDZ2)2/(1–pMZ)][3]For a statistical test we assume that under the nullhypothesis of h2= 0 (λ = 0)T = h^2/SE(h^2) ~ N(0,1)Under the alternative hypothesis, T ~ N(λ,1). Thisallows a simple prediction of power. If z1–αis the one-sided (upper tail) threshold from a standardnormal distribution corresponding to a type-I errorrate of α, and β the type-II error rate, then, for a one-sided testPower = 1–β = Prob(x > z1–α–λ)with x a standard N(0,1) random variable. Alterna-tively we can express the required power for a givenvalue of the heritability in terms of the MZ and DZsample sizezβ= z1–α–λ, or, λ = z1–β+ z1–αUsing the variance of the estimate of the heritabilityλ2= h4/ var(h^2) = (z1–α+ z1–β)2For a given proportion of MZ twins in the sample,the required total number of twins is, from Equation [3]N = 4(z1–α+z1–β)2[(1–tMZ2)2/pMZ + (1–tDZ2)2/(1–pMZ)]/ h4For example, if pMZ= 1/3 , α = 0.05, (1–β) = 0.80, h2=0.5 and c2= 0.20, then z1–α= 1.64 z1–β= 0.84 and N =172 twin pairs, n = 115 DZ and m = 57 MZ pairs.The optimal design for these parameters (fromEquation [2]) is n = 103 and m = 66, for a totalsample size of 169 twin pairs. 506Twin Research October 2004Peter M. Visscher))()((21))(21)(2()2/)var(()ˆvar(2222412WEBEnnWEnBEWBb| V))(22nWEThe expected mean squares and
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