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• W2074512077 abstract "To the Editor: In a recent article, Sham et al. (Sham et al., 2000Sham PC Cherny SS Purcell S Hewitt JK Power of linkage versus association analysis of quantitative traits, by use of variance-components models, for sibship data.Am J Hum Genet. 2000; 66: 1616-1630Abstract Full Text Full Text PDF PubMed Scopus (207) Google Scholar) investigated the power of variance-components linkage analysis by deriving an analytic expression for the noncentrality parameter (NCP) of the linkage test. The authors demonstrated that the NCP—and, hence, the power of the test to detect linkage—was determined primarily by the square of the additive and dominance genetic components of variance due to the quantitative-trait locus (QTL) and by the residual correlation between siblings. However, Sham et al. presented calculations for the univariate case only. Recently, it has been demonstrated that the power of QTL linkage analysis may be increased by use of multivariate techniques that analyze the pleiotropic action of the QTL on several variables (Boomsma Boomsma, 1996Boomsma DI Using multivariate genetic modeling to detect pleiotropic quantitative trait loci.Behav Genet. 1996; 26: 161-166Crossref PubMed Scopus (46) Google Scholar; Martin et al. Martin et al., 1997Martin N Boomsma D Machin G A twin-pronged attack on complex traits.Nat Genet. 1997; 17: 387-392Crossref PubMed Scopus (534) Google Scholar). In particular, the power of multivariate linkage analysis is strongly influenced by the correlation between the variables, being greatest when the QTL induces covariation between the variables in the direction opposite to the residual correlation (Allison et al. Allison et al., 1998Allison DB Thiel B St Jean P Elston RC Infante MC Schork NJ Multiple phenotype modeling in gene-mapping studies of quantitative traits: power advantages.Am J Hum Genet. 1998; 63: 1190-1201Abstract Full Text Full Text PDF PubMed Scopus (132) Google Scholar; Amos et al. Amos et al., 2001Amos C de Andrade M Zhu D Comparison of multivariate tests for genetic linkage.Hum Hered. 2001; 51: 133-144Crossref PubMed Scopus (79) Google Scholar). Here, I follow the methodology of Sham et al., to demonstrate analytically, for the first time, how the power of a bivariate variance-components linkage analysis depends not only on the magnitude and direction of the correlation between variables but also on the source of this correlation. The relationship between two observable variables is parameterized in terms of the path model displayed in figure 1. Observed variables for each sib pair (square boxes) are due to the combined action of several latent variables (circles), including a pleiotropic QTL (Q), polygenic and environmental effects common to each member of the sib pair (S1 and S2), and unique environmental influences specific to each sibling (E1 and E2). Causal paths between variables are represented by unidirectional arrows, whereas correlations between variables are represented by bidirectional arrows. The strength of association between each variable is measured by a path coefficient (equivalent to a partial regression coefficient), in the case of a causal path, or a correlation coefficient, in the case of a bidirectional path. The correlation between siblings for the common QTL is πˆ, the estimated proportion of genes shared identical by descent at the trait locus, whereas the correlation between siblings for shared polygenic and environmental sources (i.e., S1 and S2) is 1. Correlations between phenotypes arise because of the pleiotropic action of the QTL (represented by the product of the path coefficients q1 and q2), from polygenic and environmental effects shared between siblings (represented by the product of α, s1, and s2) and from nonshared residual effects (represented by the product of β, e1, and e2). It is assumed that each variable is standardized to unit variance. The test for linkage is computed as twice the difference in log-likelihood between a model where q1 and q2 are estimated and a model where q1 and q2 are constrained to 0. Since q1 (or, alternatively, q2) is constrained to be positive, whereas q2 has no such constraint (to allow for the possibility of a negative correlation between the observed variables), the test statistic is distributed asymptotically as a 50:50 mixture of χ21 and χ22 (Self and Liang Self and Liang, 1987Self SG Liang K Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions.J Am Stat Assoc. 1987; 82: 605-610Crossref Scopus (1897) Google Scholar). Under the null hypothesis of no linkage (N), the asymptotic parameter estimates for the covariance matrix, implied by figure 1, of the ith sib pair are ΣiN=<AR><ARR><ARC>1</ARC><ARC></ARC><ARC></ARC><ARC></ARC></ARR><ARR><ARC>q1q2+αs1s2+βe1e2</ARC><ARC>1</ARC><ARC></ARC><ARC></ARC></ARR><ARR><ARC>q122+s12</ARC><ARC>q1q22+αs1s2</ARC><ARC>1</ARC><ARC></ARC></ARR><ARR><ARC>q1q22+αs1s2</ARC><ARC>q222+s22</ARC><ARC>q1q2+αs1s2+βe1e2</ARC><ARC>1</ARC></ARR></AR> (only lower elements of the matrix are shown). Under the alternative hypothesis of linkage (L), the asymptotic parameter estimates are given by: ΣiL=<AR><ARR><ARC>1</ARC><ARC></ARC><ARC></ARC><ARC></ARC></ARR><ARR><ARC>q1q2+αs1s2+βe1e2</ARC><ARC>1</ARC><ARC></ARC><ARC></ARC></ARR><ARR><ARC>π∧iq12+s12</ARC><ARC>π∧iq1q2+αs1s2</ARC><ARC>1</ARC><ARC></ARC></ARR><ARR><ARC>π∧iq1q2+αs1s2</ARC><ARC>π∧iq22+s22</ARC><ARC>q1q2+αs1s2+βe1e2</ARC><ARC>1</ARC></ARR></AR> According to Sham et al. (Sham et al., 2000Sham PC Cherny SS Purcell S Hewitt JK Power of linkage versus association analysis of quantitative traits, by use of variance-components models, for sibship data.Am J Hum Genet. 2000; 66: 1616-1630Abstract Full Text Full Text PDF PubMed Scopus (207) Google Scholar), the NCP for linkage (λL) is equal to twice the difference in expected log-likelihoods between the alternative and null hypotheses: λL=E(2lnLL)−E(2lnLN)=ln|ΣN|−14ln|Σπ=0|−12ln|Σπ=0.5|−14ln|Σπ=1| . To evaluate this expression, note that the determinant of a matrix of order n is a sum of n! signed products, each involving n elements of the matrix. The computation is made easier, in the present case, because the variables are standardized and, therefore, the diagonal terms of the matrix are equal to 1: |Σ|=1+2r21r31r32+2r21r41r42+2r31r41r43+2r32r42r43+r212r432+r322r412+r312r422−r212−r312−r322−r412−r422−r432−2r21r32r41r43−2r21r31r42r43−2r31r32r41r42 , where rij is the element corresponding to the ith row and jth column of Σ. If we denote the right half of this equation as “1+x” and note that the first-order Taylor-series expansion of ln(1+x)≈x, then the NCP may be approximated as λL≈xS−14x2−12x1−14x0 , where xS, x2, x1, and x0 are the first-order Taylor-series approximations for the null hypothesis and the alternative hypotheses of sharing two, one, or zero alleles identical by descent at the trait locus. Evaluation of this expression in terms of the parameters in figure 1 yields λL≈q148+q248+q12q224+q14q242−q12q242−q14q222−q14s248−q24s148−q12q22s12s224+αq1q23s1s2(q12+s12−1)2+αq13q2s1s2(q22+s22−1)2+βq1q23e1e2(q12−1)2+βq13q2e1e2(q22−1)2+αβq12q22s1s2e1e2+β2q12q22e12e222 . Note particularly that the second part of the equation (i.e., the last four lines) contains terms involving the correlation between shared polygenic and environmental effects (α) and the correlation between unique environmental effects (β). The sign of these correlations contributes to the magnitude of the NCP. Consider first the terms containing the correlation between shared polygenic and environmental effects (i.e. the terms containing α). It is apparent that the parts of the expression inside parentheses must be negative. Therefore, if the QTL and shared polygenic and environmental influences produce correlations in the same direction, the terms will be negative, and therefore the NCP and the power to detect linkage will decrease. In contrast, when the QTL and shared influences induce correlations in opposite directions, the terms will become positive increasing the NCP and power. The power to detect linkage increases as the correlation between shared sources decreases (i.e., becomes more negative). A similar argument also applies to terms containing the QTL and unique sources of variation (i.e., the terms that include α), although the increase in power is more dramatic because the terms inside the parentheses are greater in magnitude and because there is an additional term containing β that is always positive (i.e., 1/2β2q21q22e21e22). The last term in the equation (i.e., αβq21q22s1s2e1e2) suggests that the increase in power will be greatest when both shared and nonshared influences induce covariance in the direction opposite to the QTL. Figure 2 displays the effect that varying the correlation between shared and unique sources of variation has on the NCP for a plausible biological model. In this model, the QTL accounts for 20% of the variance of each trait (i.e., q21=q22=0.2), and induces a positive correlation between the variables (i.e., q1 and q2 are both positive). Both shared and unique effects account for forty percent of the variance for both traits (i.e., s21=s22=0.4; e21=e22=0.4). The correlation between unique sources of variation is varied, while the shared correlation is fixed at 0 (lines with diamonds), and the correlation between shared factors is varied, whereas the unique environmental correlation is fixed at 0 (lines with triangles). Note that the graph is based on exact values for the NCP and not on the Taylor-series approximation. In both cases, the NCP increases as the correlation between the latent sources of variation decreases. However, although the increase in NCP is small and linear for the shared case, the increase is dramatic and exponential as the correlation between the unique sources of variation decreases. Thus, the power of bivariate QTL linkage analysis depends not only on the phenotypic correlation between variables but also on the source of this correlation. In conclusion, these results imply that, in a bivariate linkage analysis, one is most likely to detect a QTL that produces a correlation between variables opposite in direction to the background correlation. In particular, power is dramatically affected by the correlation between the unique environmental sources of variation. This combination of latent sources would tend to produce variables that have low or moderate phenotypic correlations, a fact that should be kept in mind when deciding which variables to include in a bivariate linkage analysis. I would like to thank Dr. David Duffy for fruitful discussions and Dr. Nick Martin for helpful comments on the manuscript." @default.
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• W2074512077 title "The Power of Multivariate Quantitative-Trait Loci Linkage Analysis Is Influenced by the Correlation between Variables" @default.
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