The Power of Multivariate Quantitative-Trait Loci Linkage Analysis Is Influenced by the Correlation between Variables

To the Editor:

In a recent article, Sham et al. (2000) 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 1996; Martin et al. 1997). 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. 1998; Amos et al. 2001). 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 , 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 1987).

Figure 1.

Path diagram showing the relationship between two observed variables (V₁ and V₂) for a pair of siblings. Covariation between the phenotypes is due to the QTL (Q), genetic and environmental sources that are shared among siblings (S₁ and S₂), and nonshared sources of variation (E₁ and E₂).

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

(only lower elements of the matrix are shown). Under the alternative hypothesis of linkage (L), the asymptotic parameter estimates are given by:

According to Sham et al. (2000), the NCP for linkage (λ_L) is equal to twice the difference in expected log-likelihoods between the alternative and null hypotheses:

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:

where r_ij 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

where x_S, 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

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.

Figure 2.

NCP as a function of either the correlation between unique sources of variation (lines with diamonds) or the correlation between shared sources of variation (lines with triangles).

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.