SUPPLEMENTAL MATERIAL
Appendix 1: model (1) to model (2)
Let and 1-be the population frequencies of ’s alleles and . Suppose that a diallelic marker is tightly linked to the QTL. We denote the marker alleles as and , with frequencies and , and is the genotype for marker and individual . The magnitude of the linkage disequilibrium between the marker and the QTL is measured by the quantity where is the frequency of the haplotype . The haplotype frequencies () of the two loci are: . The genotype frequencies are:
We note () and the number of individuals being at the ith marker genotype. Givenmodel(2), the apparent effects of and are(Sham et al, 2000):
The apparent effect sizes of the marker alleles can be expressed as follows: . Thus,
The variance of is independent from :
The same reasoning is applied to obtain the covariance between and : . Consequently, under the pleiotropism assumption,assuming independency between and , the covariance conditional to between trait 1 and trait 2 is:
Where and
Of course, cannot be directly deduced from the data. Its numerical value (also denoted in order to lighten notations) is obtained using consistent estimates of , and : with . As the observed converges almost surely to the actual by Slutsky’s lemma, the study of the distribution of can be performed conditionally to .
Appendix 2: Building of the CLIP test
The CLIP test consists in comparing the dispersion of observed around the straight line,the direction of which is defined by the eigenvector corresponding to the highest eigenvalueof ,to the highest dispersion that can be observed under the pleiotropism assumption (H0). This dispersion is given by the lowest eigenvalue of : . is maximal when is minimal.Thus, our objective is to lower-bound under H0in order to upper-bound the dispersion under pleiotropy. We note the covariance between under H0. As presented in appendix 1:
Lemma 1:
Given lemma 1, if then
The right hand side of the inequality is a meaningful lower bound of only if it is strictly positive. In other words, as, if then is the highest lower bound that we can expect for . Otherwise, the highest lower bound is 0. In this case, it is not possible to distinguish between close linkage and pleiotropism.
Given , it is possible to replace by; consequently
It is difficult to provide unbiased estimations of (due to the fact that the genotype at the QTL is unknown). Therefore, we propose to substitute them by variance components that are easier to estimate i.e. .It is established that . As decreases when increases, substituting in provides a lower bound of.We thus obtain:.
is a meaningful lower bound only if it is positive which means that:
This last condition implies the previous one concerning the positivity of the variance components (i.e.).
To summarise, the CLIP test consists of rejecting H0 if the estimations proposed for the variance component are positive, is positive and the observed covariance between is lower than :
and
where is a multiplicative coefficient higher than 1 controlling the risk of wrongly rejecting H0 when pleiotropism holds.
Appendix 3: Consistency of the CLIP test
The CLIP test is consistent if the power of the test goes to 1 when .
Let us denote and the correlation between and under the close linkage assumption (H1). The CLIP test consists of rejecting H0 if is lower than . To evaluate the consistency of the test, it is necessary to define the highest value that can take.
The marginal variance-covariance matrix of under H1 is:
Where is the correlation between and . Thus . Consequently . Therefore, since is lower than one,, which proves the consistency of the CLIP test.
Appendix 4: Building of the CLIP test including polygenic effects
Considering model(1), the variance covariance matrix for and under the pleiotropism assumption is:
With . The variance-covariance matrixis then dependent of . To overcome this problem, we define a matrix independent of so that(smallest eigenvalue). This condition implies that (which means that is positive definite for all ). Given that, if , we propose
Replacing the matrix used in appendix 2 for the demonstration of the CLIP test by this new matrix, we obtain the following conditions to reject the hypothesis of pleiotropism:
References
Slutsky, E. (1925). Über stochastische Asymptoten und Grenzwerte, Metron5: 3–89.