Mathematical Assumptions versus Biological Reality: Myths in Affected Sib Pair Linkage Analysis

Blackwelder and Elston (1985) investigated the design and analysis of sibship data to detect linkage between a genetic marker and a dichotomous trait. By considering tests for samples of affected sib pairs (ASPs), they showed that, for most rare-disease models, the mean test is the most powerful. Similarly, in the case of a rare protective allele, for most models, the mean test based on unaffected sib pairs would be the most powerful. The necessary assumptions underlying the validity of such a design were clearly stated but—until very recently—largely ignored. Vieland and Huang (2003b) stated that “as a matter of mathematical principle” two-locus heterogeneity and two-locus epistasis cannot be distinguished on the basis of ASP marker data. In this report, we consider these cases as two examples of the same unfortunate phenomenon: the misuse of mathematical proofs by ignoring the underlying assumptions necessary for their validity. In the former case, potential users of the method were fully cautioned, whereas in the latter case a critical assumption was not even mentioned by the authors under their heading “Assumptions and Notation.”

Performing a linkage analysis with only ASPs is analogous to performing an epidemiological study without controls. It is now well established that unselected sib pairs tend to share more than half their alleles identical by descent, especially in an inbred population (Leutenegger et al. 2002). Zöllner et al. (2004) found that, in their sample of 148 nuclear families from the Hutterite population of South Dakota, “the signal of increased sharing is spread broadly across the genome, and no single chromosomal location reaches genomewide significance” (p. 65), with pointwise significance at the 5% level occurring on chromosomes 1, 5, 8, 10, and 15. In his classic work on the use of sib pairs to detect linkage, Penrose (1935) pointed out that “four classes will be formed according to whether the sibs are like in both characters, unlike in both, or alike in one and unlike in the other” (p. 133), with respect to the characters under investigation for linkage. It was suggested by Elston et al. (1996) that, in any ASP study, an unaffected sib should be recruited for half the ASPs, so that, for an increase of only 25% in sample size, the concordantly affected sib pairs would be properly controlled by an equal number of discordant sib pairs (DSPs). Most recently, Lemire et al. (2004) have again pointed out the need to have DSPs as controls and have devised a statistic that contrasts the two types of sib pairs to overcome the problem. For independent sib pairs, their statistic is essentially identical to that used by Elston et al. (1973). Although it was not explicitly identified as such, this method of analysis is mathematically identical to use of the regression statistic proposed by Haseman and Elston in 1972 (HE). That method can be used for the linkage analysis of any quantitative trait; a quantitative trait subsumes a binary trait that takes on only two values, which can, without loss of generality, be taken to be 0 and 1. In the original HE method, the squared sib-pair difference (which is always zero for concordant pairs and nonzero for discordant pairs) is regressed on an estimate of the proportion of alleles the sibs share identical by descent at a marker locus and a test performed to determine whether the regression coefficient is significantly negative. This is mathematically identical to testing whether the mean proportion of marker alleles shared identical by descent is larger for concordant pairs than for DSPs, which is the method proposed by Lemire et al. (2004). As noted by Schaid et al. (2003), whether one regresses the squared sib-pair trait difference on the estimated proportion of alleles identical by descent or regresses that proportion on the squared sib-pair trait difference, the test statistic is the same. The validity of the original HE method for binary traits was again pointed out by Zeegers et al. (2003). If some sibships contain more than one unaffected sib, the method pools the concordantly affected with the concordantly unaffected sibs, comparing them with the DSPs. A recent example of this method of analysis, appropriately allowing for the dependence of sib pairs within sibships, was published by Wiesner et al. (2003). Implementation of the HE method in the computer program SIBPAL was first made generally available as part of the program package S.A.G.E. almost 20 years ago, but it is currently much extended in S.A.G.E. 5.0, in a model that allows for covariates, marker-covariate interactions, and epistasis.

Bhende et al. (1952) first demonstrated that lack of the H antigen, an intermediary point in the production of the corresponding A and B antigens at the ABO locus, results in the apparent masking of the A and B alleles. This null phenotype—discovered among the natives of Bombay, India—was called “the Bombay phenotype,” to distinguish it from that of other individuals with the O blood group. Lack of the H antigen was subsequently shown to be a rare recessive trait that segregated in families (Levine et al. 1955; Aloysia et al. 1961). The advent of molecular technology has demonstrated that at least two genes—the H-gene (FUT1) and the Secretor gene (FUT2), encoding alpha (1,2) fucosyltransferases—control the complex epistatic effect (Koda et al. 1997). Mutations in the FUT1 gene lead to lack of ABH antigen on red blood cells, whereas mutations in FUT2 suppress the formation of ABH antigen in saliva and other body fluids. The classic Indian Bombay phenotype results from a T725G mutation of FUT1 and the gene deletion of FUT2. Additional molecular variants of the classic Bombay phenotype that involve only FUT1 or FUT2 have been described.

Vieland and Huang (2003b) used a definition of epistasis that differs not only from the classical concept of epistasis, as they freely admit, but also from the modern definition of epistasis as used in quantitative genetics. In classical genetics, epistasis and hypostasis are the interlocus analogues of the intralocus concepts of dominance and recessiveness. In the case of one locus, the effect of one allele masks the effect of another allele; the masking allele is termed “dominant” and the masked allele “recessive.” In the case of two (or more) loci, the two alleles at one locus mask the effect of the two alleles at another locus; the masking alleles are termed “epistatic” and the masked ones “hypostatic.” In our example, alleles at the ABO locus are hypostatic to alleles at FUT1 and FUT2, whereas the latter are epistatic to those at the ABO locus. Farral (2003) and Cordell (2003) have discussed in detail that, under the definitions of epistasis used in quantitative genetics, it is not true that heterogeneity and epistasis cannot be distinguished on the basis of ASP marker data.

In the appendix, we show that, specifically for ASPs and their definition of epistasis, Vieland and Huang’s mathematical derivation results from the assumption of joint Hardy-Weinberg equilibrium proportions for two trait loci—which we believe is an unrealistic mathematical assumption—rather than from any mathematical principle. We do this using the same notation, and making all the same assumptions, as in the section of their paper titled “Assumptions and Notation.”

Although the assumption of joint Hardy-Weinberg equilibrium proportions—just like the assumption that unselected sib pairs, in the absence of inbreeding, share half their alleles identical by descent—may be a reasonable assumption at the time of conception, we believe it to be an unreasonably arbitrary assumption for any later time point. Most conceptions are never born (Croteau et al. 2002; Edwards 2003), so, by as early as birth, there is ample opportunity for selection or meiotic drive to disrupt both chance allele sharing at a single locus and joint equilibrium proportions at two loci. Selection may disrupt linkage equilibrium between a marker and a trait locus, or even between two marker loci—but we believe that disruption of joint equilibrium proportions for two trait loci is even more likely, especially if the trait is one for which there may be any form of selection, including meiotic drive.

In conclusion, the necessary assumption that randomly sampled sib pairs always share half their alleles identical by descent is a serious limitation of ASP linkage analysis, and we believe that epistasis is best defined as a statistical concept of dependent gene action rather than as a departure from a “fundamental heterogeneity equation” based on mathematical assumptions that do not follow from well-established biological facts. However, it is clear that any ASP design that does not include some DSPs may be unable to validly detect epistasis, regardless of the validity of the proof offered by Vieland and Huang (2003b). A limitation of using an ASP design that is much more serious than the possible confounding of heterogeneity and epistasis (which is more a question of assumptions and definitions than biological reality) is that linkage results based on ASPs alone can easily lead to false-positive results. Similarly, any test that purports to gain power by pooling the “information” available from ASPs alone with that available from comparing ASPs and DSPs (Forrest and Feingold 2000) should be used with caution.

Appendix

We show here that, without the critical assumption of joint Hardy-Weinberg equilibrium proportions, equation (1) below does not necessarily follow from a probability truism that Vieland and Huang (2003b) used to capture the biological meaning of independent gene action at each of two loci.

Assume two trait loci, each with two alleles (A and a; B and b). The allele frequencies are p_A=P(A), q_A=1-p_A=P(a), and similarly for p_B, q_B. Assume that the trait loci are unlinked to each other, that each is linked to a marker, and that the two markers are unlinked to each other. There is linkage equilibrium between each marker and the trait, as well as between the two markers. We take as our example a double recessive trait phenotype with no phenocopies. The essence of Vieland and Huang’s mathematical derivation then hinges on two further assumptions:

• 1.
  The two trait loci have a genotypic distribution that follows joint Hardy-Weinberg equilibrium proportions, with the result thatwhere f_AB is the penetrance of the double homozygote aabb, f_A is the penetrance of aa whether the genotype at the B locus is BB or Bb, and f_B is the penetrance of bb whether the genotype at the A locus is AA or Aa.
• 2.

  Two-locus epistasis is defined to be any relationship among the penetrances that does not satisfy equation (1), “on the grounds that either the genes act independently or not” (Vieland and Huang 2003a, p. 1471).

Let K be the total prevalence of a disease, K_A the prevalence due to the action of genotypes at the A locus alone, and K_B the prevalence due to genotypes at the B locus alone. It then follows, from elementary probability, that

We now show that, in the absence of phenocopies, for a double recessive trait phenotype, equation (1) only follows from equation (2) on the assumption of joint Hardy-Weinberg equilibrium proportions. (An analogous argument can be made for a double dominant or a dominant-recessive trait phenotype.)

Our specific claim is that, if f_A>0, f_B>0, and P(aa,bb)≠q²_A×q²_B, where P(aa,bb) is the joint genotypic frequency of the two-locus genotype aabb, then, for a double recessive trait phenotype with no phenocopies, equation (1) does not follow from equation (2). We prove this by contradiction: Let P(aa,bb)=q²_A×q²_B+ɛ, ɛ≠0. In the case of a double recessive trait phenotype, when there are no phenocopies and we assume Hardy-Weinberg equilibrium at each locus, K_A is defined to be q²_A×f_A, and K_B is defined to be q²_B×f_B. Therefore, we have the following result:

This, along with the assumption that equation (2) is true, implies that

or, equivalently, that

But, if equation (1) follows from equation (2), then equation (1) is true—that is,

and so equation (3) becomes

This leads to

which is a contradiction to ɛ≠0, f_A>0, and f_B>0.