Assessing Whether an Allele Can Account in Part for a Linkage Signal: The Genotype-IBD Sharing Test (GIST)

Abstract

To fine map genes, investigators often test for disease-marker association in chromosomal regions with evidence for linkage. Given a marker allele tentatively associated with disease, one would ask if this allele, or one in linkage disequilibrium (LD) with it, could account in part for the observed linkage signal. This question can be addressed by determining if families selected on the basis of the presence of the tentatively associated allele show stronger evidence of linkage as measured by increased allele sharing identical by descent (IBD) by affected family members. However, common selection strategies can be biased for or against linkage in the marker region, even given no disease-marker association. We define unbiased selection schemes and extend the definition to allow weighted selection on the basis of all genotyped family members. For affected-sibship data, we describe three genotype-based weight variables, corresponding to dominant, recessive, and additive models. We then introduce a test for association of a family weight variable with excess IBD sharing. This test allows us to determine if the linkage signal in a region can be attributed in part to the presence of a marker allele, either because of direct involvement in disease etiology or because of LD with a predisposing genetic variant. For samples of 500 affected sib pairs, the tests are powerful in detection of genotype-IBD sharing association, even for disease models with sib relative risk as low as λ_S=1.1, or when evidence for linkage is absent because of sampling variation. This makes our method a new tool for detecting linkage as well as association, especially in regions harboring a candidate gene. We have implemented these methods in the software package GIST (Genotype-IBD Sharing Test).

Introduction

In a gene-mapping study, we seek to map and identify genetic variants that predispose to a human disease. There have been many successes in identifying genes for Mendelian diseases such as Huntington disease (Huntington’s Disease Collaborative Research Group ¹⁹⁹³), or special Mendelian forms of complex diseases such as mature-onset diabetes of the young (e.g., Yamagata et al. ¹⁹⁹⁶). However, it is much more challenging to identify genetic variants that predispose to complex diseases that are multifactorial and heterogeneous.

For complex diseases, investigators often map the disease of interest first by linkage analysis, followed by association analysis in regions with evidence for linkage. In this setting, genotypes will be available for most family members for the genetic markers used in linkage analysis and will be available for some or all family members for markers used for association analysis. When an allele is found to be statistically associated with disease in our sample, we call it an “associated” allele, with an understanding that the association is tentative and may not be functional. Given an allele that is associated with disease, we wish to ask if this allele, or one in linkage disequilibrium (LD) with it, could account in part for the observed linkage signal.

One way to address this question is to determine if subsets of families selected on the basis of presence of an associated allele show stronger evidence of linkage as measured by increased allele sharing identical by descent (IBD). One possible subset is the set of families in which any genotyped affected member carries the associated allele; a second is the set of families in which the genotypes of included affected members are all homozygous for the allele (Horikawa et al. 2000). Unfortunately, if multiple affected family members are genotyped, even if the marker is not associated with disease, standard linkage analysis on the first subset is biased against linkage in the region of the marker, reducing or even eliminating possible linkage signals, whereas linkage analysis on the second subset is biased for linkage in the region of the marker, artificially generating or amplifying linkage signals. Horikawa et al. (2000) proposed a simulation-based correction for this bias under the assumption of no association.

Alternatively, one may select families in which the index case carries (or is homozygous for) the associated allele (Horikawa et al. 2000). This avoids bias (see the “Methods” section). However, if multiple affected family members are genotyped, this approach ignores families in which the index case happens not to carry (or to be homozygous for) the associated allele, while one or more of his or her affected relatives do. Moreover, inclusion of families solely on the basis of the index case seems quite arbitrary, unless the index cases represent a particular set of individuals different from their affected relatives. If multiple affected family members are genotyped, one might first randomly choose an affected member from each family and then select families on the basis of the genotype of the chosen individual. This also avoids bias; however, it usually results in different subsets of families and different numbers of families, introducing additional variability. This variation can be substantial, particularly if the associated allele is relatively infrequent. We wish to have a valid subset linkage analysis that both reflects the effect of the associated allele and does not depend on the choice of affected member in families with >1 genotyped member.

In this paper, we investigate selection schemes for affected sibships with genotype data available on ⩾1 sib. To reduce variation and make full use of the sibship genotype information, we consider all possible subsets that result on the basis of the genotype of one random sib. Each sibship is selected into a proportion of the subsets, with the proportion being determined by the genotypes of all affected sibs. We then use this proportion as a weight variable for the sibship, resulting in a weighted selection scheme. For affected sibships, we define three weight variables, corresponding to dominant, recessive, and additive models.

To determine if a marker allele can account in part for the linkage signal in a region, either due to direct predisposition or LD between the marker and a predisposing variant nearby, we introduce procedures to test for association of a genotype-based family weight variable with excess allele sharing IBD. Since for variants that predispose to complex diseases, the genetic model generally is unknown, we also consider a test based on >1 weight variables. Using simulations we show that for samples of 500 affected sib pairs (ASPs), our tests are powerful to detect genotype-IBD sharing association, even for disease models with sib relative risk as low as λ_S=1.1, or even when there is no evidence for linkage due to sampling variation. In the latter situation, an association between genotype and IBD-sharing will suggest underlying linkage. This makes our method a new tool for detecting linkage as well as association. We have implemented our methods in the software package GIST (Genotype-IBD Sharing Test).

Methods

Outline

In what follows, we first discuss the concept of IBD-sharing configuration and introduce some notation. Second, we define unbiased selection of families so that linkage analysis on the resulting subset is valid. Third, we extend our definition to allow weighted selection, in order both to reduce variation and to make full use of available genotype information when multiple affected family members are genotyped. For affected sibship data, we give three examples of unbiased weighted selection schemes, corresponding to dominant, recessive, and additive models. Fourth, to determine if the linkage signal in a chromosomal region can be attributed in part to the presence of a marker allele, we describe tests for association between a genotype-based family weight variable with excess allele sharing IBD at the marker. Finally, given a weighted selection scheme, we define the NPL (Kruglyak et al. 1996) and LOD scores (Kong and Cox 1997) for the corresponding associated families.

IBD-Sharing Configurations and Notation

An IBD-sharing configuration for a pedigree represents a pattern of allele sharing IBD among pedigree members (Thompson 1974; Whittemore and Halpern ¹⁹⁹⁴). Let S be the set of all possible configurations for a pedigree. For an ASP, S may be represented as {0, 1, 2}, where each element denotes the number of alleles the ASP shares IBD.

Among affected members of a pedigree, the distributions of IBD-sharing configurations at a locus under linkage and under no linkage are different; this serves as the basis of allele sharing-based tests for linkage (Whittemore and Halpern 1994; Kruglyak et al. ¹⁹⁹⁶; Kong and Cox ¹⁹⁹⁷). Different pedigree structures generally have different sets of possible IBD-sharing configurations. To combine signals from different pedigree structures, a scoring function is required to assign a numerical value to each IBD-sharing configuration s∈S. Examples of scoring functions include S_pairs and S_all (Whittemore and Halpern 1994); a broader set of scoring functions is described by Sengul et al. (2001). For each pedigree structure, a scoring function is standardized to have mean zero and variance one under no linkage, resulting in a family NPL score Z=Z(s) (Kruglyak et al. 1996). In real applications, we may not have complete IBD information for a locus. In this situation, we calculate the expected family NPL score conditioned on family genotype data g and assuming no linkage (Kruglyak et al. 1996). In all the simulations in this paper, we define NPL scores based on S_pairs.

Suppose there are n families. We consider an autosomal marker in Hardy-Weinberg equilibrium with two or more alleles. Let p be the frequency of allele A. Let a denote the aggregate of all the other alleles, with frequency q=1-p. Let z_i (i=0, 1, 2) be the probability that an ASP shares i alleles IBD at the marker. If the marker is unlinked to disease, z₀=.25, z₁=.5, and z₂=.25. For an ASP, if we order the sibs’ genotypes by listing the genotype of the index case before that of the other sib, an ASP has nine possible ordered genotypes at a biallelic marker; table 1 lists the joint distribution of ordered genotype and IBD sharing at the marker, allowing for linkage but assuming no disease-marker association.

Table 1.

Joint Distribution of Ordered Genotype and IBD-Sharing Configuration for ASPs at a Biallelic Marker Under No Association^[Note]

	Weight			Pr(go, IBD=i)
Ordered Genotypesgo of IndexCase and Sib	φdom	φrec	φadd	i=0	i=1	i=2
AA:
AA	1	1	1	z0p4	z1p3	z2p2
Aa	1	1/2	3/4	z02p3q	z1p2q	0
aa	1/2	1/2	1/2	z0p2q2	0	0
Aa:
AA	1	1/2	3/4	z02p3q	z1p2q	0
Aa	1	0	1/2	z04p2q2	z1pq	z22pq
aa	1/2	0	1/4	z02pq3	z1pq2	0
aa:
AA	1/2	1/2	1/2	z0p2q2	0	0
Aa	1/2	0	1/4	z02pq3	z1pq2	0
aa	0	0	0	z0q4	z1q3	z2q2
Total				z0	z1	z2

Note.— We assume a marker at Hardy-Weinberg equilibrium with two alleles, A and a, with frequencies p and q=1-p. z_i is the probability that an ASP shares i alleles IBD at the marker locus. φ_dom is the proportion of sibs carrying allele A, φ_rec the proportion of sibs homozygous for allele A, and φ_add the proportion of allele A among all four alleles in the ASP.

Unbiased Selection Schemes

When there is evidence for both linkage and association in a region, we wish to ask if the associated marker allele, or one in LD with it, could account in part for the linkage signal. One way to address this question is to determine if a subset of families selected based on presence of the allele show stronger evidence of linkage as measured by increased allele sharing IBD. In order for linkage analysis on the subset to be valid for this purpose, families similarly selected on the basis of a nonassociated allele should on average neither increase nor decrease allele sharing IBD compared to the whole data set; we say such a selection scheme is unbiased. In other words, a selection scheme is unbiased for a chromosomal locus if for any pedigree structure of affected members, the distribution of IBD-sharing configuration at the locus conditioned on being selected is the same as the unconditional distribution. Equivalently, a selection scheme is unbiased if inclusion of a family is independent of the IBD-sharing configuration among affected family members at the locus. We define several genotype-based selection schemes that are unbiased when there is no disease-marker association.

Given no disease-marker association, a single affected individual’s genotype is independent of the IBD-sharing configuration among affected members in his/her family. Hence selection of families based on the genotype of one affected member is unbiased. For example, the selection of ASPs in which the index case carries allele A (rows 1–6 in table 1) results in a conditional distribution of IBD sharing with ratio z₀:z₁:z₂ for sharing 0, 1, or 2 alleles IBD (see appendix A), the same as that for the unconditional distribution. Analogously, the selection of ASPs in which the index case is homozygous for allele A (rows 1–3 in table 1) also is unbiased. Similar unbiasedness arguments hold for sibships of size >2.

If multiple affected family members are genotyped, one may first randomly choose an individual from each family and then select families in which the chosen individual carries (or is homozygous for) allele A. It can be shown that under no disease-marker association, this selection scheme also is unbiased, although different random choices usually will result in different subsets and numbers of families, introducing additional variability. To take into account all available genotype information, one might select families based on the genotypes of >1 members. One example is to select families in which any affected member carries allele A; a second is to select families in which all affected members are homozygous for allele A. In Appendix B, we show such schemes often are biased, even under no disease-marker association.

Weighted Selection Schemes

We noted previously that under no disease-marker association, the selection of affected sibships on the basis of whether a randomly chosen sib carries allele A is unbiased but generally results in different subsets and numbers of families, introducing additional variability. This variation can be substantial, particularly if the associated allele is relatively infrequent (see the “Results” section). To reduce variation and make full use of all sibship genotype information available, we consider all possible subsets that result from this selection scheme. Each sibship is selected into a proportion of the subsets, with the proportion being determined on the basis of the genotypes of all affected sibs. This proportion then can be used as a weight variable for each sibship. For affected sibships, if the selection scheme is based on whether a randomly chosen sib carries allele A, the proportion of subsets containing a sibship equals the proportion of sibs in the sibship carrying allele A; for ASPs, the resulting weight variable is listed as column φ_dom in table 1.

This is an example of a weighted selection scheme for affected sibships in which φ_dom is the weight variable. For a class of pedigree structures, we say that a weighted selection scheme with weight variable W is unbiased for a chromosomal locus if the conditional expectation of W given an IBD-sharing configuration s, E(W|s), is a constant for all s ∈ S at the locus, and the constant is the same across all pedigree structures in the class. This constant is E(W|s)=E[E(W|s)]=E(W), the expectation of W. The unbiasedness of φ_dom for affected sibships, under the assumption of no disease-marker association, is shown in appendix A. If the weight variable W only takes values 0 and 1, so that a family is either in a subset or not, a weighted selection scheme reduces to a yes/no (unweighted) selection scheme. In this situation, the unbiasedness of a weighted selection scheme is equivalent to independence between W and IBD-sharing configuration (see appendix C), and therefore the definitions of unbiasedness in this and the last subsections are consistent.

If a weighted selection scheme based on W is unbiased for a locus, W is uncorrelated with any scoring function F defined on IBD-sharing configurations at the locus; however, W may not be independent of F (see appendix C). In particular, the correlation coefficient ρ_W,Z between W and the family NPL score Z is zero, while W and Z are not independent (see appendix C).

For affected sibships, we can define other weighted selection schemes. For selecting affected sibships in which the index case is homozygous for allele A, the corresponding weight variable φ_rec is the proportion of genotyped affected sibs homozygous for allele A. A third weight variable φ_add is the proportion of allele A among the alleles carried by the affected sibs. Note that φ_add=(φ_dom+φ_rec)/2 does not have a corresponding (unweighted) selection scheme. As indicated by the subscript, the weight variables φ_dom, φ_rec, and φ_add should be well suited for disease variants that act in a dominant, recessive, or additive manner, respectively (see the “Results” section). These weights for ASPs are listed in table 1. It can be shown easily that given no disease-marker association, φ_rec and φ_add also result in unbiased weighted selection schemes.

We have focused so far on markers for which genotypes for more than one affected family member are available. When fine mapping genes, one may choose to genotype only one individual per family. In this situation, the weight variables can be defined on the basis of the single available genotype; φ_dom and φ_rec only take values 0 and 1, thus reducing to (unweighted) selection schemes, whereas φ_add can take values 0, .5, and 1, still resulting in a weighted selection scheme. Given no disease-marker association—since the genotype of a single individual is independent of the IBD-sharing configuration among affected members in his/her family—a weight variable W defined on the basis of the genotype of one individual is not only uncorrelated with the family NPL score Z but also independent of Z.

Testing for Genotype-IBD Sharing Association

For affected sibship data, we provide test procedures to determine if the linkage signal in a region can be attributed in part to the presence of a marker allele. We first consider the situation in which we have complete IBD information and then extend to the situation with incomplete IBD information.

Given a weight variable W, 1−W also can be viewed as a weight variable. Hence, for a weighted selection scheme with weight variable W, there are two weighted subsets: family i contributes to the first subset with proportion W_i and to the second, complementary subset with proportion 1-W_i. For the two subsets, the total contributed weights are and , where and the average per-family NPL contributions are and , respectively. We wish to assess whether a₁ and a₂ differ. Note that . Since var(Z_i)=1, if W_i were constants, var(a₁-a₂) would be , and would be

where and

is the sample correlation coefficient between family weight variable W and NPL score Z. This motivates us to use r_W,Z as the basis for our test statistic.

For affected sibships, we may choose W to be any one of the weight variables φ_dom, φ_rec, and φ_add, defined on the basis of an allele A. Let ρ_W,Z be the correlation coefficient between W and Z. We showed that, given no disease-marker association, ρ_W,Z=0. If allele A is a disease-predisposing variant or is associated with one, we expect that ρ_W,Z>0. In fact, for ASPs and affected sib trios, this is true for nearly all disease models that are relevant to complex diseases (see the “Results” section). Hence, we test the null hypothesis H_0i:ρ_φi,Z=0 (i=dom, rec, add) against the alternative H_1i:ρ_φi,Z>0 to determine if the linkage signal in the region can be attributed in part to the presence of allele A.

Fisher (1921) demonstrated that if (W, Z) is distributed as bivariate normal, when the number of families n is large, is approximately distributed as standard normal, where tanh^-1(x)=0.5ln[(1+x)/(1-x)]. For a general bivariate distribution of (W, Z) with finite moments, if W and Z are independent, this result still holds; if they are not independent, large-sample theory dictates that r_W,Z is asymptotically normal, and so is tanh^-1(r_W,Z) (Ferguson 1996).

For each of our weighting schemes i (i=dom, rec, add), let and let . Since μ_i is an increasing function of ρ_W,Z, the hypotheses now are H_0i:μ_i=0 against H_1i:μ_i>0. At significance level α, we reject H_0i and accept H_1i if X_i⩾z_1-α, where z_1-α is the 100(1-α) percentile of the standard normal distribution.

The power of these tests can be estimated easily. At significance level α, for each weight variable φ_i and a disease model with correlation coefficient ρ_φi,Z, the power of the corresponding test is approximately 1-Φ(z_1-α-μ_i), where Φ is the cumulative distribution function of the standard normal distribution. The resulting power estimates agree well with the corresponding simulation-based estimates (see the “Results” subsection).

For complex diseases, we usually do not know the underlying disease model. Carrying out all three tests requires adjustment for multiple comparisons, with a likely reduction in power. As an alternative, let X_max=max(X_dom,X_rec,X_add), and the P value for an observed maximum t is Pr(X_max⩾t) under no disease-marker association.

We now estimate the distribution of X_max under no disease-marker association. Let ρ_ij=corr(X_i,X_j) be the correlation coefficient of X_i and X_j (i, j=dom, rec, add). These correlations depend on several factors: the frequency p of allele A, the distribution of IBD-sharing configurations at the locus, the sample size n, and the composition of pedigree structures in the sample. Simulations suggest that for complex diseases, when n⩾100, the three correlation coefficients vary little with respect to changes in these factors except the allele frequency p (data not shown). Hence, for various values of p, we estimate ρ_ij by simulating a large number of ASPs under no linkage. Assuming the vector (X_dom, X_rec, X_add) is approximately distributed as trivariate normal, we generate a large number of random vectors from this distribution and derive an empirical estimate of the distribution of X_max.

We have outlined our tests assuming complete IBD information. Given incomplete IBD information, we calculate the sample correlation coefficient between W and the expected NPL score conditioned on family genotype data, and carry out the above tests using . Because of the additional variation due to incomplete IBD information, the correlation between W and tends to be weaker than that between W and Z, the magnitude of E(X_i) tends to be smaller, and the power of the tests may be reduced. However, for a flanking marker density of ⩽2 cM, the reduction in power is not substantial (see the “Results” section).

Weighted-Subset NPL and LOD Scores

Given a weighted selection scheme, we define the NPL (Kruglyak et al. 1996) and LOD scores (Kong and Cox 1997) for the corresponding associated families. Again, we first consider the situation in which we have complete IBD information, and then we extend the methods to handle incomplete IBD information and to incorporate further weighting based on family size.

At a locus and for family i, let Z_i be the NPL score, which has been standardized to have mean zero and variance one under no linkage (Kruglyak et al. 1996). The NPL score for the complete sample is , where the sum ranges over all n families. For a weighted selection scheme with weight variable W={W_i}ⁿ_i=1, we define W_iZ_i as the weighted NPL contribution from family i. If the scheme is unbiased for the locus, W_i and Z_i are uncorrelated, and the mean of W_iZ_i under no linkage is E(W_iZ_i)=E(W_i)E(Z_i)=0. We might define a weighted-subset NPL score for the sample to be divided by its standard deviation under no linkage. However, the formula for var(W_iZ_i) varies for different pedigree structures and can be very complicated; it also depends on the unknown frequency p of allele A. Instead, we choose to estimate var(W_iZ_i) empirically. Since var(Z_i)=1, if W_i were a constant, var(W_iZ_i) would be W²_i. This motivates us to define a weighted-subset NPL score at the locus as . Note that if W only takes values 0 and 1, so that a family is either in a subset or not, NPL_W is the same as the NPL score calculated on the basis of the subset.

For affected sibship data, we noted previously that the selection of sibships based on whether a randomly chosen sib carries allele A usually results in different subsets, depending on the choices made. By considering all possible subsets resulting from this scheme, we defined the weight variable φ_dom for each sibship. We may expect that the NPL scores for all the subsets can be summarized by the weighted-subset NPL score NPL_φdom defined above. Indeed, simulations show that for ASPs and affected sib trios, NPL_φdom is approximately the average of NPL scores over all the subsets (see the “Results” section). Similar conclusion also holds for the weight variable φ_rec.

As an alternative to the NPL score, Kong and Cox (1997) introduced a one-parameter allele-sharing model (ASM) with parameter δ⩾0, in which δ=0 for no linkage and δ>0 for linkage. Under their model, the log-likelihood for all families is (Kong and Cox 1997). For a weighted selection scheme with weight variable W={W_i}ⁿ_i=1, we substitute W_iZ_i for Z_i and call the resulting LOD score a weighted-subset LOD score.

In the more general situation of incomplete IBD information, we can replace Z_i with the expected NPL score . In standard linkage analysis, to leverage strength of signals from families of different sizes, a weighting factor γ_i may be assigned to each family i (Kruglyak et al. 1996). When calculating weighted-subset NPL and LOD scores, we can incorporate this information by using weighted family contribution in place of Z_i. Note that, because of the way we define weighted contribution for a family, the effects of W_i and γ_i are multiplicative, and they act as if there is a single weight variable, W_iγ_i. Calculation of the weighted-subset NPL and LOD scores can easily be implemented in existing software such as GENEHUNTER (Kruglyak et al. 1996) and Allegro (Gudbjartsson et al. 2000).

We caution that the LOD scores defined in this subsection are not comparable across different loci because their magnitude depends on the effective sample size in the weighted subset, which is determined by the allele frequency. As a result, such a LOD score cannot be interpreted in the same way as scores defined over the whole sample.

Results

Variation of NPL Scores for Subsets of ASPs Selected on the Basis of the Genotype of a Random Sib

To ask whether an associated allele can account in part for a linkage signal, the simplest strategy is to perform linkage analysis on the subsets of families selected on the basis of the genotype of a single affected family member. We use the term “subset NPL score” for the NPL score that is calculated over a subset of families. We noted previously that selection of affected sibships in which a randomly chosen sib carries allele A generally results in different subsets of families and therefore different subset NPL scores in the region of the marker (see the “Methods” section). We evaluated the extent of this variation using simulation. We simulated a biallelic marker independent of disease status, either unlinked (θ=.5) or completely linked (θ=0) to disease, and further assumed that we had complete IBD information at the marker. We considered two frequencies p for marker allele A (p=.1, .5), in combination with two scenarios of IBD sharing at the marker: no linkage with (z₀,z₁,z₂)=(.25,.50,.25) (λ_S=1) and linkage with (z₀,z₁,z₂)=(.23,.50,.27) (λ_S=1.1). For each combination, we simulated 100 replicate data sets of 500 ASPs. For each data set, we generated 200 random subsets of families selected on the basis of whether a randomly chosen sib carries allele A and calculated the mean E(subset NPL) and standard deviation sd(subset NPL) of the corresponding 200 subset NPL scores. Figure 1 plots sd(subset NPL) versus E(subset NPL) for a total of 100 replicate data sets.

Figure 1.

Plots of sd(subset NPL) vs. E(subset NPL) for 100 replicate data sets of 500 ASPs. For each data set, the horizontal coordinate is the mean of NPL scores for 200 random subsets selected on the basis of whether a randomly chosen sib carries allele A, and the vertical coordinate is the standard deviation of the 200 subset NPL scores. The marker is not associated with disease variant. p is the frequency of allele A. λ_S is the sib relative risk.

Figure 1 suggests that the magnitude of variation in subset NPL scores does not depend strongly on whether the whole data set yields evidence for linkage. With the same allele frequency, the variation of subset NPL score is slightly smaller for a linked marker than for an unlinked marker, presumably because IBD sharing at a linked locus is less variable than that at an unlinked locus.

From figure 1, we also see that particularly if allele A is relatively rare, but even if it is not, the variation of NPL score for a random subset of families can be substantial. For example, when p=.1, sd(subset NPL) for most replicate data sets is ∼0.45. Thus, if a data set of 500 ASPs has average subset NPL score 2.5, the NPL scores for different subsets easily can vary from 1.6 to 3.4 (2.5±2×0.45), with the corresponding LOD scores, calculated as LOD=NPL²/(2ln10), varying from 0.56 to 2.51. Even with p=.5, sd(subset NPL) for most replicate data sets is ∼0.25, and the NPL scores for different subsets can often vary from 2.0 to 3.0, with the corresponding LOD scores varying from 0.87 to 1.95. If we had selected families on the basis of whether a randomly chosen sib is homozygous for allele A, the variation is even greater (data not shown). These results suggest that when genotype data on ⩾1 sibs are available, we should not do subset linkage analysis solely on the basis of the genotype of a single sib.

Since the weight variable φ_dom for a sibship represents the proportion of random subsets of the sample containing this sibship, we would expect that the weighted-subset NPL score NPL_φdom defined in this paper would summarize well the NPL scores for all possible subsets. For the replicate data sets previously generated, we calculated the corresponding NPL_φdom and found that it is quite comparable with E(subset NPL) (data not shown). A similar conclusion holds for affected sib trios and for the weight variable φ_rec (data not shown).

Correlation between φ_dom, φ_rec, φ_add and Z under Disease Models

We showed that for affected sibships, if W is one of the three weight variables φ_i (i=dom, rec, add) defined on the basis of allele A, and given no disease-marker association, the correlation coefficient ρ_W,Z=0. If allele A is a disease-predisposing variant or is associated with one, we would expect that ρ_W,Z>0, and we have used this assumption as the basis for using a one-sided test.

To verify this assumption, we considered the general one-locus, two-allele disease model (p_D,f₀,f₁,f₂), where p_D is the frequency of disease-predisposing allele D in the population, and f_i (i=0, 1, 2) is the penetrance for the genotype with i copies of allele D. We assumed that 0<f₀⩽f₁⩽f₂ and that the values of f_i are not all equal. Let γ_i=f_i/f₀ (i=1, 2) be the genotypic risk ratio (GRR) of the genotype with i copies of allele D versus that with 0 copies. Then, 1⩽γ₁⩽γ₂, and 1<γ₂. For ASPs and affected sib trios, we calculated ρ_W,Z, where W is one of the three weight variables φ_dom, φ_rec, φ_add defined on the basis of allele D. We considered a total of 1,701 disease models specified by (a) disease allele frequencies p_D=.1, .2,…, .9, and (b) GRRs 1⩽γ₁⩽γ₂⩽10 and 1<γ₂, in increments of 0.5. For ASPs, the correlation coefficients were >0 except for seven (0.4%) dominant models. For affected sib trios, ρ_φdom,Z>0 for all the models considered, ρ_φrec,Z>0 except for 121 (7.1%) dominant models, and ρ_φadd,Z>0 except for 4 (0.2%) models. Not only were nearly all correlation coefficients positive, but the magnitude of the negative ones all were very small. If we restricted attention to models with GRR γ₁⩽4, a likely situation for complex diseases, all correlation coefficients were >0, ranging from 0.002 to 0.44.

Type I Error Rates of the Tests

We carried out simulations to assess the properties of our tests. Under no disease-marker association, we simulated 100,000 replicate data sets of 200 ASPs for allele frequencies p=.1, .2,…, .9. We used an additive model (with p_D=.5 and λ_S=1.1) to generate the background distribution of IBD sharing at the locus; simulations showed that small changes in the background distribution of IBD sharing have little effect on the results (details not shown). Let T_i (i=dom, rec, add, max) denote the test based on X_i. Table 2 lists the estimated type I error rates at nominal significance level α=.01 using genotypes of one or both sibs and given complete IBD information.

Table 2.

Estimated Type I Error Rates for the Four Tests, at Nominal Significance Level α=1%, Using Genotypes of One or Both Sibs^[Note]

	Estimated Type I Error Rate (%) for
	One Sib				Both Sibs
p	Tdom	Trec	Tadd	Tmax	Tdom	Trec	Tadd	Tmax
.1	1.0	.3	1.0	.6	.8	.2	.8	.5
.2	1.0	.8	1.0	.9	1.0	.7	.9	.8
.3	1.0	.9	1.0	1.0	1.1	.9	.9	1.0
.4	1.0	1.0	1.0	1.0	1.2	.9	.9	1.1
.5	1.0	1.0	.9	1.0	1.3	1.0	1.0	1.2
.6	1.0	1.0	1.0	1.1	1.5	1.0	1.0	1.3
.7	1.0	1.0	1.0	1.1	1.7	1.1	1.0	1.5
.8	1.0	1.0	1.0	1.0	2.2	1.1	1.0	1.8
.9	.4	1.1	1.0	.7	1.3	1.1	1.2	1.2

Note.— Based on 100,000 replicates of 200 ASPs. p is the frequency of marker allele A. T_i is the test based on X_i (i=dom, rec, add, max).

When the tests are based on genotypes of both sibs, the type I error rates increase as p increases. All the tests are conservative when p=.1, and T_dom and hence T_max are somewhat anticonservative when p>.5, whereas T_rec and T_add are consistent with nominal values for p>.1. At significance level α=.05, the patterns are similar and the type I error rates range from 4.2% to 8.8% (data not shown).

We noted previously that under no disease-marker association, a weight variable W defined on the genotype of a single individual is independent of family NPL score Z, and the statistics are asymptotically distributed as standard normal. Hence, we expect that for tests based on genotypes of one sib, the type I error rates converge to corresponding nominal significance levels when the number of families is large. Results in table 2 support this conclusion.

For the more realistic situation of incomplete IBD information, we generated flanking markers with four equally frequent alleles (heterozygosity .75) and various intermarker distance d. When d⩽2 cM, the estimated type I error rates were very similar to those given complete IBD information (data not shown).

Power of the Tests

We next carried out simulations of ASPs to compare the power to detect genotype-IBD sharing association among the four tests T_dom, T_rec, T_add, T_max, and between tests in which the weight variables are based on the genotypes of one or both sibs. We considered two population disease allele frequencies (p_D=.2, .5) and three disease models: dominant, recessive, and additive. All models had disease prevalence 10% and single-locus relative risk λ_S=1.1 for sibs of an affected individual (table 3). For each disease model, we generated 10,000 replicate data sets of 500 ASPs. Table 3 lists the estimated power for the tests at significance level α=.01.

Table 3.

Disease Models and Estimated Power (%) for the Four Tests, at Significance Level α=1%, Using Weight Variables Defined on the Basis of Genotypes of One or Both Sibs^[Note]

	Estimated Power(%)
	Complete IBDInformation				Incomplete IBDInformation, Tmax
pD, f0/f1/f2, andNo. of Sibs	Tdom	Trec	Tadd	Tmax	d=2	d=5	d=10
.2:
.065/.162/.162:
1	85	3	72	79	71	62	49
2	95	4	86	92	86	78	65
.089/.089/.368:
1	72	100	98	100	99	98	95
2	83	100	100	100	100	100	98
.068/.148/.228:
1	80	32	83	81	74	64	51
2	92	45	93	91	86	78	66
.5:
.015/.128/.128:
1	27	1	4	19	15	11	9
2	55	1	5	45	40	32	24
.072/.072/.185:
1	28	92	86	88	84	76	63
2	40	98	94	96	93	89	78
.036/.100/.164:
1	44	51	68	64	55	46	36
2	66	68	83	83	76	67	54

Note.— Results are based on 10,000 replicates of 500 ASPs. p_D is the frequency of disease allele D in the population. f_i(i=0, 1, 2) is the penetrance for the genotype with i copies of allele D. Given incomplete IBD information, d (in cM) is the flanking marker density. All models have disease prevalence 10% and single-locus sib relative risk λ_S=1.1.

We noted previously that the definitions of weight variables φ_dom, φ_rec, and φ_add reflect dominant, recessive, and additive effects of an allele, respectively. As expected, among the first three tests, T_dom is the most powerful to detect genotype-IBD sharing association for dominant models, as are T_rec for recessive models and T_add for additive models, although the degree of advantage varies by model and allele frequency. For all the models we simulated, the power of T_max is nearly as good as the highest power that can be obtained using T_dom, T_rec, or T_add.

With the same prevalence and single-locus relative risk λ_S, it was relatively easier to detect a disease gene with the smaller disease allele frequency p_D=.2 than that with p_D=.5. When we have genotypes for only one sib per ASP, we experienced some loss of power compared to using both sibs. However, for most of the models we considered, the power is still sufficient to make our tests useful tool for screening markers in a candidate region by genotyping one sib per sibship. Models with the same values of λ_S and p_D but different disease prevalences yielded similar estimates of power (results not shown).

To assess the more realistic situation of incomplete IBD information, we generated flanking markers with four equally frequent alleles (heterozygosity .75) and at marker densities d=2, 5, and 10 cM (table 3). To simulate the scenario with the least information content, we placed the disease locus midway between the two nearest flanking markers and did not include the genotype at the disease locus in the estimation of IBD sharing. Because of the additional variability due to incomplete IBD information, the power to detect genotype-IBD sharing association is reduced, but the reduction is not substantial when the marker density is ⩽2 cM. Even when the marker density is 10 cM, which is often the situation for initial genome scans, the power to detect genotype-IBD sharing at a candidate gene is still good for some disease models.

Testing Without Prior Evidence of Linkage

Although we described our method in the context of fine mapping genes in regions with evidence of linkage, the method also may be used for regions where we do not have linkage signals. Complex diseases are likely heterogeneous. If only a small proportion of families carry a predisposing variant at a locus, a linkage signal may be small and masked by random fluctuation. Furthermore, even if a disease-predisposing variant is common, because of sampling variation, traditional linkage approaches may by chance not detect evidence of linkage in the region of the variant. This is especially true for genes with modest effect. For example, if the disease model is additive with λ_S=1.1, even with 500 ASPs, the median NPL score for the whole data set is 1.44, corresponding to a LOD score of 0.45. When no linkage is detected because of sampling variation, we want to know if our tests still have type I error rate under control and good power to detect genotype-IBD sharing association.

Simulation results comparing samples with LOD ⩽0 and those with LOD >0 showed no systematic difference in type I error rate and power of our tests between the two samples (data not shown). In other words, the overall linkage signal for the total sample has little effect on sample correlation between a family weight variable and IBD sharing at the locus. In fact, it can be shown that under the null hypothesis of no association, these two statistics are asymptotically independent. Hence, in regions for which the data fail to show evidence for underlying linkage, our method can still be used to detect the association between a tentative disease variant and excess allele sharing IBD at the locus. Such an association will suggest underlying linkage. This property makes our method a new tool for detecting linkage as well as association.

Discussion

Mapping genetic variants for complex diseases is a challenging endeavor. One common strategy is to map the disease of interest first by linkage analysis, followed by disease-marker association analysis to fine map genes in regions with evidence of linkage. In this setting, genotypes will be available for most family members for the genetic markers used in the linkage analysis, and will be available for some or all family members for markers used for LD mapping. Given an observed disease-marker association, we often ask if the tentative disease-risk allele can account for a significant portion of the linkage signal, by either direct involvement in disease etiology or LD between the marker and a predisposing variant. We addressed this question by first building a framework for unbiased family selection and extending it to allow weighted selection and then deriving a test for association of a genotype-based family weight variable with excess allele sharing IBD.

For affected sibship data, we described three genotype-based weight variables, corresponding to dominant, recessive, and additive models for a marker allele. If the allele is not associated with disease, each weight variable W is uncorrelated with the family NPL score Z. We showed that, if the allele is itself a disease-predisposing variant or is associated with one, W is almost always positively correlated with Z for a very broad class of genetic models relevant to complex diseases. Hence, a positive association between W and Z will indicate that the linkage signal in the region of the marker can be attributed in part to the presence of the marker allele.

Under the null hypothesis of no disease-marker association, although the weight variable and family NPL score are uncorrelated, they are not independent. This makes existing nonparametric tests for correlation, for example, Kendall’s τ, Spearman’s rank test, and the permutation test (Hollander and Wolfe 1999), invalid because they require independence as part of the null hypothesis, and ignoring non-independence may result in inflation in the type I error rate (Keller-McNulty and McNulty ¹⁹⁸⁷). Based on large sample theory for the sample correlation coefficient, we introduced three tests for genotype-IBD sharing association, each test being powerful for dominant, recessive, and additive models, respectively. To avoid multiple comparisons, we introduced a fourth test that is based on the maximum of all three correlation coefficients and has good overall power. With sample sizes typical for complex disease gene mapping studies, even for disease models with sib relative risk as low as λ_S=1.1, the tests are powerful to detect genotype-IBD sharing association. When the tests are based on genotypes of >1 sib, the type I error rates for T_max and T_dom are slightly higher than the nominal rate, presumably due to less accurate large-sample approximation for these two tests than for T_rec and T_add. When the tests are based on only one sib, the approximation works well for all four tests, and the type I error rates are similar to the corresponding nominal rate.

Our method also can be used for candidate genes in regions for which IBD-sharing information is available from an initial genome scan. In the situations we considered, the overall linkage signal had little effect on the power of our test. As a result, in regions with underlying linkage but for which the data do not yield evidence of linkage due to sampling variation, the test is as powerful as in the situation of having observed linkage. Such an association between genotype and IBD-sharing will suggest underlying linkage. Thus, our method can complement existing linkage analysis methods by detecting linkage, especially when the effect of a disease variant is modest. This property makes our method a new tool for detecting linkage as well as association. The marker density in the initial scan determines the IBD-sharing information content in the region, which has a direct effect on the power of our test. However, as is shown in the simulations, the power of GIST at a candidate gene is still good for some disease models, even when the marker density is 10 cM.

Horikawa et al. (2000) proposed a different approach to address a similar question. For a marker of interest, they started with a target genotype and focused on the subset of ASPs in which the two affected sibs both had the target genotype. As we have shown here, this may result in bias in linkage signals even under the null, and, thus, correction may be needed. Horikawa et al. (2000) proposed a simulation-based bias correction by generating an empirical distribution of subset LOD scores under the null hypothesis and comparing it with the subset LOD score from the real data. In our approach, we try instead to stay unbiased under the null hypothesis and derive an appropriate weight variable. We then use this weight variable in our test for association. Given an allele that is associated with disease, their approach focuses on a genotype formed by the allele, while our approach focuses on the allele itself. We plan a more detailed comparison of these methods in a separate paper.

Both our method and the transmission/disequilibrium test (TDT) (Spielman et al. 1993) are joint tests for linkage and association. However, they work in quite different situations. For a marker of interest, TDT requires genotypes for parents or unaffected sibs, in addition to affected offspring; our method requires genotypes only for affected family member(s), while taking advantage of existing IBD-sharing information estimated from genome scans.

Our method also can be used to screen typed markers in a large region or on the whole genome for genotype-IBD sharing association as long as the IBD-sharing information is available from an initial linkage scan. The test should be carried out for each marker allele. In this situation, one needs to adjust for multiple comparisons, a challenging problem that also confronts genomewide case-control association analysis. To scan the maximum number of markers in a fine mapping project, we may choose to type only one sib per family. The power of GIST is lower with one genotyped sib than with two, but the power may still be sufficient to make our test a useful tool for screening markers in a candidate region. When information on IBD sharing is already available from genome scans, only the cases need to be genotyped to test for genotype-IBD sharing association by means of GIST. In contrast, to test for disease-marker association using the traditional case-control design, an additional set of controls need to be genotyped, significantly increasing the genotyping cost. Thus, our method will potentially reduce genotyping effort compared to the case-control approach. We are currently working on comparing per-genotype efficiency between these two approaches and on combining results from these two approaches.

A limitation of our method, as for some other association analysis methods, is that it does not distinguish a disease predisposing variant from an allele that is in LD with it. As a result, if an allele is found to be significantly associated with excess allele sharing IBD at the marker, the allele still may not have a direct effect on disease pathogenesis.

Other researchers previously have used the idea of doing linkage analysis on families stratified on the basis of an associated allele, either to determine whether the allele can account for a portion of the linkage signal (Horikawa et al. 2000) or to determine whether there are additional linked disease-predisposing variants (Hugot et al. 2001). Hugot et al. (2001) reported that linkage of Crohn disease to chromosome 16 could not be entirely explained by the detected associated alleles, because the subset of families without those associated alleles had a LOD score of 1.6. In this subset, analogous to what we demonstrated in appendix B, the uniformly high degree of allele sharing identical by state at the locus would result in an upward bias for linkage. Hence, the LOD score of 1.6 may be inflated compared to the true effect of other unknown disease variants in the region.

The idea in Hugot et al. (2001) of calculating “residual” linkage signals and using them as a signpost for additional disease variants is very tempting. Unfortunately, even if one did a weighted-subset linkage analysis for families without the associated alleles, the resulting LOD score would not be sufficient to indicate if there are additional disease-predisposing variants in the region. Our simulation results showed that, even if there is no extra disease-predisposing gene or variant in the region, the weighted-subset NPL score representing the “residual” linkage signal is no longer approximately distributed as normal and behaves differently for different disease models (data not shown). Sun et al. (2002) proposed a method to test for the presence of additional causal loci in the region, given that the current marker is causative for the disease. An extension of their method to multiple causative variants may help answer this question.

The difficulty in mapping genes for complex diseases calls for new techniques in data analysis. In this paper, we provided a general framework for unbiased selection of families stratified based on a marker allele, and for affected sibship data, we introduced test procedures to determine if the linkage signal in a chromosomal region can be attributed in part to the presence of a marker allele. We believe that these procedures will prove to be a valuable addition to the geneticist's toolbox for mapping genes for complex diseases. The methods developed in this paper have been implemented in our software GIST, available at the Web site of the Vanderbilt Program in Human Genetics.

Appendix A

In the “Methods” section, we introduced two selection schemes for affected sibships: (A1) selecting sibships in which the index case carries allele A, and (A2) selecting sibships in which the index case is homozygous for allele A. For ASPs, we show that under the assumption of no disease-marker association, (A1) is unbiased. The unbiasedness of (A2) can be similarly shown.

For ASPs and for a biallelic marker with alleles A and a, table 1 lists the joint distribution of ordered genotype and IBD sharing at the marker locus under the assumption of no disease-marker association. Under that assumption, the probability that the index case carries allele A (rows 1–6 in table 1) is 1-Pr(aa)=1-q²=p(1+q), and

Thus, the conditional distribution of IBD sharing, given that the index case carries allele A, has a ratio z₀:z₁:z₂ for sharing 0, 1, 2 alleles IBD, and hence the selection scheme (A1) is unbiased.

In the “Methods” section, we also introduced three weighted selection schemes for affected sibships, with weight variables φ_dom, φ_rec, and φ_add, respectively. It can be shown that, given no disease-marker association, all three weight variables result in unbiased weighted selection schemes. For example, for ASPs and scheme φ_dom, by table 1,

The conditional expectations all equal E(φ_dom)=Pr(asibcarriesallele A)=p(1+q). Similarly, it can be easily shown that E(φ_rec∣IBD)=E(φ_rec)=Pr(asibishomozygousforallele A)=p² and E(φ_add∣IBD)=E(φ_add)=E[(φ_dom+φ_rec)/2]=p. These equations also hold for affected sibships of size >2.

Appendix B

When multiple affected family members are genotyped, with a desire to take into account all genotype information, one might select families in which any affected member carries allele A. We show that this is biased against linkage at and around the marker locus.

Consider the complement subset of families—that is, the families in which all affected members have genotype aa. These families have a uniformly high degree of allele sharing identical by state at the marker. Therefore, linkage analysis on this complement subset is biased for linkage in the region, and, consequently, linkage analysis on the original subset is biased against linkage. Breaking up allele a into several different alleles yields the same conclusion.

It is easy to quantify this bias for ASPs. Given no disease-marker association, the selection of ASPs in which any sib carries allele A (rows 1–8 in table 1) results in a conditional distribution of IBD sharing at the marker with ratio z₀(1-q⁴):z₁(1-q³):z₂(1-q²), which is different from z₀:z₁:z₂ for 0<q<1. Since (1-q⁴)/(1-q³)>1 and (1-q³)/(1-q²)>1, the conditional distribution of IBD sharing is stochastically smaller than the unconditional distribution. Therefore, this selection scheme results in a downward bias in IBD sharing at the marker locus and consequently in the region of the marker, and hence reduces evidence for linkage. For example, suppose we have n=500 ASPs, the disease model is additive with λ_S=1.1, and the frequency of allele A at a completely linked marker (recombination fraction θ=0) is p=.5. Given complete IBD information and no disease-marker association, on average, the NPL score for the complete sample is 1.44, corresponding to a LOD score of 0.45. For the subset of ASPs in which one or more sibs carries allele A (86% of families), on average, there will be fewer ASPs sharing two alleles IBD than those sharing zero alleles IBD, with corresponding LOD scores of <0.

Another possible selection scheme might be to select families in which all affected members are homozygous for allele A (Horikawa et al. 2000). A similar argument shows that this scheme results in an upward bias in IBD sharing in the region of the marker and hence amplifies evidence for linkage. In the above example, even with no disease-marker association, for the subset of ASPs in which both sibs are homozygous for allele A (14% of families), on average, the NPL score for the subset is 4.48, corresponding to a LOD score of 4.37. Horikawa et al. (2000) proposed to correct this bias by simulations under the assumption of no association.

It can be shown analogously that selecting families in which all affected members carry allele A is biased for linkage and that selecting families in which any affected member is homozygous for allele A is biased against linkage.

Appendix C

In the “Methods” section, we defined unbiased weighted selection schemes. For a class of pedigree structures, a weighted selection scheme based on weight variable W is unbiased for a chromosomal locus if E(W∣s)=E(W) for all IBD-sharing configurations s∈S at the locus, and the constant is the same across all pedigree structures in the class. Here, we prove some properties for such a scheme.

If W only takes values 0 and 1, then for any s∈S, Pr(W=1,s)=Pr(W=1∣s)·Pr(s)=E(W∣s)·Pr(s)=E(W)·Pr(s)=Pr(W=1)·Pr(s), and, similarly, Pr(W=0,s)=Pr(W=0)·Pr(s). In this situation, the unbiasedness of a weighted selection scheme implies independence between the weight variable W and IBD-sharing configuration; it can easily be shown that the reverse also is true. Thus, the two definitions of unbiasedness are consistent.

Let F be a scoring function defined on IBD-sharing configurations. If a weighted selection scheme with weight variable W is unbiased for a locus, then, for a single pedigree structure,

and W is uncorrelated with F at the locus. This property also holds when we have a mixture of pedigree structures. For example, suppose we have two types of pedigrees, and a family has pedigree type 1 with probability β and pedigree type 2 with probability 1−β. Let W_i and F_i denote the weight variable and score, respectively, for a family with pedigree type i. By the definition of unbiasedness, E(W₁)=E(W₂)=E(W). Then,

Given complete IBD information, the family NPL score Z (Kruglyak et al. 1996) itself can be viewed as a scoring function, and so W is uncorrelated with Z. However, even with complete IBD information, W may not be independent of a scoring function F at the locus. For example, for ASPs, IBD sharing can be viewed as a function of the scoring function S_pairs. The nonindependence between the weight variables in table 1 and IBD sharing is easily shown by the fact that when an ASP shares two alleles IBD, φ_dom, and φ_rec can take only values 0 and 1, and φ_add can take only values 0, 0.5, and 1.