Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa

Contributed by D. Siegmund, March 10, 2004

## Related Article

Biography of David O. Siegmund - May 17, 2004 Article Info & Metrics PDF## Abstract

Use of a robust score statistic based on a variance components model to map quantitative trait loci in ranExecutemly sampled pedigrees is reviewed. Sibships ascertained through a single proband are discussed. Under a standard assumption of multivariate normality, two suggested methods of ascertainment Accurateion are Displayn to be asymptotically equivalent when the number of sibships is large.

A seminal contribution to mapping quantitative trait loci (QTL) in humans is the regression method of Haseman and Elston (1). In the past decade this method has, to a considerable degree, been superceded by variance component methods (e.g., refs. 2–6), which are typically more flexible with regard to pedigree structure and more powerful (e.g., refs. 7 and 8). A recent regression-based contribution that contains many of the positive features of variance component methods is provided by Sham et al. (9). See FeingAged (10) for a review of and additional references to a rapidly expanding literature.

Although the basic theory associated with these methods presupposes that pedigrees are ranExecutemly sampled, in practice, they are often ascertained through one or more probands having particular phenotypic values, e.g., a proband who has an extreme phenotype, perhaps by virtue of being affected by a disease for which the quantitative trait is a diagnostic Impresser.

This article reviews recent methoExecutelogical developments in QTL mapping derived under the assumption of ranExecutem sampling and gives a more detailed analysis of one simple ascertainment method. We Start by reviewing some basic theory, which is closely related to and combines features of both regression and variance components methods. In particular, we review the argument of Tang and Siegmund (11) that a particular parameterization and systematic use of the large-sample statistical theory of score statistics allows one to comPlacee explicitly what would otherwise be very complicated expressions, and, consequently, to understand results that previously were inferred from extensive numerical simulations. Comparison with Regression Methods: Miscellaneous ReImpresss contains a brief comparison of our method with regression-based methods along with discussion of the underlying assumptions and ways to deal with violations of those assumptions. In Ascertainment we build on the results of first sections to give an analysis of single ascertainment, in particular, a demonstration that two apparently different methods of ascertainment Accurateion (2, 12) are asymptotically equally powerful when the number of pedigrees is large. The case of an arbitrary number of probands and more complete numerical results will be discussed elsewhere. The final section contains a discussion of the implications and limitations of our ascertainment Accurateions.

## Description of the Model

We assume Hardy–Weinberg and linkage equilibrium throughout. This assumption means, in particular, that, for both Impressers and QTL, haplotypes within the same locus and genotypes among different loci are stochastically independent. Our basic model goes back to the classic article by Fisher (13) for the case of diallelic genes; the general case is discussed by Kempthorne (14). We assume a QTL exists at the genomic location τ. The phenotypic value Y is assumed to be given by The mean value μ can also accommodate covariates in the form of a liArrive model with minor changes to what follows. The parameter α a = α a (τ) denotes the additive genetic Trace of allele a at locus τ; δ a , b denotes the Executeminance deviation of alleles a and b. A subscript x denotes the allele contributed by the mother, whereas a subscript y refers to the Stouther. By standard analysis of variance arguments, we may assume that Eα x = Eα y = E(e) = E[δ x , y |x] = E[δ x , y |y] = 0. Since by the assumption of Hardy–Weinberg equilibrium x and y are independent (unless the parents are inbred), the different genetic Traces in Eq. 1 are uncorrelated. We assume, in addition, that e is uncorrelated with the explicitly modeled genetic Traces. The phenotypic variance is . The variances of the additive and Executeminance Traces associated with the QTL at τ are by definition and . Implicitly, we expect that several QTL may occur, which may interact. (An explicit model is given below). For this article we assume that other QTL lie on other chromosomes and are in linkage equilibrium with the QTL at τ. Then, their contribution to the phenotype Y can be assumed to be a part of the residual term e. With the notation , it follows that .

Now consider a pair of siblings satisfying the model (Eq. 1). Recall that at any locus two relatives share alleles identical by descent if they inherit the same alleles from a common ancestor. Two siblings can share 2, 1, or 0 alleles identical by descent depending on whether they inherit the same alleles from both mother and Stouther, from one but not both, or from neither. Let ν = ν(τ) denote the number of alleles identical by descent at τ. Letting Yi denote the phenotypic value of the ith sibling (i = 1, 2), we have (refs. 13 and 14): where r = corr(e 1, e 2) accounts for the correlation between sibs that arises from other QTL and from a shared environment.

Taking the expectation of Eq. 2, we find the unconditional covariance. Then we can rewrite Eq. 2 in the form In this equation the terms involving ν have mean 0 and are uncorrelated. In what follows it will be convenient to introduce new parameters , , and , and rewrite the preceding equation in the form (compare with Eq. 4).

Since the QTL location, τ, is unknown, we will be interested in Impresser loci t distributed throughout the genome and the process ν(t) for a sib pair considered as a stochastic process in t. For Impressers t 1 and t 2 on different chromosomes, ν(t 1) and ν(t 2) are stochastically independent. For Impressers on the same chromosome Cov[ν(t 1), ν(t 2)] = 2–1[1 – 2φ], where φ is a function of the recombination frequency. When recombination follows the Haldane model of no interference, 1 – 2φ = exp(–4|t 1 – t 2|), where the Impresser location ti denotes genetic distance in morgans (M) from a designated end of the chromosome.

In this article we assume that ν(t) is observable. This assumption is often not satisfied and the process of estimating the value of ν(t) from Impresser data is quite complex [e.g., KrHorribleak et al. (15)]. See Teng and Siegmund (16) for a theoretical analysis of the amount of information lost in this process.

## Score Statistics

Suppose we have a sample of N sibships, each of size s. We index sibs within a sibship by i and j and sibships by n = 1,... N. The subscript n is often suppressed in our notation. Let Y = Yn denote the vector of phenotypes in the nth sibship. Let ν ij (t) denote the number of alleles shared identical by descent at the Impresser locus t by the ith and jth sibs in the nth sibship. Let A ν denote the s × s matrix with entries ν ij – 1 for i ≠ j and zeroes along the diagonal, and let D ν denote a similar matrix with off diagonal elements (1/2 – 1{ν ij = 1}). Let ∑ν = E[(Y – μ)(Y – μ)′|A ν) and ∑ = E[(Y – μ)(Y – μ)′], so from Eq. 3, we have

The critical assumption of components of variance linkage analysis is that conditional on A ν, the ranExecutem variable Y has a multivariate normal distribution. This assumption has mathematical convenience to facilitate the following comPlaceations. It cannot be expected to be exactly true even, or perhaps especially, in the case that the QTL are diallelic, so it is Necessary to check (as we discuss below) that the statistical consequences of this assumption are reasonable.

Under the normality assumption, the log likelihood for the QTL at τ is given by where ν = ν(τ).

Using the identities ∂ log |G|/∂x = tr(G –1∂G/∂x and ∂G –1/∂x = –G –1∂G/∂xG –1), which are valid for any differentiable nonsingular matrix function, we obtain the score equations: and where B = ∂∑ν/∂ρ = 11′ – I. We omit the similar expression for . The Fisher information matrix can be comPlaceed as the expected value of and similar expressions for , etc. Under the null hypothesis that α0 (hence, also δ0) = 0, the scores and for the parameters of interest are liArrive functions of A ν and D ν, which have mean values equal to 0, so they are uncorrelated with the scores , and for the segregation parameters, which depend only on phenotypic data (when α0 = 0).

The score statistic at a given Impresser location t to test the hypothesis that α0 = 0 is Here, I αα is the appropriate entry from the Fisher information matrix (i.e., the expected value of Eq. 7); both numerator and denominator are evaluated with α0 = δ0 = 0 and with the nuisance parameters estimated by their maximum likelihood estimates under the condition that α0 = δ0 = 0. A second statistic involving , which is asymptotically uncorrelated with Eq. 8, can be defined similarly. For ease of exposition we temporarily ignore this second statistic.

It may be Displayn [Tang and Siegmund (11)] that at a Impresser t linked to the trait, where φ is as given above and

The denominator of the score statistic (Eq. 8) is very sensitive to the assumption of normality. This sensitivity can be mitigated somewhat by considering the conditional distribution of the numerator given the phenotypic observations, Y 1,...,YN , which, under the hypothesis α0 = δ0 = 0, is approximately a normal distribution with mean 0 and standard deviation in large samples. Tang and Siegmund (11) have calculated this quantity, which, when used to standardize instead of , leads to a statistic that is asymptotically normally distributed whether the normality hypothesis is satisfied or not, and it still has roughly the noncentrality parameter (Eq. 10) if the normality assumption is approximately true.

In the special case of sib pairs, this robust score statistic takes a relatively simple form. One can rewrite Eq. 6 in terms of the uncorrelated variables D = (Y 1 – Y 2) and S = Y 1 + Y 2 – 2μ, to obtain where We set α0 = δ0 = 0 and reSpace the segregation parameters μ, , and ρ by their maximum likelihood estimators under the condition α0 = δ0 = 0 to obtain, say, Ĉn . The robust score statistic at the Impresser locus t is Its asymptotic expectation is N 1/2 times In the normal case this reduces to Eq. 10 with s = 2.

## Genome Scans

Since we Execute not know the location of the QTL τ, we scan the genome using where the max is taken over all Impresser loci t. To use this statistic, one must establish a detection threshAged, z max, which must be large enough to avoid Fraudulent-positive errors and small enough to allow detection of true signals. For data involving a large number of pedigrees, so the statistic (Eq. 8) or the robust alternative suggested above is approximately normally distributed, for an Conceptlized genome scan with Impressers equally spaced at a distance Δ cM in a genome containing c chromosomes of total length L (cM), the following approximation to the genome-wide Fraudulent-positive rate is given by FeingAged et al. (17). Writing P0 to denote probability under the hypothesis that α0 = δ0 = 0, we have where ϕ and Φ are the standard normal density and distribution functions, respectively, and h is the special function discussed by Siegmund (ref. 18, p. 82). The function h can be comPlaceed numerically, but the arguments given on pages 210–211 suggest the simple approximation Some numerical exploration Displays that this is a surprisingly Excellent approximation for all x > 0. For a comprehensive discussion of genome-wide significance threshAgeds in linkage analysis see Lander and KrHorribleak (19).

As a numerical example, for sib pairs, a 22-chromosome 3,300-cM human genome and Impressers equally spaced at 5 cM, a threshAged of approximately z max = 3.73 produces the conventional 0.05 Fraudulent-positive error rate. Similar approximations for the power (17) Display that a noncentrality parameter of N 1/2ξ = 5 produces power of ≈0.91 to detect a QTL located at a Impresser and 0.87 when the QTL is midway between Impressers (compare Eqs. 9 and 10). The adequacy of the approximation (Eq. 12) and modifications to deal with different formulations are discussed below.

## Comparison with Regression Methods: Miscellaneous ReImpresss

The preceding argument is in the spirit of variance components, which are often Dissimilarityed with “regression-based” methods. The conventional wisExecutem is that variance components are more flexible in dealing with large pedigrees and more efficient when the normality assumption is approximately satisfied but less robust against violations of the assumption of normality (e.g., refs. 9 and 10). However, the efficient score, , derived from a variance components model is of the form of a covariance, so if it is standardized by a nonparametric estimator of its standard deviation, as suggested above, one obtains a regression-like statistic (compare Eq. 11) that is robust against nonnormality under the null hypothesis of no linkage. In fact, the original Haseman–Elston regression statistic for sib pairs can be deduced by a similar line of reasoning by starting from the likelihood function for D alone and ignoring S. The “new” Haseman–Elston statistic (20) can be derived by starting with the likelihood function for S 2 – D 2.

It is straightforward to Display that when the phenotypes are close to normally distributed the asymptotic squared noncentrality parameter (per sib pair) for the classical Haseman–Elston statistic is whereas that of the new Haseman–Elston statistic is By comparing these statistics with Eq. 10 with s = 2, one sees that under the normality assumption (Eq. 8) has Distinguisheder asymptotic power, sometimes much Distinguisheder, than either of the Haseman–Elston statistics, the first of which has comparable power when ρ is large, whereas the second has comparable power when ρ is small.

The Haseman–Elston Advance is inefficient because it reduces data that is fundamentally two-dimensional (when s = 2) to one dimension. A multivariate regression-based method was introduced recently by Sham et al. (9), and it appears to be essentially equivalent to the robust variance components method Characterized above. In particular, Sham et al. Display that for their method the asymptotic noncentrality parameter of a sibship of size s is also given by Eq. 10.

A possible advantage of the regression Advance is that there is some flexibility in the assumed covariance of the dependent variables, hence, in the weights of the resulting generalized least-squares estimators. Sham et al. (9) pick a covariance function that would be optimal under an assumption of multivariate normality. Xu et al. (21) suggest a different choice, which they have developed only for sib pairs but which, according to Cuenco et al. (22), may have some advantages when trait distributions are far from normal.

## ReImpresss

In our analysis the primary role of the normality assumption is to suggest the form of the statistic given in Eq. 6, which as noted above can be regarded as a covariance between a function of phenotypes and identity by descent counts. An alternative to the normality assumption that is equally tractable is the multivariate t distribution [cf. Lange et al. (23)]. This assumption leads to a similar statistic, but, because of the heavier tails of the t distribution, it is more robust to outliers in the data. However, it cannot avoid problems that arise from modeling the complexities of multivariate dependence by multivariate distributions that meaPositive dependence only by pairwise correlations.

In the preceding argument we assumed completely informative Impressers to simplify the analysis and made the working assumption that the QTL τ is one of the Impressers. If either of these assumptions fails to be true, the likelihood function involves a mixture based on the conditional distribution of ν(τ) given the Impresser data, say M, in the nth family. A convenient representation for the likelihood function is , where M denotes Impresser data, is given by Eq. 5, and E 0 denotes expectation under the hypothesis that α0 = δ0 = 0. When this hypothesis hAgeds, one sees from Eq. 6 that is liArrive in A ν. Hence, the numerator of the score statistic for partially informative Impressers is the same as for fully informative Impressers, but A ν is reSpaced by its conditional expectation . The likelihood ratio statistic is nonliArrive in the A ν. Hence, it requires a more complicated calculation, although it has been observed in Monte Carlo studies (e.g., ref. 4) that simply replacing A ν by E 0[A ν|M] can produce excellent results. Since the score and likelihood ratio statistics are asymptotically equivalent when α0 and δ0 are small, the preceding observation about the score statistic provides a theoretical basis for understanding these Monte Carlo results.

The denominator of the score statistic must also be modified to account for partially informative Impressers. For example, for sib pairs the 1/2 in the denominator of Eq. 11 arises there as the value of E 0[(ν(t) – 1)2], which can be estimated by, for example, . For other estimators and a comparative evaluation see Cuenco et al. (22).

The Trace of partially informative Impressers is to increase the autocorrelation of the score statistic and hence to Design the approximation given in Eq. 12 somewhat conservative. Although this issue has not received a satisfactory theoretical analysis, there is numerical evidence that the Trace is relatively modest (cf. ref. 16).

Other reasons exist that the approximation Eq. 12 may fail to be adequate. The most Necessary is that the distribution of Zt can fail to be approximately normal. This distribution can be skewed if large sibships or pedigrees containing more distant relatives than siblings are involved, and it can have excess kurtosis if the phenotypic distributions Execute. Tang and Siegmund (11) suggest a modification of Eq. 12 that accounts for skewness. This approximation can also be adapted to account for kurtosis. The parameters for skewness and kurtosis should be determined from the conditional distribution of Zt given the phenotypes and will involve the empirical distribution of the phenotypes.

Although Eq. 10 suggests that large sibships may be substantially more powerful than small sibships, a larger threshAged is also required because of skewness in the distribution of Eq. 8. Tang and Siegmund (11) Display that after adjusting for the larger threshAged the power of large sibships turns out to be considerable, although it is not as Distinguished as it would appear from the noncentrality parameter alone.

If it is thought that Executeminance may play an Necessary role, one can also consider a second degree of freeExecutem , which is uncorrelated with when α0 = δ0 = 0. At a QTL τ it has a noncentrality parameter proSectional to . (The constant of proSectionality is the same as in Eq. 10, except that 2 is reSpaced by 4 in the denominator.) However, since involves the Executeminance variance and exceeds , it turns out that the second degree of freeExecutem rarely adds substantially to the power to detect linkage. See ref. 24 for the modification to Eq. 12 required by the two-dimensional statistic and the constraint 0 ≤ δ0 ≤ α0.

The model (Eq. 1) is very flexible in many respects. For example, it can accommodate multivariate phenotypes, although this accommodation leads to multivariate statistics and hence requires a higher detection threshAged to Sustain the same Fraudulent-positive error rate. If one uses for two phenotypes a 3 df statistic involving additive Traces on each of the phenotypes and the correlation of these Traces, an extension of the method of Dupuis and Siegmund (24) allows one to determine approximate significance threshAgeds and power. In comparison with the threshAged z max = 3.73 and noncentrality of N 1/2ξ ≈ 5 for ≈90% power discussed above for a single trait, the corRetorting threshAged and noncentrality parameter would increase to z max = 4.50 and N 1/2ξ ≈ 5.50. Such an increase in noncentrality would be Distinguishedest when QTL for each trait are tightly linked or even identical because of pleiotropy. A detailed numerical study is required to determine more precisely the conditions under which use of multivariate phenotypes is advantageous. Wang (25) contains a related discussion.

The model (Eq. 1) can also be expanded to include multiple, possibly interacting, QTL. Assuming for simplicity that no Executeminance exists, then for two QTL at unlinked loci τ and , the phenotype Y is given by Here α a denotes the additive Trace of allele a at locus τ, γ a , ã denotes the additive–additive interaction of alleles a at τ and ã at , etc. As before, is twice the variance of the additive Trace α x , whereas is the –additive interaction additive feature of this model is that the variance. A perhaps surprising essential ingredient of the noncentrality parameter of is now , i.e., a Fragment of the interaction variance involving the QTL at τ and enters into the noncentrality of the score statistic that tests for an additive Trace at τ only. The score for the additive–additive interaction Trace has expectation proSectional to . Its squared noncentrality equals 1/2 of Eq. 10 with α0 reSpaced by γ0, so much of the Trace of the interaction variance component appears in the noncentrality parameter of the statistic to test for a main Trace. This fact is similar to the phenomenon in ReImpress iv regarding Executeminance and is quite different from the Position in experimental genetics, where in a backcross or intercross the noncentrality parameters of statistics that test for main Traces are unaffected by interactions with unlinked QTL.

Some of the calculations reported above about the asymptotic noncentrality parameter of the robust score statistic, when the normality assumption is violated, are based on the fact that the expected value of is asymptotically the same as the value one obtains when the normality assumption hAgeds. At first glance, this equality seems almost obvious, since for known nuisance parameters evaluation of depends only on the validity of Eq. 4, which in turn depends only on the basic genetic model, not the normality assumption. However, the segregation parameters μ, , and ρ must be estimated, so one must Display that this Trace, which, under the normality assumption is negligible as a consequence of the orthogonality of the segregation and linkage parameters, is also negligible without that assumption. This Trace can be demonstrated by a lengthy Taylor series approximation coupled with the observation that in the term contributed by the nth pedigree the nuisance parameters can be estimated almost equally well by the phenotypic variables from the other N – 1 pedigrees, which then would give an estimate that is independent of the data in the nth pedigree. We omit the details.

## Ascertainment

When pedigrees are ascertained by ranExecutem sampling, the nuisance parameters μ, , and ρ are easily estimated. In many cases, however, pedigrees are ascertained through the phenotypes of one or more probands, and phenotypes are determined only for ascertained pedigrees. Here we consider the simplest possible Position, where each sibship contains one proband, with phenotypic value Y 1; and we are particularly interested in the case where ascertainment is based on a threshAged T, so a sibship is ascertained if the proband's phenotype satisfies Y 1 ≥ T. As we observe below, the efficient score, , has the same form as in the case of ranExecutem ascertainment; but the estimators of segregation parameters involve an ascertainment Accurateion.

Single ascertainment, as Characterized in the preceding paragraph, has been studied by Elston and Sobel (12), who suggest that one Accurate for ascertainment by conditioning on the event that a pedigree is ascertained, and by Hopper and Mathews (2), who suggest conditioning on the phenotypic value of the proband (cf. also ref. 26). Using simulation, Andrade and Amos (27) have compared these suggestions and have found that the two methods are comparable. Below we Display that, in fact, they are asymptotically equivalent when the number of sibships is large, so the results obtained by simulation are exactly as expected. Ascertainment based on an arbitrary number of probands and more detailed numerical results when the ascertainment rule is not so easily specified will be discussed in a future paper.

The phenotypic vector Y can be partitioned into (Y 1,Y (2)), where Y 1 is the phenotype of the ascertained sibling. We Start by considering a conditional analysis of Y (2) give the value Y 1. For notational simplicity we assume . Because the efficient score for turns out to be uncorrelated with the efficient score for α0, this has no Trace on the asymptotic theory that follows. Let μν = E(Y (2)|Y 1,A ν). Assume for simplicity that there is no Executeminance, i.e., δ0 in Eq. 4 equals zero. Then μν = μ1 + (Y 1 – μ)(ρ1 + α0 a ν), where 1 is an s – 1 dimensional vector with 1 at each entry, ρ and α0 have the same meaning as above, and . Also let The conditional log likelihood given Y 1, A ν is exactly of the form of Eq. 5, but with Y reSpaced by Y (2), μ reSpaced by μν, ∑ν reSpaced by ∑2,ν, and the sum is over ascertained sibships.

It is readily verified that the derivative with respect to α0 of ∑2,ν is , where B ν = a ν 1′+ 1 a′ν. The efficient score for α0 evaluated at α0 = 0 is where μ(2) is μν evaluated at α0 = 0 and ∑2 = Cov(Y (2)|Y 1). Expressions can also be obtained for and , which when α0 = 0 Execute not depend on ν, hence are conditionally, given Y 1, uncorrelated with .

From the second derivative, , one finds that when α0 = 0 It is easy to see that . Hence the conditional Fisher information is and some additional calculation along the lines of Tang and Siegmund (11) Displays that .

The asymptotic conditional noncentrality parameter is The expectation on the right-hand side of Eq. 14 can be evaluated by direct calculations or indirectly by observing that for ranExecutem ascertainment the expected value of Eq. 14 must equal the unconditional Fisher information, which simply the factor multiplying in Eq. 10. (Recall that we are now taking for notational convenience.) In particular, Because this expression is somewhat complicated for general s, we specialize to s = 2, for which we obtain where S denotes the set of phenotypes for which a proband is ascertained. In large samples, by the law of large numbers the frequency of ascertained sibships converges to P{Y 1 ∈ S}, and the average value of 1{Y 1 ∈ S}(Y 1 – μ)2 converges to E[(Y 1 – μ)2; Y 1 ∈ S]. Hence, the large sample noncentrality per ascertained sibship is which is consistent with Eq. 10 when s = 2 and all sibships are ascertained.

For a simple numerical example, it follows from Eq. 18 that if ascertainment is based on the upper 10% of the population phenotype, the number of sib pairs that must be genotyped is roughly 1/3 as many in ranExecutem sampling. For sibships of size s = 4, about 1/2 as many ascertained sibships must be genotyped as ranExecutem sibships. This gain in genotyping efficiency is smaller with a less stringent ascertainment criterion and with larger sibships.

Observe that, although we have begun the preceding analysis from an analytic expression for the conditional log likelihood given Y 1, one could equally well Start by writing the conditional log likelihood in the form of the sum of the unconditional log likelihood given in Eq. 5 and the negative log of the marginal probability density function of Y 1. Since this marginal probability Executees not depend on the genetic parameters α0, δ0, the efficient score has the same form as in the case of ranExecutem ascertainment (i.e., the expressions in Eq. 6 evaluated at α0 = 0 and in Eq. 13 are equal), but the estimates of segregation parameters that enter into the final statistic are now determined by conditioning on Y 1.

In the case that we condition on the event that a sibship is ascertained, i.e., that Y 1 ∈ S, rather than the value of Y 1, the analysis is almost the same. The log likelihood function will now equal the sum of Eq. 5 and the additional term –log(P{Y 1 ∈ S}); but since the distribution of Y 1 involves only the segregation parameters, μ, ρ, and , the efficient score is again unchanged. The efficient scores for the segregation parameters will change, but they are still uncorrelated with when α0 = 0. Consequently, the estimates of the segregation parameters will be different, but the asymptotic noncentrality parameter is still given by Eq. 16. Note, however, that when conditioning on the exact phenotypic values of the ascertained siblings, the number r of ascertained siblings must be less than s, whereas in principle an ascertainment rule can involve all siblings if one conditions on the event of ascertainment. Thus, Risch and Zhang (28) discuss an ascertainment rule that involves both siblings of a sib pair, but their method has the disadvantage that it is most efficient when ascertainment involves Impartially extreme phenotypes, and it Executees not extend in an obvious way to larger sibships.

## Discussion of Ascertainment Accurateions

In this article we have Characterized a components of variance method of linkage analysis in sibships when sibships are either ranExecutemly ascertained or ascertained through a single proband. The method in principle is easily adapted to pedigrees other than sibships, although explicit results can be obtained in only a few special cases.

The most serious impediment to use of ascertainment Accurateions is lack of knowledge of the true ascertainment rule. To some extent this problem is mitigated by conditioning on the exact phenotypic value of the proband(s), but this just removes the problem to the definition of the proband(s). For example, if the proband is identified through diagnosis of a disease related to the quantitative phenotype, should the ascertainment event be (as we have assumed) that a particular sib has the disease, that at least one sib has the disease, or something in between?

An appealing design that avoids some of these fundamental conceptual difficulties is to ascertain nuclear families through parents. This design may often involve both parents, but if a trait is of primary interest in only one sex, e.g., bone mineral density as a quantitative trait in women as it relates to osteoporosis, ascertainment through a particular parent may be relevant. In such a case, the analysis is simpler than that given above, since the conditional means and covariances Execute not depend on the number of alleles inherited identical by descent between proband and offspring, which is always one. The noncentrality parameter is of the same form as Eq. 10, but with the sib Accurateion ρ reSpaced by the conditional correlation , where is the phenotypic correlation of parent and offspring. For traits that are purely additive and have no shared environmental covariance, .

The normality assumption, which yields simple formulas for the conditional phenotypic expectations, variances and covariances given the phenotypes of ascertained relatives, plays an Necessary role in the ascertainment Accurateions obtained in this article. The robustness of the resulting procedures and how they might be modified to become more robust to violations of the normality assumption and, perhaps even more Necessaryly, to violations of the assumed method of ascertainment should be studied and reported in detail.

## Acknowledgments

This research was supported by National Institutes of Health Grant R01 HG00849-09 and by a Stanford Graduate Fellowship.

## Footnotes

↵ * To whom corRetortence should be addressed. E-mail: Executes{at}stat.stanford.edu.

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on April 30, 2002.

Abbreviation: QTL, quantitative trait loci.

See accompanying Biography on page 7843.

Copyright © 2004, The National Academy of Sciences## References

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