Likelihood-ratio affected sib-pair tests applied to multiply affected sibships: issues of power and type I error rate

"All-pairs" likelihood-ratio analyses, such as those performed by MAPMAKER/SIBS [Kruglyak and Lander, 1995], require that a sibship containing N affected siblings be split into N(N - 1)/2 sibships, each containing a different pair of affected sibs, before analysis. Each of these N(N - 1)/2 sibships may also contain the other affected sibs from the original sibship, coded as unaffected, to infer missing parental genotypes, as is done automatically in MAPMAKER/SIBS. Then, the use of the same individuals both as affecteds to test for linkage and, elsewhere, as unaffecteds to infer missing parental genotypes leads to negative correlations in the estimated identity by descent sharing among affected pairs from the same original multiplex sibship. This gives a conservative test of linkage, even when no downweighting is applied. Conversely, if the other affected sibs from the original sibship are omitted, the correlations are positive and the linkage test is anticonservative in the absence of weighting. True type I error probability also depends on marker informativity, typed parents, number of affected sibs included in the analysis, and the weighting scheme. This suggests the use of simulation, rather than asymptotic theory, to assess significance levels. The power of multiplex sibships relative to affected pairs increases with increasing phenocopy percentage, but the presence of typed unaffected sibs improves the relative power of multiplex sibships greatly only when penetrance is high. It was found that the 2/N weighting proposed by Suarez and Hodge [1979] increased power over an unweighted analysis in many situations, provided significance levels were adjusted appropriately by simulation.     

A comparison of sib-pair linkage tests for disease susceptibility loci

An analytical study is conducted of the properties of statistical tests to detect linkage between a disease locus and a very polymorphic marker locus when data on sib pairs are available. In most instances the most powerful test is the test based on the mean number of marker alleles shared identical by descent by the two members of a sib pair, and the most efficient sampling strategy is almost always to sample only pairs with both sibs affected. We show it is valid to use the information from all possible sib pairs as though they came from separate families when data on sibships of size three or larger are available, though more power may be obtained if different weights are given to the different sibship sizes.     

Sibship analysis of associations between SNP haplotypes and a continuous trait with application to mammographic density

Haplotype‐based association studies have been proposed as a powerful comprehensive approach to identify causal genetic variation underlying complex diseases. Data comparisons within families offer the additional advantage of dealing naturally with complex sources of noise, confounding and population stratification. Two problems encountered when investigating associations between haplotypes and a continuous trait using data from sibships are (i) the need to define within‐sibship comparisons for sibships of size greater than two and (ii) the difficulty of resolving the joint distribution of haplotype pairs within sibships in the absence of parental genotypes. We therefore propose first a method of orthogonal transformation of both outcomes and exposures that allow the decomposition of between‐ and within‐sibship regression effects when sibship size is greater than two. We conducted a simulation study, which confirmed analysis using all members of a sibship is statistically more powerful than methods based on cross‐sectional analysis or using subsets of sib‐pairs. Second, we propose a simple permutation approach to avoid errors of inference due to the within‐sibship correlation of any errors in haplotype assignment. These methods were applied to investigate the association between mammographic density (MD), a continuously distributed and heritable risk factor for breast cancer, and single nucleotide polymorphisms (SNPs) and haplotypes from the VDR gene using data from a study of 430 twins and sisters. We found evidence of association between MD and a 4‐SNP VDR haplotype. In conclusion, our proposed method retains the benefits of the between‐ and within‐pair analysis for pairs of siblings and can be implemented in standard software. Genet. Epidemiol. 34: 309–318, 2010. © 2009 Wiley‐Liss, Inc.     

A generalized estimating equations approach to linkage analysis in sibships in relation to multiple markers and exposure factors

We describe a multiple regression approach to nonparametric linkage analysis in sibships incorporating multiple genetic loci, environmental covariates, and interactions. The covariance in trait residuals between sib pairs is treated as the dependent variable, regressed upon identical-by-descent sharing probabilities and interaction effects, using generalized estimating equations to allow for the correlations among multiple sib pairs within a sibship. Individual covariates can also be introduced in the model for the trait means. An application to the GAW11 simulated data revealed linkage with each of the four simulated loci, as well as gene x environment interactions of E1 with loci C and D and gene x gene interactions among the cluster of loci A, B, and D.     

Interpretation of simultaneous linkage and family-based association tests in genome screens

Linkage and association analyses have played important roles in identifying susceptibility genes for complex diseases. Linkage tests and family-based tests of association are often applied in the same data to help fine-map disease loci or validate results. This paradigm increases efficiency by making maximal use of family data sets. However, it is not intuitively clear under what conditions association and linkage tests performed in the same data set may be correlated. Understanding this relationship is important for interpreting the combined results of both tests. We used computer simulations and theoretical statements to estimate the correlation between linkage statistics (affected sib pair maximum LOD scores) and family-based association statistics (pedigree disequilibrium test (PDT) and association in the pressure of linkage (APL)) under various hypotheses. Different types of pedigrees were studied: nuclear families with affected sib pairs, extended pedigrees and incomplete pedigrees. Both simulation and theoretical results showed that when there is no linkage or no association, the linkage and association tests are not correlated. When there is linkage and association in the data, the two tests have a positive correlation. We concluded that when linkage and association tests are applied in the same data, the type I error rate of neither test will be affected and that power can be increased by applying tests conditionally.     

Comparison of four sib-pair linkage methods for analyzing sibships with more than two affecteds: Interest of the binomial maximum likelihood approach

Family samples collected for sib-pair linkage studies usually include some sibships with more than two affecteds (multiplex sibships). Several methods have been proposed to take into account these multiplex sibships, and four of them are discussed in this work. Two methods, which are the most widely used, are based on the number of alleles shared by the sib-pairs constitutive of the multiplex sibship, with the first using the total number of these shared alleles (“all possible pairs” method) and the second considering a weighted number of these alleles (weighted method). The two other approaches considered the sibship as a whole, with in particular a likelihood method based on a binomial distribution of parental alleles among affected offspring. We theoretically show that, in the analysis of sibships with two affecteds, this likelihood method is expected to be more powerful than the classical mean test when a common asymptotic type I error is used. The variation of the sibship informativeness (assessed by the proportion of heterozygous parents) according to the number of affected sibs is investigated under various genetic models. Simulations under the null hypothesis of no linkage indicate that the “all possible pairs” is anticonservative, especially for type I errors ≤ 0.001, whereas the weighted method generally provides satisfactory results. The likelihood method shows very consistent results in terms of type I errors, whatever the sample size, and provides power levels similar to those of the other methods. This binomial likelihood approach, which accounts in a natural way for multiplex sibships and provides a simple likelihood-ratio test for linkage involving a single parameter, appears to be a quite interesting alternative to analyze sib-pair studies. Genet. Epidemiol. 15:371–390,1998. © 1998 Wiley-Liss, Inc.     

Correcting for ascertainment bias of relative-risk estimates obtained using affected-sib-pair linkage data

Locus-specific sibling relative risk is often estimated using affected-sib-pair lod score analysis of affected sibships and may be used to decide whether to continue or discontinue the search for additional susceptibility genes. We showed that relative-risk estimates obtained using affected-sib-pair data are asymptotically unbiased when each pair is given a weight inversely proportional to the sibship ascertainment probability. Here we show by simulation that the extent of the bias of relative risks estimated using the incorrect ascertainment weights is small for dominant models, but large for single-locus recessive models and some two-locus heterogeneity models. Since in practice the ascertainment scheme is often unknown, we investigate methods for jointly estimating ascertainment and relative risks from affected-sibship data. Given a sufficient sample size, a reasonable estimate of relative risk may always be obtained using only affected pairs from sibships with two affected and no unaffected siblings. This estimate, which has a large variance, may then be used in a three-stage procedure (which we call the alpha method) to estimate consistently both the ascertainment probabilities and the relative risks with greater precision. We additionally propose correction factors to eliminate small-sample bias of relative risks and investigate the bias due to error in the estimate of disease locus location.     

Informative-transmission disequilibrium test (i-TDT): combined linkage and association mapping that includes unaffected offspring as well as affected offspring

To date, there is no test valid for the composite null hypothesis of no linkage or no association that utilizes transmission information from heterozygous parents to their unaffected offspring as well as the affected offspring from ascertained nuclear families. Since the unaffected siblings also provide information about linkage and association, we introduce a new strategy called the informative-transmission disequilibrium test (i-TDT), which uses transmission information from heterozygous parents to all of the affected and unaffected offspring in ascertained nuclear families and provides a valid chi-square test for both linkage and association. The i-TDT can be used in various study designs and can accommodate all types of independent nuclear families with at least one affected offspring. We show that the transmission/disequilibrium test (TDT) (Spielman et al. [1993] Am. J. Hum. Genet. 52:506-516) is a special case of the i-TDT, if the study sample contains only case-parent trios. If the sample contains only affected and unaffected offspring without parental genotypes, the i-TDT is equivalent to the sibship disequilibrium test (SDT) (Horvath and Laird [1998] Am. J. Hum. Genet. 63:1886-1897. In addition, the test statistic of i-TDT is simple, explicit and can be implemented easily without intensive computing. Through computer simulations, we demonstrate that power of the i-TDT can be higher in many circumstances compared to a method that uses affected offspring only. Applying the i-TDT to the Framingham Heart Study data, we found that the apolipoprotein E (APOE) gene is significantly linked and associated with cross-sectional measures and longitudinal changes in total cholesterol.     

Contrasting effects of maternal fertility and birth rank on the occurrence of neural tube defects.

The relationships between the occurrence of anencephalus and spina bifida, sibship size and birth rank were examined, using linked records for births in British Columbia. Comparison of 414 sibships in which at least one infant had a neural tube defect with 1362 randomly chosen unaffected sibships showed that the affected sibships were larger. There were both more births than expected after the affected birth, and shorter intervals between births before the affected birth. Within sibships, the risk of anencephalus or spina bifida decreased strongly with increasing birth rank. No associations were seen with maternal age at first birth.     

Score test for age at onset genetic linkage analysis in selected sibling pairs

A new score statistic is derived, which uses information from registries (age‐specific incidences) and family studies (sib–sib marginal correlation) to weight affected sibling pairs according to their age at onset. Age at onset of sibling pairs is modelled by a gamma frailty model. From this model we derive a bivariate survival function, which depends on the marginal survival and on the marginal correlation. The score statistic for linkage is a classical nonparametric linkage (NPL) statistic where the identical by descent sharing is weighted by a particular function of the age at onset data. Since the statistic is based on survival models, it can also be applied to discordant and healthy sibling pairs. Simulation studies show that the proposed method is robust and more powerful than standard NPL methods. As illustration we apply the new score statistic to data from a breast cancer study. Copyright © 2009 John Wiley & Sons, Ltd.     

Testing association and linkage using affected-sib-parent study designs

We have developed a method for jointly testing linkage and association using data from affected sib pairs and their parents. We specify a conditional logistic regression model with two covariates, one that quantifies association (either direct association or indirect association via linkage disequilibrium), and a second that quantifies linkage. The latter covariate is computed based on expected identity-by-descend (ibd) sharing of marker alleles between siblings. In addition to a joint test of linkage and association, our general framework can be used to obtain a linkage test comparable to the mean test (Blackwelder and Elston [1985] Genet. Epidemiol. 2:85-97), and an association test comparable to the Family-Based Association Test (FBAT; Rabinowitz and Laird [2000] Hum. Hered. 50:211-223). We present simulation results demonstrating that our joint test can be more powerful than some standard tests of linkage or association. For example, with a relative risk of 2.7 per variant allele at a disease locus, the estimated power to detect a nearby marker with a modest level of LD was 58.1% by the mean test (linkage only), 69.8% by FBAT, and 82.5% by our joint test of linkage and association. Our model can also be used to obtain tests of linkage conditional on association and association conditional on linkage, which can be helpful in fine mapping.     

A general class of association tests for family-based data using weight functions

Based on the symmetry of transmitted/nontransmitted alleles from heterozygous parents under the null hypothesis of no association, the work proposed here establishes a general statistical framework for constructing association tests with data from nuclear families with multiple affected children. A class of association tests is proposed for both diallelic and multiallelic markers. The proposed test statistics reduce to the transmission disequilibrium test for trios, to T(su) by Martin et al. ([1997] Am. J. Hum. Genet. 61:439-448) for affected sib pairs, and to the pedigree disequilibrium test by Martin et al. ([2000] Am. J. Hum. Genet. 67:146-154); [2001] Am. J. Hum. Genet. 68:1065-1067) when using affected sibships only. The association test used in simulation and for real data (sitosterolemia) is the one which has the best overall power in detecting association. This association test is generally more powerful than the association tests proposed by Martin et al. ([2000] Am. J. Hum. Genet. 67:146-154); [2001] Am. J. Hum. Genet. 68:1065-1067) when using only affected sibships. For the sitosterolemia data set, the association test has its most significant result (P-value=0.0012) for the marker locus on the same bacterial artificial chromosome as the disease locus.     

A family study of blood pressure in Polynesians.

The family aggregation of blood pressure was studied in Tokelau Island children aged 5-14 years and their parents resident on their home islands in 1971. Five hundred and two (97 per cent) of the children had a recorded blood pressure and they formed 210 sibling groups. The sibship similarity of blood pressure z scores adjusted for year of age and sex was examined by analysis of variance between and within sibships in the 133 sibships with more than one member. For both systolic and diastolic pressure a statistically significant sibship similarity exists which is independent of family size, level of pressure, and the sibship similarity of Quetelet Index. The correlation coefficient of the z score of one index child chosen at random and the remaining siblings is 0-14 (n = 282, p = 0-017). Of the parental variables studied the mother's systolic pressure is the best, and only, predictor of the child's systolic z score. These results suggest that in the Tokelau islanders a family similarity of blood pressure is established relatively early in life.     

Affected sib pair identity by state analyses

Four methods using identity by state (IBS) data from affected sib pairs are compared for their ability to detect linkage between a diallelic marker and disease. A joint null hypothesis of no linkage and no linkage disequilibrium between the marker and disease must be considered. Two tests have undesirable properties in the case of linkage disequilibrium. Which of the other two tests has more power is dependent on the presence or not of linkage disequilibrium. The procedure of choice when possible is to type parents of affected sib pairs: the null hypothesis of no linkage can then be tested using identity by descent (IBD) values from informative parents, and the null hypothesis of no marker association with disease (linkage equilibrium) can be tested independently using the marker allele frequencies in the affected sib pairs.     

Genetics of IDDM: evidence for complex inheritance with HLA

Analysis of the Fifth Genetic Analysis Workshop (GAW5) insulin-dependent diabetes mellitus (IDDM) data leads to the following conclusions: 1) With a maximum-likelihood affected sib pair method, there is strong evidence for linkage with HLA and no evidence for linkage with INS, Gm, or Km. 2) Susceptibility as defined by HLA genotypes is very complex. Each DR allele has a unique susceptibility, and DR3 and DR4 haplotype associations for DR 3/4 genotypes are different from those for 3/X and 4/X. 3) Risk is substantially higher in sibships with an affected father compared to those with an affected mother. This excess cannot be attributed to transmission distortion at HLA.     

Effect of maternal and infant covariates on sibship correlation in birth weight

Birth weight on 12,644 singleton infants from 6,196 sibships born in Maryland between 1980 and 1984 were used to estimate the effects of nine maternal and infant covariates on the sibship correlation in birth weight. Assuming a homogeneous correlation across all families, the estimated intraclass correlation was 0.4664 (+/- 0.0099). This high sibship correlation makes it possible to predict, with reasonable accuracy, the birth weight of a child given information on previous sibs, as well as covariates on the mother and/or infant pertinent to a given pregnancy. The reduction in variance associated with incorporating information on the nine covariates used here was approximately equal to that obtained by conditioning on a single previous sib. Testing for heterogeneity in correlation among different groups of families showed that a crude measure of parity (first live birth vs. other), time between births, mother's marital status, and maternal age at the birth of the last child significantly influenced the sibship correlation in birth weight.     

Analysis of Swedish male breast cancer family data: A simple way to incorporate a common sibling effect

Based on a population-based cohort study, Olsson et al. [1993] found significant evidence for elevated incidence of breast and ovarian cancers among female first-degree relatives of men with breast cancer. Using an extension of logistic regressive models we investigate whether, after allowing for multifactorial familial correlations, single locus segregation could be the cause of the elevated incidence in these families. The logit for a given sib in the class D logistic regressive model depends on the order in which affected sibs occur in a sibship. That makes the model less appropriate for the situation where a polygenic component or a common sibling environment may be present, as well as being computationally cumbersome. In this paper, we propose to use the proportion of siblings in a sibship who are affected to quantify a sibling correlation. That not only relaxes the interchangeability problem but also makes the model computationally efficient. We then use this modified class D logistic regressive model for our segregation analysis. Using the proportion of siblings in a sibship who are affected as a covariate resulted in a significantly higher likelihoods in all the models we investigated. We found evidence for a dominant Mendelian gene leading to early age of onset of breast and/or ovarian cancer. This could either be a germline mutation of BRCA2 or a mutation in a gene different from BRCA2. Genet. Epidemiol. 15:201–212,1998. © 1998 Wiley-Liss, Inc.     

Meta‐Analysis of Sib Pair Linkage Studies of Asthma and the Interleukin‐9 Gene (IL9)

Asthma is a complex disease, with an etiology that includes both genetic and environmental influences that may interact. The moderate heritability of asthma has led researchers to investigate its molecular genetic basis using both exploratory investigations of linkage via genome scans, as well as targeted studies of specific candidate genes. Promising candidate genes include the cytokine genes on chromosome 5q23–31. Both genome scans and association studies of these candidate genes/genomic regions have yielded mixed findings, which raise the possibilities of a true relation that emerges more strongly in certain samples simply due to sampling variability, as well as of genetic heterogeneity. Meta‐analytic approaches that combine data across samples and examine how findings may vary as a function of effect modifiers can address both of these possibilities. In this study, we used a meta‐analytic approach to examine linkage between the interleukin‐9 gene (IL9), one of the cytokine genes located on chromosome 5q31, and asthma diagnoses and serum IgE levels in four samples. We analyzed IBD allele sharing for affected, unaffected, and discordant sib pairs, and as a function of sibling differences in IgE levels. We used a recently developed logistic regression‐based method that allows for the inclusion of covariates and/or effect modifiers in the analysis of allele sharing in sib‐pairs [Rice et al., Genet Epidemiol 17(Suppl. 1):S691–5, 1999]. Sex of the siblings and transmitting parent were considered both as covariates and effect modifiers in analyses. The results provided little evidence for linkage, or for heterogeneity therein due to sex or transmitting parent, either within or across samples. © 2001 Wiley‐Liss, Inc.     

Sibling sex ratio and male homosexuality

As an explanation for the increased sex ratio observed in the sibships of male homosexuals, Slater has hypothesized that the increase, itself, may be a causative factor in predisposing a male to homosexual behavior. As a test of this hypothesis, an additive and a multiplicative risk model relating male homosexuality to the sexual configuration of the homosexual's sibship were formulated and tested against published data. Both models were found to predict a curvilinear decrease in sibling sex ratio as sibship size increased, and both models were found to fit the observed sibship size sex ratio data closely. The analysis suggests that approximately 10% of the variance in male homosexual behavior can be accounted for by the sexual configuration of the homosexual's sibship.This work was supported in part by USPHS grants MH31302 and MH14677.     

The impact of varying diagnostic thresholds on affected sib pair linkage analysis

For Mendelian disorders, it is usually simple to classify individuals as either affected or unaffected. By contrast, for “complex” phenotypes, the diagnostic boundary of the disorder is often uncertain. This paper explores the following question: to most efficiently detect linkage in such a complex phenotype by the affected sib pair method, where should the diagnostic threshold be drawn? The model assumes that the disorder is due to a generalized two-allele single major locus (SML) where liability in each genotype is normally distributed. Evidence for linkage between the marker and disease loci is highly dependent on the location of the threshold. The relationship between the placement of the threshold and population linkage information (PLI) is Gaussian-like. At high thresholds, linkage efficiency (LE) (or the amount of linkage information per affected sib pair) is high but PLI is low because the number of affected sib pairs is very small. At low thresholds, the number of affected sib pairs is high, but PLI is low because LE is very low. The model is applied to published SML parameters for schizophrenia, and maximal PLI is achieved at thresholds broader than those for schizophrenia alone.