Comparison of tests for association and linkage in incomplete families

To analyze incomplete families, the following statistical tests can be used: LRAT-a simple likelihood-based association test, TRANSMIT, SIBASSOC/STDT, and RCTDT. We compared these four tests, for the diallelic case, on simulated data sets. The comparisons focused on the power to detect linkage and association when different familial structures, resistance to population stratification, resistance to misclassification of the disease status of the healthy sib, and the effect of nonpaternity were considered. The simulations lead to the following conclusions. The type I errors of TRANSMIT, SIBASSOC/STDT, and RCTDT were not affected by population stratification. LRAT showed bias under strong population stratification. High nonpaternity rates can lead to inflated type I errors, highlighting the importance of identification of half sibs. Under different homogeneous models, the power of TRANSMIT was very similar to that of LRAT, and, similarly, no difference in power was observed between SIBASSOC/STDT and RCTDT. Under various recessive and additive models, TRANSMIT was slightly more powerful than SIBASSOC/STDT when monoparental families with one affected and one unaffected sib were analyzed. Under various dominant models, SIBASSOC/STDT was slightly more powerful than TRANSMIT. Misclassification of the disease status of healthy sibs, as well as the discarding of incomplete families, resulted in a consistent loss of power.     

Statistical properties of Teng and Risch's sibship type tests for detecting an association between disease and a candidate allele

Risch and Teng [Genome Res 1998;8:1273-1288] and Teng and Risch [Genome Res 1999;9:234-241] proposed a class of transmission/disequilibrium test-like statistical tests based on the difference between the estimated allele frequencies in the affected and control populations. They evaluated the power of a variety of family-based and nonfamily-based designs for detecting an association between a candidate allele and disease. Because they were concerned with diseases with low penetrances, their power calculations assumed that unaffected individuals can be treated as a random sample from the population. They predicted that this assumption rendered their sample size calculations slightly conservative. We generalize their partial ascertainment conditioning by including the status of the unaffected sibs in the calculations of the distribution and power of the statistic used to compare the allele frequency in affected offspring to the estimated frequency in the parents, based on sibships with genotyped affected and unaffected sibs. Sample size formulas for our full ascertainment methods are presented. The sample sizes for our procedure are compared to those of Teng and Risch. The numerical results and simulations indicate that the simplifying assumption used in Teng and Risch can produce both conservative and anticonservative results. The magnitude of the difference between the sample sizes needed by their partial ascertainment approximation and the full ascertainment is small in the circumstances they focused on but can be appreciable in others, especially when the baseline penetrances are moderate. Two other statistics, using different estimators for the variance of the basic statistic comparing the allele frequencies in the affected and unaffected sibs are introduced. One of them incorporates an estimate of the null variance obtained from an auxiliary sample and appears to noticeably decrease the sample sizes required to achieve a prespecified power.     

Use of parents, sibs, and unrelated controls for detection of associations between genetic markers and disease.

Detecting the association between genetic markers and complex diseases can be a critical first step toward identification of the genetic basis of disease. Misleading associations can be avoided by choosing as controls the parents of diseased cases, but the availability of parents often limits this design to early-onset disease. Alternatively, sib controls offer a valid design. A general multivariate score statistic is presented, to detect the association between a multiallelic genetic marker locus and affection status; this general approach is applicable to designs that use parents as controls, sibs as controls, or even unrelated controls whose genotypes do not fit Hardy-Weinberg proportions or that pool any combination of these different designs. The benefit of this multivariate score statistic is that it will tend to be the most powerful method when multiple marker alleles are associated with affection status. To plan these types of studies, we present methods to compute sample size and power, allowing for varying sibship sizes, ascertainment criteria, and genetic models of risk. The results indicate that sib controls have less power than parental controls and that the power of sib controls can be increased by increasing either the number of affected sibs per sibship or the number of unaffected control sibs. The sample-size results indicate that the use of sib controls to test for associations, by use of either a single-marker locus or a genomewide screen, will be feasible for markers that have a dominant effect and for common alleles having a recessive effect. The results presented will be useful for investigators planning studies using sibs as controls.     

Score tests of genetic association in the presence of linkage based on the additive genetic gamma frailty model

Nuclear families with multiple affected sibs are often collected for genetic linkage analysis of complex diseases. Once linkage evidence is established, dense markers are often typed in the linked region for genetic association analysis based on linkage disequilibrium (LD). Detection of association in the presence of linkage localizes disease genes more accurately than the methods that rely on linkage alone. However, test of association due to LD in the linked region needs to account for dependency of the allele transmissions to different sibs within a family. In this paper, we define a joint model for genetic linkage and association and derive the corresponding joint survival function of age of onset for the sibs within a sibship. The joint survival function is a function of both the inheritance vector and the genotypes at the candidate marker locus. Based on this joint survival function, we derive score tests for genetic association. The proposed methods utilize the phenotype data of all the sibs and have the advantages of family-based designs which can avoid the potential spurious association caused by population admixture. In addition, the methods can account for variable age of onset or age at censoring and possible covariate effects, and therefore provide important tools for modelling disease heterogeneity. Simulation studies and application to the data sets from the 12th Genetic Analysis Workshop indicate that the proposed methods have correct type 1 error rates and increased power over other existing methods for testing allelic association.     

Absence of kin discrimination in cannibalistic Anuran tadpoles of the frog Hoplobatrachus tigerinus (Daudin)

Kin discrimination was tested in the cannibalistic H. tigerinus tadpoles to know whether cannibalism is selectively directed towards non-kin members or it is indiscriminate. The association choice tests were conducted using satiated as well as starved subjects with the assumption that they will associate near non-sibs rather than near sibs with the intention of preferentially cannibalizing them. However, test tadpoles, fed or starved showed a random association choice with sibs and non-sibs, as in the end-bias stimulus blank tests. Therefore it is suggested that cannibalistic H. tigerinus tadpoles do not discriminate sibs from non-sibs and cannibalize on both rather indiscriminately.     

Transmission/Disequilibrium Tests Incorporating Unaffected Offspring

We propose a new method for family-based tests of association and linkage called transmission/disequilibrium tests incorporating unaffected offspring (TDTU). This new approach, constructed based on transmission/disequilibrium tests for quantitative traits (QTDT), provides a natural extension of the transmission/disequilibrium test (TDT) to utilize transmission information from heterozygous parents to their unaffected offspring as well as the affected offspring from ascertained nuclear families. TDTU can be used in various study designs and can accommodate all types of independent nuclear families with at least one affected offspring. When the study sample contains only case-parent trios, the TDTU is equivalent to TDT. Informative-transmission disequilibrium test (i-TDT) and generalized disequilibrium test(GDT) are another two methods that can use information of both unaffected offspring and affected offspring. In contract to i-TDT and GDT, the test statistic of TDTU is simpler and more explicit, and can be implemented more easily. Through computer simulations, we demonstrate that power of the TDTU is slightly higher compared to i-TDT and GDT. All the three methods are more powerful than method that uses affected offspring only, suggesting that unaffected siblings also provide information about linkage and association.     

A Discordant-Sibship Test for Disequilibrium and Linkage: No Need for Parental Data

The sibship disequilibrium test (SDT) is designed to detect both linkage in the presence of association and association in the presence of linkage (linkage disequilibrium). The test does not require parental data but requires discordant sibships with at least one affected and one unaffected sibling. The SDT has many desirable properties: it uses all the siblings in the sibship; it remains valid if there are misclassifications of the affectation status; it does not detect spurious associations due to population stratification; asymptotically it has a chi2 distribution under the null hypothesis; and exact P values can be easily computed for a biallelic marker. We show how to extend the SDT to markers with multiple alleles and how to combine families with parents and data from discordant sibships. We discuss the power of the test by presenting sample-size calculations involving a complex disease model, and we present formulas for the asymptotic relative efficiency (which is approximately the ratio of sample sizes) between SDT and the transmission/disequilibrium test (TDT) for special family structures. For sib pairs, we compare the SDT to a test proposed both by Curtis and, independently, by Spielman and Ewens. We show that, for discordant sib pairs, the SDT has good power for testing linkage disequilibrium relative both to Curtis''s tests and to the TDT using trios comprising an affected sib and its parents. With additional sibs, we show that the SDT can be more powerful than the TDT for testing linkage disequilibrium, especially for disease prevalence >.3.     

A test for linkage and association in general pedigrees: the pedigree disequilibrium test

Family-based tests of linkage disequilibrium typically are based on nuclear-family data including affected individuals and their parents or their unaffected siblings. A limitation of such tests is that they generally are not valid tests of association when data from related nuclear families from larger pedigrees are used. Standard methods require selection of a single nuclear family from any extended pedigrees when testing for linkage disequilibrium. Often data are available for larger pedigrees, and it would be desirable to have a valid test of linkage disequilibrium that can use all potentially informative data. In this study, we present the pedigree disequilibrium test (PDT) for analysis of linkage disequilibrium in general pedigrees. The PDT can use data from related nuclear families from extended pedigrees and is valid even when there is population substructure. Using computer simulations, we demonstrated validity of the test when the asymptotic distribution is used to assess the significance, and examined statistical power. Power simulations demonstrate that, when extended pedigree data are available, substantial gains in power can be attained by use of the PDT rather than existing methods that use only a subset of the data. Furthermore, the PDT remains more powerful even when there is misclassification of unaffected individuals. Our simulations suggest that there may be advantages to using the PDT even if the data consist of independent families without extended family information. Thus, the PDT provides a general test of linkage disequilibrium that can be widely applied to different data structures.     

A sibship test for linkage in the presence of association: the sib transmission/disequilibrium test.

Linkage analysis with genetic markers has been successful in the localization of genes for many monogenic human diseases. In studies of complex diseases, however, tests that rely on linkage disequilibrium (the simultaneous presence of linkage and association) are often more powerful than those that rely on linkage alone. This advantage is illustrated by the transmission/disequilibrium test (TDT). The TDT requires data (marker genotypes) for affected individuals and their parents; for some diseases, however, data from parents may be difficult or impossible to obtain. In this article, we describe a method, called the "sib TDT" (or "S-TDT"), that overcomes this problem by use of marker data from unaffected sibs instead of from parents, thus allowing application of the principle of the TDT to sibships without parental data. In a single collection of families, there might be some that can be analyzed only by the TDT and others that are suitable for analysis by the S-TDT. We show how all the data may be used jointly in one overall TDT-type procedure that tests for linkage in the presence of association. These extensions of the TDT will be valuable for the study of diseases of late onset, such as non-insulin-dependent diabetes, cardiovascular diseases, and other diseases associated with aging.     

Inclusion of unaffected sibs increases power in model-free linkage analysis of a behavioral trait

Study design strategies are of critical importance in the search for genes underlying complex diseases. Two important design choices in planning gene mapping studies are the analytic strategy to be used, which will have an impact on the type of data to be collected, and the choice of genetic markers. In the present paper, we used the simulated behavioral trait data provided in the Genetic Analysis Workshop 14 to: 1) investigate the usefulness of incorporating unaffected sibs in model-free linkage analysis and, 2) compare linkage results of genome scans using a 7-cM microsatellite map with a 3-cM single nucleotide polymorphisms map. To achieve these aims, we used the maximum-likelihood-binomial method with two different coding approaches. We defined the unaffected sibs as those totally free of phenotypes correlated to the disease. Without prior knowledge of the answers, we were able to correctly localize 2 out of 5 loci (LOD > 3) in a sample of 200 families that included the unaffected sibs but only one locus when based on an affected-only strategy, using either microsatellite or SNPs genome scan. LOD scores were considerably higher using the analytic strategy which incorporated the unaffected sibs. In conclusion, including unaffected sibs in model-free linkage analysis of complex binary traits is helpful, at least when complete parental data are available, whereas there are no striking advantages in using single nucleotide polymorphisms over microsatellite map at marker densities used in the current study.     

Transmission disequilibrium test with discordant sib pairs when parents are available

The transmission disequilibrium test (TDT) has been employed to map disease susceptibility loci (DSL), while being immune to the problem of population admixture. The customary TDT test (TDT(D)) was developed for affected child(ren) and their parents and was most often applied to case-parent trios. Recently, the TDT has been extended to the situations when (1) parents are not available but affected and nonaffected sibs from each family are available, (2) unrelated control-parent trios are available for combined analyses with case-parent trios (TDT(DC)), and (3) large pedigrees. For many diseases, affected children in the case-parent trios enlisted into the TDT(D) have unaffected sibs who can be recruited. We present an extension of the TDT by effectively incorporating one unaffected sib of each of the affected children in the case-parent trios into a single analysis (TDT(DS), where DS denotes discordant sib pairs). We have developed a general analytical method for computing the statistical power of the TDT(DS) under any genetic model, the accuracy of which is validated by computer simulations. We compare the power of the TDT(D), TDT(DC), and TDT(DS) under a range of parameter space and genetic models. We find that the TDT(DS) is generally more powerful than the TDT(DC) and TDT(D), particularly when the disease is prevalent (>30%) in the population. The relative power of the TDT(D) and the TDT(DS) largely depends upon the allele frequencies and genetic effects at the DSL, whereas the recombination rate, the degree of linkage disequilibrium, and the marker allele frequencies have little effect. Importantly, the TDT(DS) not only may be more powerful, it also has the advantage of being able to test for segregation distortion that may yield false linkage/association in the TDT(D).     

Power evaluations for family-based tests of association with incomplete parental genotypes

While a variety of methods have been developed to deal with incomplete parental genotype information in family-based association tests, sampling design issues with incomplete parental genotype data still have not received much attention. In this article, we present simulation studies with four genetic models and various sampling designs and evaluate power in family-based association studies. Efficiency depends heavily on disease prevalence. With rare diseases, sampling affecteds and their parents is preferred, and three sibs will be required to have close power if parents are unavailable. With more common diseases, sampling affecteds and two sibs will generally be more efficient than trios. When parents are unavailable, siblings need not be phenotyped if the disease is rare, but a loss of power will result with common diseases. Finally, for a class of complex traits where other genetic and environmental factors also cause phenotypic correlation among siblings, little loss of efficiency occurs to rare disease, but substantial loss of efficiency occurs to common disease.     

Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.     

Comparing strategies for association mapping in samples with related individuals

In this paper, different strategies to test for association in samples with related individuals designed for linkage studies are compared. Because no independent controls are available, a family-based association test and case-control tests corrected for the presence of related individuals in which unaffected relatives are used as controls were tested. When unrelated controls are available, additional strategies including selection of a single case per family considering either all families or a subset of linked families, are also considered. Analyses are performed on the simulated dataset, blind to the answers. The case-control test corrected for the presence of related individuals is the most powerful strategy to detect three loci associated with the disease under study. Using a correction factor for the case-control test performed conditional on the marker information rather than unconditional does not impact the power significantly.     

Trend tests for case-control studies of genetic markers: power,sample size and robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.     

Identification of Mendelian inconsistencies between SNP and pedigree information of sibs

Background

Using SNP genotypes to apply genomic selection in breeding programs is becoming common practice. Tools to edit and check the quality of genotype data are required. Checking for Mendelian inconsistencies makes it possible to identify animals for which pedigree information and genotype information are not in agreement.

Methods

Straightforward tests to detect Mendelian inconsistencies exist that count the number of opposing homozygous marker (e.g. SNP) genotypes between parent and offspring (PAR-OFF). Here, we develop two tests to identify Mendelian inconsistencies between sibs. The first test counts SNP with opposing homozygous genotypes between sib pairs (SIBCOUNT). The second test compares pedigree and SNP-based relationships (SIBREL). All tests iteratively remove animals based on decreasing numbers of inconsistent parents and offspring or sibs. The PAR-OFF test, followed by either SIB test, was applied to a dataset comprising 2,078 genotyped cows and 211 genotyped sires. Theoretical expectations for distributions of test statistics of all three tests were calculated and compared to empirically derived values. Type I and II error rates were calculated after applying the tests to the edited data, while Mendelian inconsistencies were introduced by permuting pedigree against genotype data for various proportions of animals.

Results

Both SIB tests identified animal pairs for which pedigree and genomic relationships could be considered as inconsistent by visual inspection of a scatter plot of pairwise pedigree and SNP-based relationships. After removal of 235 animals with the PAR-OFF test, SIBCOUNT (SIBREL) identified 18 (22) additional inconsistent animals.Seventeen animals were identified by both methods. The numbers of incorrectly deleted animals (Type I error), were equally low for both methods, while the numbers of incorrectly non-deleted animals (Type II error), were considerably higher for SIBREL compared to SIBCOUNT.

Conclusions

Tests to remove Mendelian inconsistencies between sibs should be preceded by a test for parent-offspring inconsistencies. This parent-offspring test should not only consider parent-offspring pairs based on pedigree data, but also those based on SNP information. Both SIB tests could identify pairs of sibs with Mendelian inconsistencies. Based on type I and II error rates, counting opposing homozygotes between sibs (SIBCOUNT) appears slightly more precise than comparing genomic and pedigree relationships (SIBREL) to detect Mendelian inconsistencies between sibs.     

HLA and disease: predictions for HLA haplotype sharing in families.

An analysis of published data on the segregation of HLA haplotypes in families with more than one individual affected with insulin-dependent diabetes mellitus or multiple sclerosis yields three conclusions: (1) In families with unaffected parents, affected sib pairs are much more often HLA haplotype identical in sibships with two affected sibs than in sibships with three or four affected sibs (P less than .01). (2) In families with unaffected parents and HLA half-identical affected sibs, well siblings more often receive the single haplotype not found in the affected sibs than is expected by chance (P less than .05). (3) In families with one affected parent, well siblings of affected individuals may share with the affected child a haplotype from the unaffected parent less than 50% of the time (P less than .10). These results are consistent with the premise that in some non-Mendelian, familial, HLA-associated disease more than one gene may contribute to susceptibility to the disorder.     

Increasing the power and efficiency of disease-marker case-control association studies through use of allele-sharing information

Case-control disease-marker association studies are often used in the search for variants that predispose to complex diseases. One approach to increasing the power of these studies is to enrich the case sample for individuals likely to be affected because of genetic factors. In this article, we compare three case-selection strategies that use allele-sharing information with the standard strategy that selects a single individual from each family at random. In affected sibship samples, we show that, by carefully selecting sibships and/or individuals on the basis of allele sharing, we can increase the frequency of disease-associated alleles in the case sample. When these cases are compared with unrelated controls, the difference in the frequency of the disease-associated allele is therefore also increased. We find that, by choosing the affected sib who shows the most evidence for pairwise allele sharing with the other affected sibs in families, the test statistic is increased by >20%, on average, for additive models with modest genotype relative risks. In addition, we find that the per-genotype information associated with the allele sharing-based strategies is increased compared with that associated with random selection of a sib for genotyping. Even though we select sibs on the basis of a nonparametric statistic, the additional gain for selection based on the unknown underlying mode of inheritance is minimal. We show that these properties hold even when the power to detect linkage to a region in the entire sample is negligible. This approach can be extended to more-general pedigree structures and quantitative traits.     

Effect of two- and three-locus linkage disequilibrium on the power to detect marker/phenotype associations

There has been much recent interest in describing the patterns of linkage disequilibrium (LD) along a chromosome. Most empirical studies that have examined this issue have concentrated on LD between collections of pairs of markers and have not considered the joint effect of a group of markers beyond these pairwise connections. Here, we examine many different patterns of LD defined by both pairwise and joint multilocus LD terms. The LD patterns we considered were chosen in part by examining those seen in real data. We examine how changes in these patterns affect the power to detect association when performing single-marker and haplotype-based case-control tests, including a novel haplotype test based on contrasting LD between affected and unaffected individuals. Through our studies we find that differences in power between single-marker tests and haplotype-based tests in general do not appear to be large. Where moderate to high levels of multilocus LD exist, haplotype tests tend to be more powerful. Single-marker tests tend to prevail when pairwise LD is high. For moderate pairwise values and weak multilocus LD, either testing strategy may come out ahead, although it is also quite likely that neither has much power.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.