Power calculations for genetic association studies using estimated probability distributions

The determination of the power of-or of an appropriate sample size for-genetic association studies that exploit linkage disequilibrium requires many assumptions. Some of the more important assumptions include the linkage-disequilibrium strength among alleles at the observed marker-locus sites and a potential trait-influencing locus, the frequencies of the marker locus and trait-influencing alleles, and the ultimate density of the marker locus "map" (i.e., the number of bases between marker loci) necessary in order to identify, with some confidence, trait-influencing alleles. I consider an approach to assessment of the power and sample-size requirements of genetic case-control association study designs that makes use of empirically derived estimates of the distributions of important parameters often assumed to take on arbitrary values. My proposed methodology is extremely general and flexible and ultimately can provide realistic answers to questions such as "How many markers and/or how many individuals might it take to identify, with confidence, a disease gene, via linkage-disequilibrium and association methods from a candidate region or whole genome perspective?" I showcase aspects of the proposed methodology, using information abstracted from the literature.     

Linkage Disequilibrium between Allozymes in Natural Populations of Lodgepole Pine

Pairwise linkage disequilibrium values (D) were estimated for 14 allozyme loci in two natural populations of lodgepole pine (Pinus contorta ssp. latifolia). Maternal multilocus genotypes were inferred from samples of (haploid) megagametophytic seed-endosperms. Coupling/repulsion double heterozygotes were distinguished for closely linked pairs of loci. Assays of seven of the loci in seed embryos allowed estimates of D for these loci in the outcross pollen pool (estimates of outcrossing rates indicate no significant departures from random mating in either population). No disequilibrium was observed between unlinked loci in either maternal genotypes or outcross pollen. However, significant disequilibrium was observed within and between gametes for some allelic combinations of four tightly linked loci; the assumption of random association of gamete types within individuals is thus invalid for some loci in lodgepole pine. Possible causes of the observed D were examined using the noncentrality parameter of the general noncentral chi square distribution. We concluded, from estimates of population size, linkage and measurements of population substructure, that neither drift nor population subdivision was responsible for the significant values of D which were observed and that epistatic selection was the most likely cause of the disequilibrium observed.     

Use of restriction fragment length polymorphisms for genetic counseling: Population genetic considerations

Two-locus population genetic models are analyzed to evaluate the utility of restriction fragment length polymorphisms for purposes of genetic counseling. It is shown that the linkage disequilibrium between a neutral marker and a tightly linked overdominant mutant will increase rapidly as the mutant moves to its polymorphic equilibrium. The linkage disequilibrium decays for deleterious recessive mutants. Two measures involving the linkage disequilibrium are investigated to determine how much information the transmission of the neutral marker provides about the transmission of the selected gene. In certain kinds of matings, where the parental two-locus genotypes and linkage phases are known, it is possible to determine whether or not a progeny is homozygous for the selected gene on the basis of the fetal genotype at the marker locus. A quantity of primary interest is the fraction of matings between individuals heterozygous for the selected gene in which exact diagnosis can be made in this way. The expected proportion of such matings, taken over all two-locus matings involving heterozygotes at the selected locus, is calculated as a function of the gene frequencies at the two loci and the linkage disequilibrium between them. This expected value is maximized when the linkage disequilibrium is at its maximum in absolute value. Fewer than half of all matings are informative if the linkage disequilibrium is small in magnitude or if the gene frequencies at the two loci are quite different. Consideration is also given to various conditional measures of association that may be useful when the parental two-locus genotypes are unknown. The results suggest that the utility of tightly linked neutral marker genes in predicting the transmission of a selected gene is generally less when selection acts against a recessive gene than for overdominant selection.     

Paralogs are revealed by proportion of heterozygotes and deviations in read ratios in genotyping‐by‐sequencing data from natural populations

Whole‐genome duplications have occurred in the recent ancestors of many plants, fish, and amphibians, resulting in a pervasiveness of paralogous loci and the potential for both disomic and tetrasomic inheritance in the same genome. Paralogs can be difficult to reliably genotype and are often excluded from genotyping‐by‐sequencing (GBS) analyses; however, removal requires paralogs to be identified which is difficult without a reference genome. We present a method for identifying paralogs in natural populations by combining two properties of duplicated loci: (i) the expected frequency of heterozygotes exceeds that for singleton loci, and (ii) within heterozygotes, observed read ratios for each allele in GBS data will deviate from the 1:1 expected for singleton (diploid) loci. These deviations are often not apparent within individuals, particularly when sequence coverage is low; but, we postulated that summing allele reads for each locus over all heterozygous individuals in a population would provide sufficient power to detect deviations at those loci. We identified paralogous loci in three species: Chinook salmon (Oncorhynchus tshawytscha) which retains regions with ongoing residual tetrasomy on eight chromosome arms following a recent whole‐genome duplication, mountain barberry (Berberis alpina) which has a large proportion of paralogs that arose through an unknown mechanism, and dusky parrotfish (Scarus niger) which has largely rediploidized following an ancient whole‐genome duplication. Importantly, this approach only requires the genotype and allele‐specific read counts for each individual, information which is readily obtained from most GBS analysis pipelines.     

Microsatellites for tree of heaven (Ailanthus altissima)

Nine (CT)_n microsatellites were developed for tree of heaven, Ailanthus altissima, from invasive populations on the Mediterranean islands. These loci had seven to 12 alleles in 96 trees from five islands. Two loci had significant deficits of heterozygotes within islands while the other loci were in Hardy–Weinberg equilibrium, and four pairs of loci had significant linkage disequilibrium within a single island. These loci were also polymorphic in one to three individuals of the tree of heaven varieties, Ailanthus altissima erythrocarpa, Ailanthus altissima sutchuenensis and Ailanthus altissima tanakai, and the related species Ailanthus giraldii and Ailanthus vilmariniana.     

Properties of the multiallelic trend test

Disease genes can be mapped on the basis of associations between genetic markers and disease status, with the case-control design having the advantage of not requiring individuals from different generations. When the marker loci have multiple alleles, there has been debate on whether the power of tests for association increases or decreases. We show here that the multiple-allele version of Armitage's trend test has increased power over the two-allele version under the requirement of equifrequent alleles, but not in general. The trend test has the advantage of remaining valid even when the sampled population is not in Hardy-Weinberg equilibrium. A departure from Hardy-Weinberg means that association tests depend on gametic and nongametic linkage disequilibrium between marker and disease loci, and we illustrate the magnitude of these effects with simulated data.     

Multilocus linkage analysis of affected sib pairs

OBJECTIVE: The conventional affected sib pair methods evaluate the linkage information at a locus by considering only marginal information. We describe a multilocus linkage method that uses both the marginal information and information derived from the possible interactions among several disease loci, thereby increasing the significance of loci with modest effects. METHODS: Our method is based on a statistic that quantifies the linkage information contained in a set of markers. By a marker selection-reduction process, we screen a set of polymorphisms and select a few that seem linked to disease. RESULTS: We test our approach on genome scan data for inflammatory bowel disease (InfBD) and on simulated data. On real data we detect 6 of the 8 known InfBD loci; on simulated data we obtain improvements in power of up to 40% compared to a conventional single-locus method. CONCLUSION: Our extensive simulations and the results on real data show that our method is in general more powerful than single-locus methods in detecting disease loci responsible for complex traits. A further advantage of our approach is that it can be extended to make use of both the linkage and the linkage disequilibrium between disease loci and nearby markers.     

Novel measures of linkage disequilibrium that correct the bias due to population structure and relatedness

Among the several linkage disequilibrium measures known to capture different features of the non-independence between alleles at different loci, the most commonly used for diallelic loci is the r(2) measure. In the present study, we tackled the problem of the bias of r(2) estimate, which results from the sample structure and/or the relatedness between genotyped individuals. We derived two novel linkage disequilibrium measures for diallelic loci that are both extensions of the usual r(2) measure. The first one, r(S)(2), uses the population structure matrix, which consists of information about the origins of each individual and the admixture proportions of each individual genome. The second one, r(V)(2), includes the kinship matrix into the calculation. These two corrections can be applied together in order to correct for both biases and are defined either on phased or unphased genotypes.We proved that these novel measures are linked to the power of association tests under the mixed linear model including structure and kinship corrections. We validated them on simulated data and applied them to real data sets collected on Vitis vinifera plants. Our results clearly showed the usefulness of the two corrected r(2) measures, which actually captured 'true' linkage disequilibrium unlike the usual r(2) measure.     

General equations for Pt, Ps, and the power of the TDT and the affected-sib-pair test

Several equations are highlighted here, whose algebraic symmetries and generality make them very useful for understanding and comparing the properties of the transmission disequilibrium test (TDT) and affected sib-pair test. Methods using the equations are also presented that yield precise estimates of sample sizes needed for genome scans or for testing a single candidate gene, and these power methods are shown to compare favorably with alternative approaches recently described by Knapp (1999) and by Tu and Whittemore (1999). Simple relationships are also noted that summarize the relative sample sizes required for equivalent power to detect association by the TDT or case-control designs. As single-nucleotide polymorphism (SNP) maps revolutionize the search for disease-causing genes, the equations should prove useful for planning and evaluating studies of linkage and association across a broad range of possible disease models and relationships between markers and linked disease loci.     

Efficient study designs for test of genetic association using sibship data and unrelated cases and controls

Linkage mapping of complex diseases is often followed by association studies between phenotypes and marker genotypes through use of case-control or family-based designs. Given fixed genotyping resources, it is important to know which study designs are the most efficient. To address this problem, we extended the likelihood-based method of Li et al., which assesses whether there is linkage disequilibrium between a disease locus and a SNP, to accommodate sibships of arbitrary size and disease-phenotype configuration. A key advantage of our method is the ability to combine data from different family structures. We consider scenarios for which genotypes are available for unrelated cases, affected sib pairs (ASPs), or only one sibling per ASP. We construct designs that use cases only and others that use unaffected siblings or unrelated unaffected individuals as controls. Different combinations of cases and controls result in seven study designs. We compare the efficiency of these designs when the number of individuals to be genotyped is fixed. Our results suggest that (1) when the disease is influenced by a single gene, the one sibling per ASP-control design is the most efficient, followed by the ASP-control design, and familial cases contribute more association information than singleton cases; (2) when the disease is influenced by multiple genes, familial cases provide more association information than singleton cases, unless the effect of the locus being tested is much smaller than at least one other untested disease locus; and (3) the case-control design can be useful for detecting genes with small effect in the presence of genes with much larger effect. Our findings will be helpful for researchers designing and analyzing complex disease-association studies and will facilitate genotyping resource allocation.     

Mapping autosomal recessive vitamin D dependency type I to chromosome 12q14 by linkage analysis.

Linkage analysis in French-Canadian families with vitamin D dependency type I (VDD1) demonstrated that the gene responsible for the disease is linked to polymorphic RFLP markers in the 12q14 region. We studied 76 subjects in 14 sibships which included 17 affected individuals and 17 obligate heterozygotes. Significant results for linkage were obtained with the D12S17 locus at the male recombination fraction (theta m) .018 (Z[theta m theta f] = 3.20) and with D126 at (theta m = .025 (Z[theta m theta f] = 3.07). Multipoint linkage analysis and studies of haplotypes and recombinants strongly suggest the localization of the VDD1 locus between the collagen type II alpha 1 (COL2A1) locus and clustered loci D12S14, D12S17, and D12S6, which segregate as a three-marker haplotype. Linkage disequilibrium between VDD1 and this three-marker haplotype supports the notion of a founder effect in the studied population. The current status of the localization of the disease allows for carrier detection in the families at risk.     

Enzyme Polymorphism and Cyclic Parthenogenesis in DAPHNIA MAGNA. II. Heterosis following Sexual Reproduction

Cyclical parthenogenesis exaggerates the force of selection relative to recombination and will therefore enhance interlocus effects. Observations of electrophoretic variation in a natural population of Daphnia magna Straus (Crustacea: Cladocera) are interpreted in this light. Sexual reproduction led to Hardy-Weinberg equilibrium, but heterozygote excesses rapidly developed at each of three observed loci during subsequent parthenogenesis. Homozygote fecundity was often lower than that of heterozygotes; this may have been the cause of some of the observed frequency changes. The superior fitness of the enzyme heterozygotes does not imply that selection was necessarily acting on the enzyme loci themselves, since apparent heterosis is the expected result of linkage disequilibrium.     

Characterization of 10 novel microsatellite loci for the brown marbled grouper, Epinephelus fuscoguttatus (Serranidae)

Epinephelus fuscoguttatus is a commercially important marine fish species in southeast Asia. Due to overfishing and water pollution, this species has been declared as near-threatened. Thus, to provide information to help maintain and preserve the species, microsatellites were developed, using an enriched genomic library method. Thirty individuals were collected from the hatchery of the Fishery Research Institute, Terengganu, Malaysia. These individuals, from four to six years old, originated from Sabah and are maintained in captive culture as broodstock. Genomic DNA was extracted from the fins of selected individuals that weighed 3-8 kg. Ten microsatellite loci were found to be polymorphic in this population, with 5 to 21 alleles per locus. Observed and expected heterozygosities ranged from 0.53 to 0.97 and 0.59 to 0.95, respectively. Only one locus deviated significantly from Hardy-Weinberg equilibrium and no significant linkage disequilibrium was found among the pairs of loci. These polymorphic microsatellite loci will be used by the Malaysian Fishery Research Institute for investigating genetic diversity and for developing breeding strategies.     

Incorporating genotypes of relatives into a test of linkage disequilibrium.

Genetic data from autosomal loci in diploids generally consist of genotype data for which no phase information is available, making it difficult to implement a test of linkage disequilibrium. In this paper, we describe a test of linkage disequilibrium based on an empirical null distribution of the likelihood of a sample. Information on the genotypes of related individuals is explicitly used to help reconstruct the gametic phase of the independent individuals. Simulation studies show that the present approach improves on estimates of linkage disequilibrium gathered from samples of completely independent individuals but only if some offspring are sampled together with their parents. The failure to incorporate some parents sharply decreases the sensitivity and accuracy of the test. Simulations also show that for multiallelic data (more than two alleles) our testing procedure is not as powerful as an exact test based on known haplotype frequencies, owing to the interaction between departure from Hardy-Weinberg equilibrium and linkage disequilibrium.     

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.     

Linkage disequilibrium fine mapping of quantitative trait loci: A simulation study

Recently, the use of linkage disequilibrium (LD) to locate genes which affect quantitative traits (QTL) has received an increasing interest, but the plausibility of fine mapping using linkage disequilibrium techniques for QTL has not been well studied. The main objectives of this work were to (1) measure the extent and pattern of LD between a putative QTL and nearby markers in finite populations and (2) investigate the usefulness of LD in fine mapping QTL in simulated populations using a dense map of multiallelic or biallelic marker loci. The test of association between a marker and QTL and the power of the test were calculated based on single-marker regression analysis. The results show the presence of substantial linkage disequilibrium with closely linked marker loci after 100 to 200 generations of random mating. Although the power to test the association with a frequent QTL of large effect was satisfactory, the power was low for the QTL with a small effect and/or low frequency. More powerful, multi-locus methods may be required to map low frequent QTL with small genetic effects, as well as combining both linkage and linkage disequilibrium information. The results also showed that multiallelic markers are more useful than biallelic markers to detect linkage disequilibrium and association at an equal distance.     

Maximum-likelihood estimation of gene location by linkage disequilibrium.

Linkage disequilibrium, D, between a polymorphic disease and mapped markers can, in principle, be used to help find the map position of the disease gene. Likelihoods are therefore derived for the value of D conditional on the observed number of haplotypes in the sample and on the population parameter Nc, where N is the effective population size and c the recombination fraction between the disease and marker loci. The likelihood is computed explicitly for the case of two loci with heterozygote superiority and, more generally, by computer simulations assuming a steady state of constant population size and selective pressures or neutrality. It is found that the likelihood is, in general, not very dependent on the degree of selection at the loci and is very flat. This suggests that precise information on map position will not be obtained from estimates of linkage disequilibrium.     

Multilocus association testing of quantitative traits based on partial least-squares analysis

Because of combining the genetic information of multiple loci, multilocus association studies (MLAS) are expected to be more powerful than single locus association studies (SLAS) in disease genes mapping. However, some researchers found that MLAS had similar or reduced power relative to SLAS, which was partly attributed to the increased degrees of freedom (dfs) in MLAS. Based on partial least-squares (PLS) analysis, we develop a MLAS approach, while avoiding large dfs in MLAS. In this approach, genotypes are first decomposed into the PLS components that not only capture majority of the genetic information of multiple loci, but also are relevant for target traits. The extracted PLS components are then regressed on target traits to detect association under multilinear regression. Simulation study based on real data from the HapMap project were used to assess the performance of our PLS-based MLAS as well as other popular multilinear regression-based MLAS approaches under various scenarios, considering genetic effects and linkage disequilibrium structure of candidate genetic regions. Using PLS-based MLAS approach, we conducted a genome-wide MLAS of lean body mass, and compared it with our previous genome-wide SLAS of lean body mass. Simulations and real data analyses results support the improved power of our PLS-based MLAS in disease genes mapping relative to other three MLAS approaches investigated in this study. We aim to provide an effective and powerful MLAS approach, which may help to overcome the limitations of SLAS in disease genes mapping.     

Isolation and characterization of 30 novel polymorphic microsatellite loci from Japanese halfbeak, Hyporhamphus sajori (Temminck et Schlegel, 1846)

The construction of genetic linkage map and evaluation of population genetic diversity both require large numbers of polymorphic molecular markers. In the study, 60 microsatellite loci were isolated from a dinucleotide-enriched genomic library of Japanese halfbeak (Hyporhamphus sajori). And 30 polymorphic microsatellite loci were found to be polymorphic between 2 and 11 alleles. The number of observed, expected heterozygosity and polymorphism information content per locus in 24 individuals ranged from 0.1667 to 1.000, 0.1828 to 0.9220, 0.1828 to 0.8945, respectively. Three loci significantly deviated from Hardy–Weinberg equilibrium after Bonferroni correction analysis and there was no significant linkage disequilibrium found between pairs of loci. As a result, 30 microsatellite loci probably should provide sufficient level of genetic diversity to investigate the fine-scale population structure, stock management and enhancement, genetic linkage map construction and molecular marker-assisted breeding in H. sajori.