Power and sample size calculations for case-control genetic association tests when errors are present: application to single nucleotide polymorphisms

The purpose of this work is to quantify the effects that errors in genotyping have on power and the sample size necessary to maintain constant asymptotic Type I and Type II error rates (SSN) for case-control genetic association studies between a disease phenotype and a di-allelic marker locus, for example a single nucleotide polymorphism (SNP) locus. We consider the effects of three published models of genotyping errors on the chi-square test for independence in the 2 x 3 table. After specifying genotype frequencies for the marker locus conditional on disease status and error model in both a genetic model-based and a genetic model-free framework, we compute the asymptotic power to detect association through specification of the test's non-centrality parameter. This parameter determines the functional dependence of SSN on the genotyping error rates. Additionally, we study the dependence of SSN on linkage disequilibrium (LD), marker allele frequencies, and genotyping error rates for a dominant disease model. Increased genotyping error rate requires a larger SSN. Every 1% increase in sum of genotyping error rates requires that both case and control SSN be increased by 2-8%, with the extent of increase dependent upon the error model. For the dominant disease model, SSN is a nonlinear function of LD and genotyping error rate, with greater SSN for lower LD and higher genotyping error rate. The combination of lower LD and higher genotyping error rates requires a larger SSN than the sum of the SSN for the lower LD and for the higher genotyping error rate.     

基于选择基因型对数量性状进行关联分析

数量性状的遗传分析可以通过＂选择基因型＂的方式完成。本文提出了一个利用极端样本来对数量性状位点（QTL）进行关联分析的统计量T。统计量T比较上极端群体样本中具有纯合子标记的性状值差异。通过计算机模拟考察了无关联情形时T的分布和Ⅰ型错误率,结果表明,在各种样本选择策略下,T的分布近似于χ^2-分布,Ⅰ型错误率接近设定的显著性水平。同时,考察了各种遗传模型下不同遗传率,不同样本大小,及不同样本选择阈值对T的统计功效的影响,结果表明,T的功效随着标记和QTL间连锁不平衡程度的增强及遗传率和样本大小的增大而增大,当样本选择阈值更严格时,功效也越大。     

A family-based likelihood ratio test for general pedigree structures that allows for genotyping error and missing data

The purpose of this work is the development of a family-based association test that allows for random genotyping errors and missing data and makes use of information on affected and unaffected pedigree members. We derive the conditional likelihood functions of the general nuclear family for the following scenarios: complete parental genotype data and no genotyping errors; only one genotyped parent and no genotyping errors; no parental genotype data and no genotyping errors; and no parental genotype data with genotyping errors. We find maximum likelihood estimates of the marker locus parameters, including the penetrances and population genotype frequencies under the null hypothesis that all penetrance values are equal and under the alternative hypothesis. We then compute the likelihood ratio test. We perform simulations to assess the adequacy of the central chi-square distribution approximation when the null hypothesis is true. We also perform simulations to compare the power of the TDT and this likelihood-based method. Finally, we apply our method to 23 SNPs genotyped in nuclear families from a recently published study of idiopathic scoliosis (IS). Our simulations suggest that this likelihood ratio test statistic follows a central chi-square distribution with 1 degree of freedom under the null hypothesis, even in the presence of missing data and genotyping errors. The power comparison shows that this likelihood ratio test is more powerful than the original TDT for the simulations considered. For the IS data, the marker rs7843033 shows the most significant evidence for our method (p = 0.0003), which is consistent with a previous report, which found rs7843033 to be the 2nd most significant TDTae p value among a set of 23 SNPs.     

Dye shift: a neglected source of genotyping error in molecular ecology

Molecular ecologists must be vigilant in detecting and accounting for genotyping error, yet potential errors stemming from dye-induced mobility shift (dye shift) may be frequently neglected and largely unknown to researchers who employ 3-primer systems with automated genotyping. When left uncorrected, dye shift can lead to mis-scoring alleles and even to falsely calling new alleles if different dyes are used to genotype the same locus in subsequent reactions. When we used four different fluorophore labels from a standard dye set to genotype the same set of loci, differences in the resulting size estimates for a single allele ranged from 2.07 bp to 3.68 bp. The strongest effects were associated with the fluorophore PET, and relative degree of dye shift was inversely related to locus size. We found little evidence in the literature that dye shift is regularly accounted for in 3-primer studies, despite knowledge of this phenomenon existing for over a decade. However, we did find some references to erroneous standard correction factors for the same set of dyes that we tested. We thus reiterate the need for strict quality control when attempting to reduce possible sources of genotyping error, and in cases where different dyes are applied to a single locus, perhaps mistakenly, we strongly discourage researchers from assuming generic correction patterns.     

The impact of missing and erroneous genotypes on tagging SNP selection and power of subsequent association tests

OBJECTIVE: Single nucleotide polymorphisms (SNPs) serve as effective markers for localizing disease susceptibility genes, but current genotyping technologies are inadequate for genotyping all available SNP markers in a typical linkage/association study. Much attention has recently been paid to methods for selecting the minimal informative subset of SNPs in identifying haplotypes, but there has been little investigation of the effect of missing or erroneous genotypes on the performance of these SNP selection algorithms and subsequent association tests using the selected tagging SNPs. The purpose of this study is to explore the effect of missing genotype or genotyping error on tagging SNP selection and subsequent single marker and haplotype association tests using the selected tagging SNPs. METHODS: Through two sets of simulations, we evaluated the performance of three tagging SNP selection programs in the presence of missing or erroneous genotypes: Clayton's diversity based program htstep, Carlson's linkage disequilibrium (LD) based program ldSelect, and Stram's coefficient of determination based program tagsnp.exe. RESULTS: When randomly selected known loci were relabeled as 'missing', we found that the average number of tagging SNPs selected by all three algorithms changed very little and the power of subsequent single marker and haplotype association tests using the selected tagging SNPs remained close to the power of these tests in the absence of missing genotype. When random genotyping errors were introduced, we found that the average number of tagging SNPs selected by all three algorithms increased. In data sets simulated according to the haplotype frequecies in the CYP19 region, Stram's program had larger increase than Carlson's and Clayton's programs. In data sets simulated under the coalescent model, Carlson's program had the largest increase and Clayton's program had the smallest increase. In both sets of simulations, with the presence of genotyping errors, the power of the haplotype tests from all three programs decreased quickly, but there was not much reduction in power of the single marker tests. CONCLUSIONS: Missing genotypes do not seem to have much impact on tagging SNP selection and subsequent single marker and haplotype association tests. In contrast, genotyping errors could have severe impact on tagging SNP selection and haplotype tests, but not on single marker tests.     

An examination of the genotyping error detection function of SIMWALK2

This investigation was undertaken to assess the sensitivity and specificity of the genotyping error detection function of the computer program SIMWALK2. We chose to examine chromosome 22, which had 7 microsatellite markers, from a single simulated replicate (330 pedigrees with a pattern of missing genotype data similar to the Framingham families). We created genotype errors at five overall frequencies (0.0, 0.025, 0.050, 0.075, and 0.100) and applied SIMWALK2 to each of these five data sets, respectively assuming that the total error rate (specified in the program), was at each of these same five levels. In this data set, up to an assumed error rate of 10%, only 50% of the Mendelian-consistent mistypings were found under any level of true errors. And since as many as 70% of the errors detected were false-positives, blanking suspect genotypes (at any error probability) will result in a reduction of statistical power due to the concomitant blanking of correctly typed alleles. This work supports the conclusion that allowing for genotyping errors within likelihood calculations during statistical analysis may be preferable to choosing an arbitrary cut-off.     

Linkage analysis in the presence of errors II: marker-locus genotyping errors modeled with hypercomplex recombination fractions

It is well known that genotyping errors lead to loss of power in gene-mapping studies and underestimation of the strength of correlations between trait- and marker-locus genotypes. In two-point linkage analysis, these errors can be absorbed in an inflated recombination-fraction estimate, leaving the test statistic quite robust. In multipoint analysis, however, genotyping errors can easily result in false exclusion of the true location of a disease-predisposing gene. In a companion article, we described a "complex-valued" extension of the recombination fraction to accommodate errors in the assignment of trait-locus genotypes, leading to a multipoint LOD score with the same robustness to errors in trait-locus genotypes that is seen with the conventional two-point LOD score. Here, a further extension of this model to "hypercomplex-valued" recombination fractions (hereafter referred to as "hypercomplex recombination fractions") is presented, to handle random and systematic sources of marker-locus genotyping errors. This leads to a multipoint method (either "model-based" or "model-free") with the same robustness to marker-locus genotyping errors that is seen with conventional two-point analysis but with the advantage that multiple marker loci can be used jointly to increase meiotic informativeness. The cost of this increased robustness is a decrease in fine-scale resolution of the estimated map location of the trait locus, in comparison with traditional multipoint analysis. This probability model further leads to algorithms for the estimation of the lower bounds for the error rates for genomewide and locus-specific genotyping, based on the null-hypothesis distribution of the LOD-score statistic in the presence of such errors. It is argued that those genome scans in which the LOD score is 0 for >50% of the genome are likely to be characterized by high rates of genotyping errors in general.     

Significant deviations from Hardy-Weinberg equilibrium caused by low levels of microsatellite genotyping errors

Microsatellite genotyping from samples with varying quality can result in an uneven distribution of errors. Previous studies reporting error rates have focused on estimating the effects of both randomly distributed and locus‐specific errors. Sample‐specific errors, however, can also significantly affect results in population studies despite a large sample size. From two studies including six microsatellite markers genotyped from 272 sperm whale DNA samples, and 33 microsatellites genotyped from 213 bowhead whales, we investigated the effects of sample‐ and locus‐specific errors on calculations of Hardy–Weinberg equilibrium. The results of a jackknife analysis in these two studies identified seven individuals that were highly influential on estimates of Hardy–Weinberg equilibrium for six different markers. In each case, the influential individual was homozygous for a rare allele. Our results demonstrate that Hardy–Weinberg P values are very sensitive to homozygosity in rare alleles for single individuals, and that > 50% of these cases involved genotype errors likely due to low sample quality. This raises the possibility that even small, normal levels of laboratory errors can result in an overestimate of the degree to which markers are out of Hardy–Weinberg equilibrium and hence overestimate population structure. To avoid such bias, we recommend routine identification of influential individuals and multiple replications of those samples.     

Effect of genotyping error in model-free linkage analysis using microsatellite or single-nucleotide polymorphism marker maps

Errors while genotyping are inevitable and can reduce the power to detect linkage. However, does genotyping error have the same impact on linkage results for single-nucleotide polymorphism (SNP) and microsatellite (MS) marker maps? To evaluate this question we detected genotyping errors that are consistent with Mendelian inheritance using large changes in multipoint identity-by-descent sharing in neighboring markers. Only a small fraction of Mendelian consistent errors were detectable (e.g., 18% of MS and 2.4% of SNP genotyping errors). More SNP genotyping errors are Mendelian consistent compared to MS genotyping errors, so genotyping error may have a greater impact on linkage results using SNP marker maps. We also evaluated the effect of genotyping error on the power and type I error rate using simulated nuclear families with missing parents under 0, 0.14, and 2.8% genotyping error rates. In the presence of genotyping error, we found that the power to detect a true linkage signal was greater for SNP (75%) than MS (67%) marker maps, although there were also slightly more false-positive signals using SNP marker maps (5 compared with 3 for MS). Finally, we evaluated the usefulness of accounting for genotyping error in the SNP data using a likelihood-based approach, which restores some of the power that is lost when genotyping error is introduced.     

Quantifying the percent increase in minimum sample size for SNP genotyping errors in genetic model-based association studies

Kang et al. [Genet Epidemiol 2004;26:132-141] addressed the question of which genotype misclassification errors are most costly, in terms of minimum percentage increase in sample size necessary (%MSSN) to maintain constant asymptotic power and significance level, when performing case/control studies of genetic association in a genetic model-free setting. They answered the question for single nucleotide polymorphisms (SNPs) using the 2 x 3 chi2 test of independence. We address the same question here for a genetic model-based framework. The genetic model parameters considered are: disease model (dominant, recessive), genotypic relative risk, SNP (marker) and disease allele frequency, and linkage disequilibrium. %MSSN coefficients of each of the six possible error rates are determined by expanding the non-centrality parameter of the asymptotic distribution of the 2 x 3 chi2 test under a specified alternative hypothesis to approximate %MSSN using a linear Taylor series in the error rates. In this work we assume errors misclassifying one homozygote as another homozygote are 0, since these errors are thought to rarely occur in practice. Our findings are that there are settings of the genetic model parameters that lead to large total %MSSN for both dominant and recessive models. As SNP minor allele approaches 0, total %MSSN increases without bound, independent of other genetic model parameters. In general, %MSSN is a complex function of the genetic model parameters. Use of SNPs with small minor allele frequency requires careful attention to frequency of genotyping errors to insure that power specifications are met. Software to perform these calculations for study design is available, and an example of its use to study a disease is given.     

Estimation of genotyping error rate from repeat genotyping,unintentional recaptures and known parent–offspring comparisons in 16 microsatellite loci for brown rockfish (Sebastes auriculatus)

Genotyping errors are present in almost all genetic data and can affect biological conclusions of a study, particularly for studies based on individual identification and parentage. Many statistical approaches can incorporate genotyping errors, but usually need accurate estimates of error rates. Here, we used a new microsatellite data set developed for brown rockfish (Sebastes auriculatus) to estimate genotyping error using three approaches: (i) repeat genotyping 5% of samples, (ii) comparing unintentionally recaptured individuals and (iii) Mendelian inheritance error checking for known parent–offspring pairs. In each data set, we quantified genotyping error rate per allele due to allele drop‐out and false alleles. Genotyping error rate per locus revealed an average overall genotyping error rate by direct count of 0.3%, 1.5% and 1.7% (0.002, 0.007 and 0.008 per allele error rate) from replicate genotypes, known parent–offspring pairs and unintentionally recaptured individuals, respectively. By direct‐count error estimates, the recapture and known parent–offspring data sets revealed an error rate four times greater than estimated using repeat genotypes. There was no evidence of correlation between error rates and locus variability for all three data sets, and errors appeared to occur randomly over loci in the repeat genotypes, but not in recaptures and parent–offspring comparisons. Furthermore, there was no correlation in locus‐specific error rates between any two of the three data sets. Our data suggest that repeat genotyping may underestimate true error rates and may not estimate locus‐specific error rates accurately. We therefore suggest using methods for error estimation that correspond to the overall aim of the study (e.g. known parent–offspring comparisons in parentage studies).     

Incorporating individual error rate into association test of unmatched case-control design

OBJECTIVES: Genotyping error commonly occurs and could reduce the power and bias statistical inference in genetics studies. In addition to genotypes, some automated biotechnologies also provide quality measurement of each individual genotype. We studied the relationship between the quality measurement and genotyping error rate. Furthermore, we propose two association tests incorporating the genotyping quality information with the goal to improve statistical power and inference. METHODS: 50 pairs of DNA sample duplicates were typed for 232 SNPs by BeadArray technology. We used scatter plot, smoothing function and generalized additive models to investigate the relationship between genotype quality score (q) and inconsistency rate (?) among duplicates. We constructed two association tests: (1) weighted contingency table test (WCT) and (2) likelihood ratio test (LRT) to incorporate individual genotype error rate (epsilon(i)), in unmatched case-control setting. RESULTS: In the 50 duplicates, we found q and ? were in strong negative association, suggesting the genotypes with low quality score were more likely to be mistyped. The WCT improved the statistical power and partially corrects the bias in point estimation. The LRT offered moderate power gain, but was able to correct the bias in odds ratio estimation. The two new methods also performed favorably in some scenarios when epsilon(i) was mis-specified. CONCLUSIONS: With increasing number of genetic studies and application of automated genotyping technology, there is a growing need to adequately account for individual genotype error rate in statistical analysis. Our study represents an initial step to address this need and points out a promising direction for further research.     

Associative overdominance: Evaluating the effects of inbreeding and linkage disequilibrium

Expressions are obtained for the expected phenotypic values of homozygous and heterozygous genotypes for a neutral marker locus linked to a locus segregating for a recessive deleterious gene. The phenotypic values are functions of the allele frequencies at the marker locus, the inbreeding coefficient and the degree of association of the deleterious gene with the marker alleles. The analysis is extended to more than two alleles at the marker locus. Either linkage disequilibrium or inbreeding alone can produce an apparent superiority of heterozygotes for the marker locus (unless specified otherwise, the terms ‘homozygote’ and ‘heterozygote’ will refer to the marker locus). The effect of linkage disequilibrium on the difference between the heterozygote and homozygote values can be positive (associative overdominance) or negative (associative underdominance), depending on the frequencies of the marker alleles and the degree of their association with the deleterious gene. Inbreeding has always a positive effect. In general, the expected value of a homozygote is a positive function of its allele frequency. When the various homozygous genotypes are combined into one class and the various heterozygous genotypes into another, the phenotypic difference of the two classes is a function of the evenness of the allelic frequency distribution. Inbreeding is a more likely explanation of associative overdominance if the frequency of the deleterious gene is low, but its effect on the character high. Conversely, linkage disequilibrium is more likely if the frequency is high and the effect low. The degrees of association between marker alleles and the deleterious gene can, in principle, be estimated from the observed phenotypic scores and used to calculate expected multi-locus genotype scores. This could provide the basis for statistical tests of the associative overdominance hypothesis as an explanation of observed correlations between multi-locus heterozygosity and phenotypic traits.     

TECHNICAL ADVANCES: Effects of genotyping protocols on success and errors in identifying individual river otters (Lontra canadensis) from their faeces

In noninvasive genetic sampling, when genotyping error rates are high and recapture rates are low, misidentification of individuals can lead to overestimation of population size. Thus, estimating genotyping errors is imperative. Nonetheless, conducting multiple polymerase chain reactions (PCRs) at multiple loci is time-consuming and costly. To address the controversy regarding the minimum number of PCRs required for obtaining a consensus genotype, we compared consumer-style the performance of two genotyping protocols (multiple-tubes and 'comparative method') in respect to genotyping success and error rates. Our results from 48 faecal samples of river otters (Lontra canadensis) collected in Wyoming in 2003, and from blood samples of five captive river otters amplified with four different primers, suggest that use of the comparative genotyping protocol can minimize the number of PCRs per locus. For all but five samples at one locus, the same consensus genotypes were reached with fewer PCRs and with reduced error rates with this protocol compared to the multiple-tubes method. This finding is reassuring because genotyping errors can occur at relatively high rates even in tissues such as blood and hair. In addition, we found that loci that amplify readily and yield consensus genotypes, may still exhibit high error rates (7-32%) and that amplification with different primers resulted in different types and rates of error. Thus, assigning a genotype based on a single PCR for several loci could result in misidentification of individuals. We recommend that programs designed to statistically assign consensus genotypes should be modified to allow the different treatment of heterozygotes and homozygotes intrinsic to the comparative method.     

megasat: automated inference of microsatellite genotypes from sequence data

megasat is software that enables genotyping of microsatellite loci using next‐generation sequencing data. Microsatellites are amplified in large multiplexes, and then sequenced in pooled amplicons. megasat reads sequence files and automatically scores microsatellite genotypes. It uses fuzzy matches to allow for sequencing errors and applies decision rules to account for amplification artefacts, including nontarget amplification products, replication slippage during PCR (amplification stutter) and differential amplification of alleles. An important feature of megasat is the generation of histograms of the length–frequency distributions of amplification products for each locus and each individual. These histograms, analogous to electropherograms traditionally used to score microsatellite genotypes, enable rapid evaluation and editing of automatically scored genotypes. megasat is written in Perl, runs on Windows, Mac OS X and Linux systems, and includes a simple graphical user interface. We demonstrate megasat using data from guppy, Poecilia reticulata. We genotype 1024 guppies at 43 microsatellites per run on an Illumina MiSeq sequencer. We evaluated the accuracy of automatically called genotypes using two methods, based on pedigree and repeat genotyping data, and obtained estimates of mean genotyping error rates of 0.021 and 0.012. In both estimates, three loci accounted for a disproportionate fraction of genotyping errors; conversely, 26 loci were scored with 0–1 detected error (error rate ≤0.007). Our results show that with appropriate selection of loci, automated genotyping of microsatellite loci can be achieved with very high throughput, low genotyping error and very low genotyping costs.     

An entropy-based genome-wide transmission/disequilibrium test

Availability of a large collection of single nucleotide polymorphisms (SNPs) and efficient genotyping methods enable the extension of linkage and association studies for complex diseases from small genomic regions to the whole genome. Establishing global significance for linkage or association requires small P-values of the test. The original TDT statistic compares the difference in linear functions of the number of transmitted and nontransmitted alleles or haplotypes. In this report, we introduce a novel TDT statistic, which uses Shannon entropy as a nonlinear transformation of the frequencies of the transmitted or nontransmitted alleles (or haplotypes), to amplify the difference in the number of transmitted and nontransmitted alleles or haplotypes in order to increase statistical power with large number of marker loci. The null distribution of the entropy-based TDT statistic and the type I error rates in both homogeneous and admixture populations are validated using a series of simulation studies. By analytical methods, we show that the power of the entropy-based TDT statistic is higher than the original TDT, and this difference increases with the number of marker loci. Finally, the new entropy-based TDT statistic is applied to two real data sets to test the association of the RET gene with Hirschsprung disease and the Fcγ receptor genes with systemic lupus erythematosus. Results show that the entropy-based TDT statistic can reach p-values that are small enough to establish genome-wide linkage or association analyses.     

Identification and analysis of error types in high-throughput genotyping

Although it is clear that errors in genotyping data can lead to severe errors in linkage analysis, there is as yet no consensus strategy for identification of genotyping errors. Strategies include comparison of duplicate samples, independent calling of alleles, and Mendelian-inheritance-error checking. This study aimed to develop a better understanding of error types associated with microsatellite genotyping, as a first step toward development of a rational error-detection strategy. Two microsatellite marker sets (a commercial genomewide set and a custom-designed fine-resolution mapping set) were used to generate 118,420 and 22,500 initial genotypes and 10,088 and 8,328 duplicates, respectively. Mendelian-inheritance errors were identified by PedManager software, and concordance was determined for the duplicate samples. Concordance checking identifies only human errors, whereas Mendelian-inheritance-error checking is capable of detection of additional errors, such as mutations and null alleles. Neither strategy is able to detect all errors. Inheritance checking of the commercial marker data identified that the results contained 0.13% human errors and 0.12% other errors (0.25% total error), whereas concordance checking found 0.16% human errors. Similarly, Mendelian-inheritance-error checking of the custom-set data identified 1.37% errors, compared with 2.38% human errors identified by concordance checking. A greater variety of error types were detected by Mendelian-inheritance-error checking than by duplication of samples or by independent reanalysis of gels. These data suggest that Mendelian-inheritance-error checking is a worthwhile strategy for both types of genotyping data, whereas fine-mapping studies benefit more from concordance checking than do studies using commercial marker data. Maximization of error identification increases the likelihood of linkage when complex diseases are analyzed.     

The effect that genotyping errors have on the robustness of common linkage-disequilibrium measures

The rapid development of a dense single-nucleotide-polymorphism marker map has stimulated numerous studies attempting to characterize the magnitude and distribution of background linkage disequilibrium (LD) within and between human populations. Although genotyping errors are an inherent problem in all LD studies, there have been few systematic investigations documenting their consequences on estimates of background LD. Therefore, we derived simple deterministic formulas to investigate the effect that genotyping errors have on four commonly used LD measures-D', r, Q, and d-in studies of background LD. We have found that genotyping error rates as small as 3% can have serious affects on these LD measures, depending on the allele frequencies and the assumed error model. Furthermore, we compared the robustness of D', r, Q, and d, in the presence of genotyping errors. In general, Q and d are more robust than D' and r, although exceptions do exist. Finally, through stochastic simulations, we illustrate how genotyping errors can lead to erroneous inferences when measures of LD between two samples are compared.     

Disequilibrium mapping of a quantitative-trait locus in an expanding population.

Linkage disequilibrium (LD) mapping can be successful if there is strong nonrandom association between marker alleles and an allele affecting a trait of interest. The principles of LD mapping of dichotomous traits are well understood, but less is known about LD mapping of a quantitative-trait locus (QTL). It is shown in this report that selective genotyping can increase the power to detect and map a rare allele of large effect at a QTL. Two statistical tests of the association between an allele and a quantitative character are proposed. These tests are approximately independent, so information from them can be combined. Analytic theory is developed to show that these two tests are effective in detecting the presence of a low-frequency allele with a relatively large effect on the character when the QTL is either already a candidate locus or closely linked to a marker locus that is in strong LD with the QTL. The latter situation is expected in a rapidly growing population in which the allele of large effect was present initially in one copy. Therefore, the proposed tests are useful under the same conditions as those for successful LD mapping of a dichotomous trait or disease. Simulations show that, for detection of the presence of a QTL, these tests are more powerful than a simple t-test. The tests also provide a basis for defining a measure of association, gamma, between a low-frequency allele at a putative QTL and a low-frequency allele at a marker locus.