Statistical evaluation of age-at-onset anticipation: a new test and evaluation of its behavior in realistic applications.

The discovery that microsatellite repeat expansions can cause clinical disease has fostered renewed interest in testing for age-at-onset anticipation (AOA). A commonly used procedure is to sample affected parent-child pairs (APCPs) from available data sets and to test for a difference in mean age at onset between the parents and the children. However, standard statistical methods fail to take into account the right truncation of both the parent and child age-at-onset distributions under this design, with the result that type I error rates can be inflated substantially. Previously, we had introduced a new test, based on the correct, bivariate right-truncated, age-at-onset distribution. We showed that this test has the correct type I error rate for random APCPs, even for quite small samples. However, in that paper, we did not consider two key statistical complications that arise when the test is applied to realistic data. First, affected pairs usually are sampled from pedigrees preferentially selected for the presence of multiple affected individuals. In this paper, we show that this will tend to inflate the type I error rate of the test. Second, we consider the appropriate probability model under the alternative hypothesis of true AOA due to an expanding microsatellite mechanism, and we show that there is good reason to believe that the power to detect AOA may be quite small, even for substantial effect sizes. When the type I error rate of the test is high relative to the power, interpretation of test results becomes problematic. We conclude that, in many applications, AOA tests based on APCPs may not yield meaningful results.     

The affected-/discordant-sib-pair design can guarantee validity of multipoint model-free linkage analysis of incomplete pedigrees when there is marker-marker disequilibrium

Genomewide linkage studies are tending toward the use of single-nucleotide polymorphisms (SNPs) as the markers of choice. However, linkage disequilibrium (LD) between tightly linked SNPs violates the fundamental assumption of linkage equilibrium (LE) between markers that underlies most multipoint calculation algorithms currently available, and this leads to inflated affected-relative-pair allele-sharing statistics when founders' multilocus genotypes are unknown. In this study, we investigate the impact that the degree of LD, marker allele frequency, and association type have on estimating the probabilities of sharing alleles identical by descent in multipoint calculations and hence on type I error rates of different sib-pair linkage approaches that assume LE. We show that marker-marker LD does not inflate type I error rates of affected sib pair (ASP) statistics in the whole parameter space, and that, in any case, discordant sib pairs (DSPs) can be used to control for marker-marker LD in ASPs. We advocate the ASP/DSP design with appropriate sib-pair statistics that test the difference in allele sharing between ASPs and DSPs.     

ASP – a simulation-based power calculator for genetic linkage studies of qualitative traits, using sib-pairs

An easy-to-use, simulation-based power calculator (ASP) for linkage analysis using sib-pair designs (concordant or discordant) has been developed and made publicly available via the Internet. The program employs a diallelic model for the trait locus, at which parental/offspring genotypes are simulated, assuming Hardy-Weinberg equilibrium in the parental generation. Genotypes at a linked multiallelic marker locus are simulated conditional upon the inheritance pattern at the trait locus, allowing for recombination. Marker genotypes are tested for non-Mendelian identity-by-descent sharing, using both an unrestricted and a restricted likelihood ratio test, the latter representing an extension of the "mean test" from fully to partially informative families. The power of user-defined datasets is estimated by the number of simulations giving significant results at varying type I error levels.     

Effect of genotyping error on type-I error rate of affected sib pair studies with genotyped parents

OBJECTIVE: In affected sib pair studies without genotyped parents the effect of genotyping error is generally to reduce the type I error rate and power of tests for linkage. The effect of genotyping error when parents have been genotyped is unknown. We investigated the type I error rate of the single-point Mean test for studies in which genotypes of both parents are available. METHODS: Datasets were simulated assuming no linkage and one of five models for genotyping error. In each dataset, Mendelian-inconsistent families were either excluded or regenotyped, and then the Mean test applied. RESULTS: We found that genotyping errors lead to an inflated type I error rate when inconsistent families are excluded. Depending on the genotyping-error model assumed, regenotyping inconsistent families has one of several effects. It may produce the same type I error rate as if inconsistent families are excluded; it may reduce the type I error, but still leave an anti-conservative test; or it may give a conservative test. Departures of the type I error rate from its nominal level increase with both the genotyping error rate and sample size. CONCLUSION: We recommend that markers with high error rates either be excluded from the analysis or be regenotyped in all families.     

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

A method for incorporating ages at onset in affected sibpair linkage studies

Data with varying age at disease onset arise frequently in studies of mapping disease associated genes. Naively combining affected subjects with different ages at onset may result in a much reduced power in detecting the disease genes. In this paper we present a weighted score test statistic to detect the linkage between marker and latent disease loci using affected sibpairs, where the weight is used for assigning differential contribution due to the varying age at onset of each affected sibpair to the test statistic. We show that the weighted test has a correct type I error rate asymptotically. For an illustrative purpose, we analyze a data set from the 12th Genetic Analysis Workshop. The result shows that the weighted tests appear to be able to pinpoint the location of latent disease genes better than the mean IBD test with equal weight with respect to the age at onset. To avoid the potential power loss due to the improper weight, we propose to use a combined test statistic, taking the maximum of two tests, one that is weighted by the age-dependent penetrance function and the other that may be invariant to the age. We conduct an analytical study, comparing the combined test with weighted and equal weight with respect to age test. It shows that the combined test retains the most power of the better one of the two tests being combined.     

A simple nonparametric multipoint procedure to test for linkage through mothers or fathers as well as imprinting effects in the presence of linkage

A simple multipoint procedure to test for parent-of-origin effects in samples of affected siblings is discussed. The procedure consists of artificially changing all full sibs to half-sibs, with distinct mothers or fathers depending on the parental origin to be evaluated, then analyzing these families with commonly used statistics and software. The procedure leads to tests for linkage through mothers or fathers and also leads to a test for imprinting effects in the presence of linkage. Moreover, simulations illustrate that in regions unlinked to susceptibility genes this multipoint procedure does not have an inflated type I error if a sex-averaged genetic map is used, even when large differences exist between male-specific and female-specific maps. In regions linked with susceptibility genes, the test of imprinting is biased under the null hypothesis if differences exist between sex-specific maps, irrespective of the map used in the analysis. The procedure is applied to the Collaborative Study on the Genetics of Alcoholism dataset from the Genetic Analysis Workshop 14. Results indicate that brothers categorized as affected according to the DMS-III-R and Feighner classification show evidence of linkage through fathers to the 6q25 region (p = 0.00038) as well as modest evidence of imprinting (p = 0.018). This region harbors OPRM1, a candidate gene for substance dependence.     

Two-locus disease models with two marker loci: the power of affected-sib-pair tests.

Recently, Schork et al. found that two-trait-locus, two-marker-locus (parametric) linkage analysis can provide substantially more linkage information than can standard one-trait-locus, one-marker-locus methods. However, because of the increased burden of computation, Schork et al. do not expect that their approach will be applied in an initial genome scan. Further, the specification of a suitable two-locus segregation model can be crucial. Affected-sibpair tests are computationally simple and do not require an explicit specification of the disease model. In the past, however, these tests mainly have been applied to data with a single marker locus. Here, we consider sib-pair tests that make it possible to analyze simultaneously two marker loci. The power of these tests is investigated for different (epistatic and heterogeneous) two-trait-locus models, each trait locus being linked to one of the marker loci. We compare these tests both with the test that is optimal for a certain model and with the strategy that analyzes each marker locus separately. The results indicate that a straightforward extension of the well-known mean test for two marker loci can be much more powerful than single-marker-locus analysis and that is power is only slightly inferior to the power of the optimal test.     

Combined linkage and association mapping of quantitative trait loci by multiple markers

Using multiple diallelic markers, variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL) based on nuclear families. The objective is to build a model that may fully use marker information for fine association mapping of QTL in the presence of prior linkage. The measures of linkage disequilibrium and the genetic effects are incorporated in the mean coefficients and are decomposed into orthogonal additive and dominance effects. The linkage information is modeled in variance-covariance matrices. Hence, the proposed methods model both association and linkage in a unified model. On the basis of marker information, a multipoint interval mapping method is provided to estimate the proportion of allele sharing identical by descent (IBD) and the probability of sharing two alleles IBD at a putative QTL for a sib-pair. To test the association between the trait locus and the markers, both likelihood-ratio tests and F-tests can be constructed on the basis of the proposed models. In addition, analytical formulas of noncentrality parameter approximations of the F-test statistics are provided. Type I error rates of the proposed test statistics are calculated to show their robustness. After comparing with the association between-family and association within-family (AbAw) approach by Abecasis and Fulker et al., it is found that the method proposed in this article is more powerful and advantageous based on simulation study and power calculation. By power and sample size comparison, it is shown that models that use more markers may have higher power than models that use fewer markers. The multiple-marker analysis can be more advantageous and has higher power in fine mapping QTL. As an application, the Genetic Analysis Workshop 12 German asthma data are analyzed using the proposed methods.     

A note on exact tests of Hardy-Weinberg equilibrium

Deviations from Hardy-Weinberg equilibrium (HWE) can indicate inbreeding, population stratification, and even problems in genotyping. In samples of affected individuals, these deviations can also provide evidence for association. Tests of HWE are commonly performed using a simple chi2 goodness-of-fit test. We show that this chi2 test can have inflated type I error rates, even in relatively large samples (e.g., samples of 1,000 individuals that include approximately 100 copies of the minor allele). On the basis of previous work, we describe exact tests of HWE together with efficient computational methods for their implementation. Our methods adequately control type I error in large and small samples and are computationally efficient. They have been implemented in freely available code that will be useful for quality assessment of genotype data and for the detection of genetic association or population stratification in very large data sets.     

Impact of non-ignorable missingness on genetic tests of linkage and/or association using case-parent trios

The transmission/disequilibrium test was introduced to test for linkage disequilibrium between a marker and a putative disease locus using case-parent trios. However, parental genotypes may be incomplete in such a study. When parental information is non-randomly missing, due, for example, to death from the disease under study, the impact on type I error and power under dominant and recessive disease models has been reported. In this paper, we examine non-ignorable missingness by assigning missing values to the genotypes of affected parents. We used unrelated case-parent trios in the Genetic Analysis Workshop 14 simulated data for the Danacaa population. Our computer simulations revealed that the type I error of these tests using incomplete trios was not inflated over the nominal level under either recessive or dominant disease models. However, the power of these tests appears to be inflated over the complete information case due to an excess of heterozygous parents in dyads.     

Which strategy is better for linkage analysis: single-nucleotide polymorphisms or microsatellites? Evaluation by identity-by-state – identity-by-descent transformation affected sib-pair method on GAW14 data

The central issue for Genetic Analysis Workshop 14 (GAW14) is the question, which is the better strategy for linkage analysis, the use of single-nucleotide polymorphisms (SNPs) or microsatellite markers? To answer this question we analyzed the simulated data using Duffy's SIB-PAIR program, which can incorporate parental genotypes, and our identity-by-state – identity-by-descent (IBS-IBD) transformation method of affected sib-pair linkage analysis which uses the matrix transformation between IBS and IBD. The advantages of our method are as follows: the assumption of Hardy-Weinberg equilibrium is not necessary; the parental genotype information maybe all unknown; both IBS and its related IBD transformation can be used in the linkage analysis; the determinant of the IBS-IBD transformation matrix provides a quantitative measure of the quality of the marker in linkage analysis. With the originally distributed simulated data, we found that 1) for microsatellite markers there are virtually no differences in types I and II error rates when parental genotypes were or were not used; 2) on average, a microsatellite marker has more power than a SNP marker does in linkage detection; 3) if parental genotype information is used, SNP markers show lower type I error rates than microsatellite markers; and 4) if parental genotypes are not available, SNP markers show considerable variation in type I error rates for different methods.     

Comparison of multivariate tests for genetic linkage

OBJECTIVES: Multivariate tests for linkage can provide improved power over univariate tests but the type I error rates and comparative power of commonly used methods have not previously been compared. Here we studied the behavior of bivariate formulations of the variance component (VC) and Haseman-Elston (H-E) approaches. METHODS: We compared through simulation studies the bivariate H-E test with the unconstrained bivariate VC approach and with a VC approach in which the major-gene correlation is constrained to +/-1. We also compared these methods to univariate methods. RESULTS: Bivariate approaches are more powerful than univariate analyses unless the traits are very highly positively correlated. The power of the bivariate H-E test was less than the VC procedures. The constrained test was often less powerful than the unconstrained test. The empirical distributions of the bivariate H-E test and the unconstrained bivariate VC test conformed with asymptotic distributions for samples of 100 or more sibships of size 4. CONCLUSIONS: The unconstrained VC test is valuable for testing for preliminary linkages using multivariate phenotypes. The bivariate H-E test was less powerful than the bivariate VC tests.     

Imprinting detection by extending a regression-based QTL analysis method

We present an extension of a regression-based quantitative-trait linkage analysis method to incorporate parent-of-origin effects. We separately regressed total, paternal, and maternal IBD sharing on traits’ squared sums and differences. We also developed a test for imprinting that indicates whether there is any difference between the paternal and maternal regression coefficients. Since this method treats the identity-by-descent information as the dependent variable that is conditioned on the trait, it can be readily applied to data from complex ascertainment processes. We performed a simulation study to examine the performance of the method. We found that when using empirical critical values, the method shows identical or higher power compared to existing methods for evaluation of parent-of-origin effect in linkage analysis of quantitative traits. Missing parental genotypes increase the type I error rate of the linkage test and decrease the power of the imprinting test. When the major gene has a low heritability, the power of the method decreases considerably, but the statistical tests still perform well. We also applied a permutation algorithm, which ensures the appropriate type I error rate for the test for imprinting. The method was applied to a data from a study of 6 body size related measures and 23 loci on chromosome 7 for 255 nuclear families. Multipoint identities-by-descent (IBD) were obtained using a modification of the SIMWALK 2 program. A parent-of-origin effect consistent with maternal imprinting was suggested at 99.67–111.26 Mb for body mass index, bioelectrical impedance analysis, waist circumference, and leptin concentration. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. An erratum to this article can be found at     

A tournament of linkage tests in complex inheritance

The performance of some weakly parametric linkage tests in common use was compared on 200 replicates of oligogenic inheritance from Genetic Analysis Workshop 10. Each random sample for the quantitative trait was dichotomized at different thresholds and also selected through 2 affected sibs, generating 8 combinations of sample and variable. The variance component program SOLAR performed best with a continuous trait, even in selected samples, when the population mean was used. The sib-pair program SIBPAL2 was best in most other cases when the phenotype product, population mean, and empirical estimates of pair correlations were used. The BETA program that introduced phenotype products was slightly more powerful than maximum likelihood scores under the null hypothesis and approached but did not exceed SIBPAL2 under its optimal conditions. Type I errors generally exceeded expectations from a chi(2) test, but were conservative with respect to bounds on lods. All methods can be improved by use of the population mean, empirical correlations, logistic representation for affection status, and correct lods for samples that favour the null hypothesis. It remains uncertain whether all information can be extracted by weakly parametric methods and whether correction for ascertainment bias demands a strongly parametric model. Performance on a standard set of simulated data is indispensable for recognising optimal methods.     

Transmission-disequilibrium tests for quantitative traits.

The transmission-disequilibrium test (TDT) of Spielman et al. is a family-based linkage-disequilibrium test that offers a powerful way to test for linkage between alleles and phenotypes that is either causal (i.e., the marker locus is the disease/trait allele) or due to linkage disequilibrium. The TDT is equivalent to a randomized experiment and, therefore, is resistant to confounding. When the marker is extremely close to the disease locus or is the disease locus itself, tests such as the TDT can be far more powerful than conventional linkage tests. To date, the TDT and most other family-based association tests have been applied only to dichotomous traits. This paper develops five TDT-type tests for use with quantitative traits. These tests accommodate either unselected sampling or sampling based on selection of phenotypically extreme offspring. Power calculations are provided and show that, when a candidate gene is available (1) these TDT-type tests are at least an order of magnitude more efficient than two common sib-pair tests of linkage; (2) extreme sampling results in substantial increases in power; and (3) if the most extreme 20% of the phenotypic distribution is selectively sampled, across a wide variety of plausible genetic models, quantitative-trait loci explaining as little as 5% of the phenotypic variation can be detected at the .0001 alpha level with <300 observations.     

Robustness and power of the maximum-likelihood-binomial and maximum-likelihood-score methods,in multipoint linkage analysis of affected-sibship data.

The maximum-likelihood-binomial (MLB) method, based on the binomial distribution of parental marker alleles among affected offspring, recently was shown to provide promising results by two-point linkage analysis of affected-sibship data. In this article, we extend the MLB method to multipoint linkage analysis, using the general framework of hidden Markov models. Furthermore, we perform a large simulation study to investigate the robustness and power of the MLB method, compared with those of the maximum-likelihood-score (MLS) method as implemented in MAPMAKER/SIBS, in the multipoint analysis of different affected-sibship samples. Analyses of multiple-affected sibships by means of the MLS were conducted by consideration of all possible sib pairs, with (weighted MLS [MLSw]) or without (unweighted MLS [MLSu]) application of a classic weighting procedure. In simulations under the null hypothesis, the MLB provided very consistent type I errors regardless of the type of family sample (sib pairs or multiple-affected sibships), as did the MLS for samples with sib pairs only. When samples included multiple-affected sibships, the MLSu led to inflation of low type I errors, whereas the MLSw yielded very conservative tests. Power comparisons showed that the MLB generally was more powerful than the MLS, except in recessive models with allele frequencies <.3. Missing parental marker data did not strongly influence type I error and power results in these multipoint analyses. The MLB approach, which in a natural way accounts for multiple-affected sibships and which provides a simple likelihood-ratio test for linkage, is an interesting alternative for multipoint analysis of sibships.     

Analysis of affected sib pairs, with covariates--with and without constraints

Covariate models have previously been developed as an extension to affected-sib-pair methods in which the covariate effects are jointly estimated with the degree of excess allele sharing. These models can estimate the differences in sib-pair allele sharing that are associated with measurable environment or genes. When there are no covariates, the pattern of identical-by-descent allele sharing in affected sib pairs is expected to fall within a small triangular region of the potential parameter space, under most genetic models. By restriction of the estimated allele sharing to this triangle, improved power is obtained in tests for genetic linkage. When the affected-sib-pair model is generalized to allow for covariates that affect allele sharing, however, new constraints and new methods for the application of constraints are required. Three generalized constraint methods are proposed and evaluated by use of simulated data. The results compare the power of the different methods, with and without covariates, for a single-gene model with age-dependent onset and for quantitative and qualitative gene-environment and gene-gene interaction models. Covariates can improve the power to detect linkage and can be particularly valuable when there are qualitative gene-environment interactions. In most situations, the best strategy is to assume that there is no dominance variance and to obtain constrained estimates for covariate models under this assumption.     

Associating Multivariate Quantitative Phenotypes with Genetic Variants in Family Samples with a Novel Kernel Machine Regression Method

The recent development of sequencing technology allows identification of association between the whole spectrum of genetic variants and complex diseases. Over the past few years, a number of association tests for rare variants have been developed. Jointly testing for association between genetic variants and multiple correlated phenotypes may increase the power to detect causal genes in family-based studies, but familial correlation needs to be appropriately handled to avoid an inflated type I error rate. Here we propose a novel approach for multivariate family data using kernel machine regression (denoted as MF-KM) that is based on a linear mixed-model framework and can be applied to a large range of studies with different types of traits. In our simulation studies, the usual kernel machine test has inflated type I error rates when applied directly to familial data, while our proposed MF-KM method preserves the expected type I error rates. Moreover, the MF-KM method has increased power compared to methods that either analyze each phenotype separately while considering family structure or use only unrelated founders from the families. Finally, we illustrate our proposed methodology by analyzing whole-genome genotyping data from a lung function study.     

Accounting for linkage in family-based tests of association with missing parental genotypes

In studies of complex diseases, a common paradigm is to conduct association analysis at markers in regions identified by linkage analysis, to attempt to narrow the region of interest. Family-based tests for association based on parental transmissions to affected offspring are often used in fine-mapping studies. However, for diseases with late onset, parental genotypes are often missing. Without parental genotypes, family-based tests either compare allele frequencies in affected individuals with those in their unaffected siblings or use siblings to infer missing parental genotypes. An example of the latter approach is the score test implemented in the computer program TRANSMIT. The inference of missing parental genotypes in TRANSMIT assumes that transmissions from parents to affected siblings are independent, which is appropriate when there is no linkage. However, using computer simulations, we show that, when the marker and disease locus are linked and the data set consists of families with multiple affected siblings, this assumption leads to a bias in the score statistic under the null hypothesis of no association between the marker and disease alleles. This bias leads to an inflated type I error rate for the score test in regions of linkage. We present a novel test for association in the presence of linkage (APL) that correctly infers missing parental genotypes in regions of linkage by estimating identity-by-descent parameters, to adjust for correlation between parental transmissions to affected siblings. In simulated data, we demonstrate the validity of the APL test under the null hypothesis of no association and show that the test can be more powerful than the pedigree disequilibrium test and family-based association test. As an example, we compare the performance of the tests in a candidate-gene study in families with Parkinson disease.