Order‐restricted Scores Test for the Evaluation of Population‐based Case–control Studies when the Genetic Model is Unknown

The Cochran–Armitage (CA) linear trend test for proportions is often used for genotype‐based analysis of candidate gene association. Depending on the underlying genetic mode of inheritance, the use of model‐specific scores maximises the power. Commonly, the underlying genetic model, i.e. additive, dominant or recessive mode of inheritance, is a priori unknown. Association studies are commonly analysed using permutation tests, where both inference and identification of the underlying mode of inheritance are important. Especially interesting are tests for case–control studies, defined by a maximum over a series of standardised CA tests, because such a procedure has power under all three genetic models. We reformulate the test problem and propose a conditional maximum test of scores‐specific linear‐by‐linear association tests. For maximum‐type, sum and quadratic test statistics the asymptotic expectation and covariance can be derived in a closed form and the limiting distribution is known. Both the limiting distribution and approximations of the exact conditional distribution can easily be computed using standard software packages. In addition to these technical advances, we extend the area of application to stratified designs, studies involving more than two groups and the simultaneous analysis of multiple loci by means of multiplicity‐adjusted p‐values for the underlying multiple CA trend tests. The new test is applied to reanalyse a study investigating genetic components of different subtypes of psoriasis. A new and flexible inference tool for association studies is available both theoretically as well as practically since already available software packages can be easily used to implement the suggested test procedures.     

Robust genomic control for association studies

Population-based case-control studies are a useful method to test for a genetic association between a trait and a marker. However, the analysis of the resulting data can be affected by population stratification or cryptic relatedness, which may inflate the variance of the usual statistics, resulting in a higher-than-nominal rate of false-positive results. One approach to preserving the nominal type I error is to apply genomic control, which adjusts the variance of the Cochran-Armitage trend test by calculating the statistic on data from null loci. This enables one to estimate any additional variance in the null distribution of statistics. When the underlying genetic model (e.g., recessive, additive, or dominant) is known, genomic control can be applied to the corresponding optimal trend tests. In practice, however, the mode of inheritance is unknown. The genotype-based chi (2) test for a general association between the trait and the marker does not depend on the underlying genetic model. Since this general association test has 2 degrees of freedom (df), the existing formulas for estimating the variance factor by use of genomic control are not directly applicable. By expressing the general association test in terms of two Cochran-Armitage trend tests, one can apply genomic control to each of the two trend tests separately, thereby adjusting the chi (2) statistic. The properties of this robust genomic control test with 2 df are examined by simulation. This genomic control-adjusted 2-df test has control of type I error and achieves reasonable power, relative to the optimal tests for each model.     

Direct power comparisons between simple LOD scores and NPL scores for linkage analysis in complex diseases.

Several methods have been proposed for linkage analysis of complex traits with unknown mode of inheritance. These methods include the LOD score maximized over disease models (MMLS) and the "nonparametric" linkage (NPL) statistic. In previous work, we evaluated the increase of type I error when maximizing over two or more genetic models, and we compared the power of MMLS to detect linkage, in a number of complex modes of inheritance, with analysis assuming the true model. In the present study, we compare MMLS and NPL directly. We simulated 100 data sets with 20 families each, using 26 generating models: (1) 4 intermediate models (penetrance of heterozygote between that of the two homozygotes); (2) 6 two-locus additive models; and (3) 16 two-locus heterogeneity models (admixture alpha = 1.0,.7,.5, and.3; alpha = 1.0 replicates simple Mendelian models). For LOD scores, we assumed dominant and recessive inheritance with 50% penetrance. We took the higher of the two maximum LOD scores and subtracted 0.3 to correct for multiple tests (MMLS-C). We compared expected maximum LOD scores and power, using MMLS-C and NPL as well as the true model. Since NPL uses only the affected family members, we also performed an affecteds-only analysis using MMLS-C. The MMLS-C was both uniformly more powerful than NPL for most cases we examined, except when linkage information was low, and close to the results for the true model under locus heterogeneity. We still found better power for the MMLS-C compared with NPL in affecteds-only analysis. The results show that use of two simple modes of inheritance at a fixed penetrance can have more power than NPL when the trait mode of inheritance is complex and when there is heterogeneity in the data set.     

Genomic control for association studies under various genetic models

Case-control studies are commonly used to study whether a candidate allele and a disease are associated. However, spurious association can arise due to population substructure or cryptic relatedness, which cause the variance of the trend test to increase. Devlin and Roeder derived the appropriate variance inflation factor (VIF) for the trend test and proposed a novel genomic control (GC) approach to estimate VIF and adjust the test statistic. Their results were derived assuming an additive genetic model and the corresponding VIF is independent of the candidate allele frequency. We determine the appropriate VIFs for recessive and dominant models. Unlike the additive test, the VIFs for the optimal tests for these two models depend on the candidate allele frequency. Simulation results show that, when the null loci used to estimate the VIF have allele frequencies similar to that of the candidate gene, the GC tests derived for recessive and dominant models remain optimal. When the underlying genetic model is unknown or the null loci and candidate gene have quite different allele frequencies, the GC tests derived for the recessive or dominant models cannot be used while the GC test derived for the additive model can be.     

Segregation and linkage analyses of 72 leprosy pedigrees

Data on 72 families with multiple cases of leprosy were analyzed for a susceptibility gene linked to the HLA loci. We conducted segregation analysis with the program POINTER and identity of HLA types by descent analysis to determine the most likely mode of inheritance. We then conducted linkage analysis with the program LINKAS, first assuming linkage equilibrium and then allowing for linkage disequilibrium and etiological heterogeneity. Segregation results suggest a recessive mode of inheritance, especially for the tuberculoid forms of leprosy. The linkage results, limited to tuberculoid forms and assuming a recessive model, suggest a hypothesis of loose linkage with no unlinked locus. When an additive model is assumed, the best fit is obtained with a hypothesis of complete linkage (theta = 0.0) with heterogeneity. We currently favor the additive model as the more plausible one.     

Using lod-score differences to determine mode of inheritance: a simple, robust method even in the presence of heterogeneity and reduced penetrance.

Determining the mode of inheritance is often difficult under the best of circumstances, but when segregation analysis is used, the problems of ambiguous ascertainment procedures, reduced penetrance, heterogeneity, and misdiagnosis make mode-of-inheritance determinations even more unreliable. The mode of inheritance can also be determined using a linkage-based method (maximized maximum lod score or mod score) and association-based methods, which can overcome many of these problems. In this work, we determined how much information is necessary to reliably determine the mode of inheritance from linkage data when heterogeneity and reduced penetrance are present in the data set. We generated data sets under both dominant and recessive inheritance with reduced penetrance and with varying fractions of linked and unlinked families. We then analyzed those data sets, assuming reduced penetrance, both dominant and recessive inheritance, and no heterogeneity. We investigated the reliability of two methods for determining the mode of inheritance from the linkage data. The first method examined the difference (delta) between the maximum lod scores calculated under the two mode-of-inheritance assumptions. We found that if delta was > 1.5, then the higher of the two maximum lod scores reflected the correct mode of inheritance with high reliability and that a delta of 2.5 appeared to practically guarantee a correct mode-of-inheritance inference. Furthermore, this reliability appeared to be virtually independent of alpha, the fraction of linked families in the data set, although the reliability decreased slightly as alpha fell below .50.(ABSTRACT TRUNCATED AT 250 WORDS)     

Evidence of inbreeding depression on human height

Stature is a classical and highly heritable complex trait, with 80%-90% of variation explained by genetic factors. In recent years, genome-wide association studies (GWAS) have successfully identified many common additive variants influencing human height; however, little attention has been given to the potential role of recessive genetic effects. Here, we investigated genome-wide recessive effects by an analysis of inbreeding depression on adult height in over 35,000 people from 21 different population samples. We found a highly significant inverse association between height and genome-wide homozygosity, equivalent to a height reduction of up to 3 cm in the offspring of first cousins compared with the offspring of unrelated individuals, an effect which remained after controlling for the effects of socio-economic status, an important confounder (χ(2) = 83.89, df = 1; p = 5.2 × 10(-20)). There was, however, a high degree of heterogeneity among populations: whereas the direction of the effect was consistent across most population samples, the effect size differed significantly among populations. It is likely that this reflects true biological heterogeneity: whether or not an effect can be observed will depend on both the variance in homozygosity in the population and the chance inheritance of individual recessive genotypes. These results predict that multiple, rare, recessive variants influence human height. Although this exploratory work focuses on height alone, the methodology developed is generally applicable to heritable quantitative traits (QT), paving the way for an investigation into inbreeding effects, and therefore genetic architecture, on a range of QT of biomedical importance.     

A Fast Method that Uses Polygenic Scores to Estimate the Variance Explained by Genome-wide Marker Panels and the Proportion of Variants Affecting a Trait

Several methods have been proposed to estimate the variance in disease liability explained by large sets of genetic markers. However, current methods do not scale up well to large sample sizes. Linear mixed models require solving high-dimensional matrix equations, and methods that use polygenic scores are very computationally intensive. Here we propose a fast analytic method that uses polygenic scores, based on the formula for the non-centrality parameter of the association test of the score. We estimate model parameters from the results of multiple polygenic score tests based on markers with p values in different intervals. We estimate parameters by maximum likelihood and use profile likelihood to compute confidence intervals. We compare various options for constructing polygenic scores, based on nested or disjoint intervals of p values, weighted or unweighted effect sizes, and different numbers of intervals, in estimating the variance explained by a set of markers, the proportion of markers with effects, and the genetic covariance between a pair of traits. Our method provides nearly unbiased estimates and confidence intervals with good coverage, although estimation of the variance is less reliable when jointly estimated with the covariance. We find that disjoint p value intervals perform better than nested intervals, but the weighting did not affect our results. A particular advantage of our method is that it can be applied to summary statistics from single markers, and so can be quickly applied to large consortium datasets. Our method, named AVENGEME (Additive Variance Explained and Number of Genetic Effects Method of Estimation), is implemented in R software.     

Association Testing Strategy for Data from Dense Marker Panels

Genome wide association studies have been usually analyzed in a univariate manner. The commonly used univariate tests have one degree of freedom and assume an additive mode of inheritance. The experiment-wise significance of these univariate statistics is obtained by adjusting for multiple testing. Next generation sequencing studies, which assay 10-20 million variants, are beginning to come online. For these studies, the strategy of additive univariate testing and multiple testing adjustment is likely to result in a loss of power due to (1) the substantial multiple testing burden and (2) the possibility of a non-additive causal mode of inheritance. To reduce the power loss we propose: a new method (1) to summarize in a single statistic the strength of the association signals coming from all not-very-rare variants in a linkage disequilibrium block and (2) to incorporate, in any linkage disequilibrium block statistic, the strength of the association signals under multiple modes of inheritance. The proposed linkage disequilibrium block test consists of the sum of squares of nominally significant univariate statistics. We compare the performance of this method to the performance of existing linkage disequilibrium block/gene-based methods. Simulations show that (1) extending methods to combine testing for multiple modes of inheritance leads to substantial power gains, especially for a recessive mode of inheritance, and (2) the proposed method has a good overall performance. Based on simulation results, we provide practical advice on choosing suitable methods for applied analyses.     

Model-specific tests on variance heterogeneity for detection of potentially interacting genetic loci

ABSTRACT: BACKGROUND: Trait variances among genotype groups at a locus are expected to differ in the presence of an interaction between this locus and another locus or environment. A simple maximum test on variance heterogeneity can thus be used to identify potentially interacting single nucleotide polymorphisms (SNPs). RESULTS: We propose a multiple contrast test for variance heterogeneity that compares the mean of Levene residuals for each genotype group with their average as an alternative to a global Levene test. We applied this test to a Bogalusa Heart Study dataset to screen for potentially interacting SNPs across the whole genome that influence a number of quantitative traits. A user-friendly implementation of this method is available in the R statistical software package multcomp. CONCLUSIONS: We show that the proposed multiple contrast test of model-specific variance heterogeneity can be used to test for potential interactions between SNPs and unknown alleles, loci or covariates and provide valuable additional information compared with traditional tests. Although the test is statistically valid for severely unbalanced designs, care is needed in interpreting the results at loci with low allele frequencies.     

Family-based tests of association and linkage that use unaffected sibs, covariates, and interactions

We extend the methodology for family-based tests of association and linkage to allow for both variation in the phenotypes of subjects and incorporation of covariates into general-score tests of association. We use standard association models for a phenotype and any number of predictors. We then construct a score statistic, using likelihoods for the distribution of phenotype, given genotype. The distribution of the score is computed as a function of offspring genotypes, conditional on parental genotypes and trait values for offspring and parents. This approach provides a natural extension of the transmission/disequilibrium test to any phenotype and to multiple genes or environmental factors and allows the study of gene-gene and gene-environment interaction. When the trait varies among subjects or when covariates are included in the association model, the score statistic depends on one or more nuisance parameters. We suggest two approaches for obtaining parameter estimates: (1) choosing the estimate that minimizes the variance of the test statistic and (2) maximizing the statistic over a nuisance parameter and using a corrected P value. We apply our methods to a sample of families with attention-deficit/hyperactivity disorder and provide examples of how covariates and gene-environment and gene-gene interactions can be incorporated.     

A test of linkage for complex discrete and continuous traits in nuclear families

This article presents a score test for genetic linkage in nuclear families which applies to any trait having a distribution belonging to the exponential family, which includes binary and normal distributions, and distributions which are skewed or have nonnormal kurtosis. The specific distribution need not be specified and the method applies to sibships of arbitrary size. Tests of complex genetic effects are given, including unspecified mode of inheritance or additive, dominant, overdominant, and recessive modes of inheritance, covariates, multiple-locus models, including gene-gene interactions, and gene-environment interactions. The relation of our method to the Haseman-Elston methods is studied theoretically and by simulation.     

X-inactivation informs variance-based testing for X-linked association of a quantitative trait

Background

The X chromosome plays an important role in human diseases and traits. However, few X-linked associations have been reported in genome-wide association studies, partly due to analytical complications and low statistical power.

Results

In this study, we propose tests of X-linked association that capitalize on variance heterogeneity caused by various factors, predominantly the process of X-inactivation. In the presence of X-inactivation, the expression of one copy of the chromosome is randomly silenced. Due to the consequent elevated randomness of expressed variants, females that are heterozygotes for a quantitative trait locus might exhibit higher phenotypic variance for that trait. We propose three tests that build on this phenomenon: 1) A test for inflated variance in heterozygous females; 2) A weighted association test; and 3) A combined test. Test 1 captures the novel signal proposed herein by directly testing for higher phenotypic variance of heterozygous than homozygous females. As a test of variance it is generally less powerful than standard tests of association that consider means, which is supported by extensive simulations. Test 2 is similar to a standard association test in considering the phenotypic mean, but differs by accounting for (rather than testing) the variance heterogeneity. As expected in light of X-inactivation, this test is slightly more powerful than a standard association test. Finally, test 3 further improves power by combining the results of the first two tests. We applied the these tests to the ARIC cohort data and identified a novel X-linked association near gene AFF2 with blood pressure, which was not significant based on standard association testing of mean blood pressure.

Conclusions

Variance-based tests examine overdispersion, thereby providing a complementary type of signal to a standard association test. Our results point to the potential to improve power of detecting X-linked associations in the presence of variance heterogeneity.     

Simultaneous inference for ratios of linear combinations of general linear model parameters

Consider a general linear model with p -dimensional parameter vector beta and i.i.d. normal errors. Let K(1), ..., K(k ), and L be linearly independent vectors of constants such that L(T)beta not equal 0. We describe exact simultaneous tests for hypotheses that Ki(T)beta/L(T)beta equal specified constants using one-sided and two-sided alternatives, and describe exact simultaneous confidence intervals for these ratios. In the case where the confidence set is a single bounded contiguous set, we describe what we claim are the best possible conservative simultaneous confidence intervals for these ratios - best in that they form the minimum k -dimensional hypercube enclosing the exact simultaneous confidence set. We show that in the case of k = 2, this "box" is defined by the minimum and maximum values for the two ratios in the simultaneous confidence set and that these values are obtained via one of two sources: either from the solutions to each of four systems of equations or at points along the boundary of the simultaneous confidence set where the correlation between two t variables is zero. We then verify that these intervals are narrower than those previously presented in the literature.     

Common major gene inheritance of extreme overweight

We studied 3925 individuals in 961 families to determine the mode of inheritance of overweight. As an index of overweight, we examined body mass index. Our analyses indicate that the most likely genetic model for susceptibility to overweight included moderate polygenic inheritance (34% of variance resulting from many genes with small effects) and common (21% frequency) recessively expressed major genes (a few genes with large effects on the individuals who possess them). Standard statistical criteria for accepting both polygenic and major gene inheritance were met, including tests of Mendelian transmission. These results suggest that recessive major gene inheritance of overweight may be common and that homozygosity for overweight susceptibility alleles often results in overweight. Clinical, biologic, and empirical observations all suggest genetic heterogeneity, that is, more than one predisposing gene.     

Application of a mixed inheritance model to the detection of quantitative trait loci in swine

The primary goal of this study was to investigate statistical properties of a mixed inheritance model for the localization of quantitative trait loci (QTL). This is based on the analysis of phenotypic data for the amount of intramuscular fat (IMF) scored on 305 individuals originating from a cross between Duroc and Norwegian Landrace breeds. Marker genotype information is available for F1 and F2 generations. Statistical procedures compared involve i) the interval mapping, ii) the composite interval mapping, iii) a regression method, and iv) a mixed inheritance model accounting for a random animal additive genetic effect and relationships between individuals. The basic statistical properties of the latter approach are then assessed using Monte Carlo simulations showing slight unconservativeness as compared to chi(2)2df and reasonable power to detect QTL of moderate effects. In the analysis of IMF data, the significant evidence for the existing QTL is detected on chromosome 6. A chromosomal region recommended for a second-step fine mapping analysis is identified between markers SW1823 and S0228, based on three types of confidence intervals derived by using: i) the Jackknife algorithm, ii) the numerical variance approximation, and iii) the LOD score approach. The Jackknife algorithm was additionally used to quantify each family's contribution to the test statistic and to the estimate of QTL position.     

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.     

Complex segregation analysis of dental morphological variants

A set of 20 morphological variants of the dental crowns and four characteristics of the jaws are tested for probable mode of inheritance using the complex segregation analysis method of Morton et al. (Am. J. Hum. Genet. 23:602-611, 1971). Models tested include three two-allele single-locus models (dominant, codominant, and recessive) and a model employing the polychotomized normal distribution of liability (an additive polygenic model), with transmissibility estimated via maximum likelihood. Most of the traits studied are observed using ordinal scales with several grades, and many are tested using more than one dichotomy of their scale. These multiple analyses allow for an examination of such factors as trait incidence on the results of the statistical analysis. The results of the analysis yield propositions of major genes for 13 of the 24 traits examined. Two traits give good evidence of being polygenic in origin. The remaining nine characters present methodological problems that do not allow for a definite conclusion on their mode of inheritance at this time. The ability to test varying levels of transmissibility in the polygenic model allows for an estimation of the percentage of trait variance determined by familial factors. Estimates of transmissibility for all characters examined range from 0 to 1, with a mean of 0.36. These findings may suggest a large environmental role in the development of dental crown morphology. However, the possibility exists that difficulties in the ability to classify the expression of certain traits consistently result in overestimates of the environmental influences on the development of those characters.     

Robust genomic control and robust delta centralization tests for case-control association studies

The population-based case-control design is a powerful approach for detecting susceptibility markers of a complex disease. However, this approach may lead to spurious association when there is population substructure: population stratification (PS) or cryptic relatedness (CR). Two simple approaches to correct for the population substructure are genomic control (GC) and delta centralization (DC). GC uses the variance inflation factor to correct for the variance distortion of a test statistic, and the DC centralizes the non-central chi-square distribution of the test statistic. Both GC and DC have been studied for case-control association studies mainly under a specific genetic model (e.g. recessive, additive or dominant), under which an optimal trend test is available. The genetic model is usually unknown for many complex diseases. In this situation, we study the performance of three robust tests based on the GC and DC corrections in the presence of the population substructure. Our results show that, when the genetic model is unknown, the DC- (or GC-) corrected maximum and Pearson's association test are robust and have good control of Type I error and high power relative to the optimal trend tests in the presence of PS (or CR).     

Association between the ENPP1 K121Q Polymorphism and Risk of Diabetic Kidney Disease: A Systematic Review and Meta-Analysis

The potential association between the K121Q (A/C, rs1044498) polymorphism in the ectonucleotide pyrophosphatase/phosphodiesterase (ENPP1) gene and risk of diabetic kidney disease (DKD) has been investigated. Nevertheless, the effect of this variant on DKD risk is still under debate, and conflicting results have been reported. To this date, no meta-analysis has evaluated the association of the K121Q polymorphism with DKD. This paper describes the first meta-analysis conducted to evaluate whether the ENPP1K121Q polymorphism is associated with DKD. A literature search was conducted to identify all case-control or cross-sectional studies that evaluated associations between the ENPP1K121Q polymorphism and DKD. Pooled odds ratios (OR) and 95% confidence intervals (95% CI) were calculated for allele contrast, additive, dominant and recessive inheritance models. Seven studies were eligible for inclusion in the meta-analysis, providing data on 3571 type 1 or type 2 diabetic patients (1606 cases with DKD and 1965 diabetic controls without this complication). No significant heterogeneity was observed among the studies included in the meta-analysis when assuming different inheritance models (I^² < 50% or P > 0.10 for the entire sample and after stratification by ethnicity). Meta-analysis results revealed significant associations between the K121Q polymorphism and risk of DKD in Asians and Europeans when assuming the different inheritance models analyzed. The most powerful association was observed for the additive model (OR = 1.74, 95% CI 1.27-2.38 for the total sample). In conclusion, the present meta-analysis detected a significant association between the ENPP1K121Q polymorphism and increased susceptibility of DKD in European and Asian populations.