Ascertainment-Adjusted Maximum Likelihood Estimation for the Additive Genetic Gamma Frailty Model

The additive genetic gamma frailty model has been proposed for genetic linkage analysis for complex diseases to account for variable age of onset and possible covariates effects. To avoid ascertainment biases in parameter estimates, retrospective likelihood ratio tests are often used, which may result in loss of efficiency due to conditioning. This paper considers when the sibships are ascertained by having at least two affected sibs with the disease before a given age and provides two approaches for estimating the parameters in the additive gamma frailty model. One approach is based on the likelihood function conditioning on the ascertainment event, the other is based on maximizing a full ascertainment-adjusted likelihood. Explicit forms for these likelihood functions are derived. Simulation studies indicate that when the baseline hazard function can be correctly pre-specified, both approaches give accurate estimates of the model parameters. However, when the baseline hazard function has to be estimated simultaneously, only the ascertainment-adjusted likelihood method gives an unbiased estimate of the parameters. These results imply that the ascertainment-adjusted likelihood ratio test in the context of the additive genetic gamma frailty may be used for genetic linkage analysis.     

Empirical likelihood-based robust tests for genetic association analysis with quantitative traits

Genome-wide association studies (GWAS) are effective in investigating the loci related with complex diseases. For most of these studies, the genetic inheritance model is not known in advance and therefore robust tests are preferred. Empirical likelihood (EL) method is well known for its flexibility and nonparametric properties, but is rarely investigated in GWAS. In this study, we develop EL-based test statistics to detect the association of a disease and genetic loci while the genetic model is unknown. The performance of proposed tests is evaluated by simulations and compared with several existing methods. For illustration, we apply these tests to identify the single nucleotide polymorphisms associated with alkaline phosphatase level on mouse chromosome 6.     

Power and robustness of sib-pair linkage tests and extension to larger sibships

We investigate the power and robustness of Haseman and Elstonfs sib-pair test for genetic linkage between a marker locus and a locus affecting a quantitative trait, and compare the test to that of Penrose. The Haseman-Elston test is more powerful than Penrrose's test; its power is acceptable for cases of tight linkage and high heritability due to the hypothesized quantitative trait locus, but is quite low in other situations. Computer simulations indicate that both tests are valid for normally distributed trait values, and that the Haseman-Elston test is robust for a variety of continuous distributions of the trait values. Several linkage tests are developed for sib trios that are much more powerful , for the same total number of sibs, than the test on independent sib pairs. The Haseman-Elston test on all possible sib pairs is suggested for sibships of size larger than three and for samples including sibships of various sizes.     

Sample sizes for the transmission disequilibrium tests: tdt,s-tdt and 1-tdt

The transmission/disequilibrium test (TDT) is widely used to detect the linkage disequilibrium between a candidate locus (a marker) and a disease locus. The TDT is a family-based design and has the advantage that it is a valid test when population stratification exist. The TDT requires the marker genotypes of affected individuals and their parents. For diseases with late age of onset, it is difficult or impossible to obtain the marker genotype of the parents. Therefore, when both parents marker genotypes are unavailable, Ewex and Spielman extended the TDT to the S-TDT for use in sibships with at least one affected individual and one unaffected individual. When only one of the parents' genotype is available. Sun et al. proposed a test the 1-TDT, for use with niarker genotypes of affected individuals and only one available parent. Here, we study the saniple sizes of TDT, S-TDT, and 1-TDT. We show that the sample size needed for the 1-TDT is rogghly the same as the sample size needed for the S-TDT with two sibs and is about twice the sample size for the TDT.     

Modeling the Effects of Genetic Factors on Late-Onset Diseases in Cohort Studies

Many late-onset diseases are caused by what appears to be a combination of a genetic predisposition to disease and environmental factors. The use of existing cohort studies provides an opportunity to infer genetic predisposition to disease on a representative sample of a study population, now that many such studies are gathering genetic information on the participants. One feature to using existing cohorts is that subjects may be censored due to death prior to genetic sampling, thereby adding a layer of complexity to the analysis. We develop a statistical framework to infer parameters of a latent variables model for disease onset. The latent variables model describes the role of genetic and modifiable risk factors on the onset ages of multiple diseases, and accounts for right-censoring of disease onset ages. The framework also allows for missing genetic information by inferring a subject's unknown genotype through appropriately incorporated covariate information. The model is applied to data gathered in the Framingham Heart Study for measuring the effect of different Apo-E genotypes on the occurrence of various cardiovascular disease events.     

Locating genes involved in human diseases

The increasing amount of information that is becoming available about the structure and composition of the DNA constituting the human chromosomes has provided new opportunities to locate genes that affect susceptibilities to a range of diseases. The accurate location of these genes is important in genetic counselling and in understanding the effects of genes that may result in disease. Various methods of analysing the data when DNA information is available at a single marker locus for an affected child and his or her parents are reviewed and applied to data on insulin-dependent diabetes mellitus . The importance of distinguishing between the association of alleles at a marker locus and at a disease locus resulting from chromosomal linkage from that resulting from other causes is emphasized.     

A frailty model for (interval) censored family survival data,applied to the age at onset of non-physical problems

Family survival data can be used to estimate the degree of genetic and environmental contributions to the age at onset of a disease or of a specific event in life. The data can be modeled with a correlated frailty model in which the frailty variable accounts for the degree of kinship within the family. The heritability (degree of heredity) of the age at a specific event in life (or the onset of a disease) is usually defined as the proportion of variance of the survival age that is associated with genetic effects. If the survival age is (interval) censored, heritability as usually defined cannot be estimated. Instead, it is defined as the proportion of variance of the frailty associated with genetic effects. In this paper we describe a correlated frailty model to estimate the heritability and the degree of environmental effects on the age at which individuals contact a social worker for the first time and to test whether there is a difference between the survival functions of this age for twins and non-twins.     

A Novel Powerful Joint Analysis with Data Fusion in Two-stage Case–Control Genome-wide Association Studies

In genome-wide association studies (GWASs) to detect the disease-associated genetic variants, two-stage design has received much attention because of its cost effectiveness and high efficiency. Under the framework of a two-stage design, it has been shown that joint analysis is more powerful than replication-based analysis. Several robust tests have been proposed for joint analysis to handle the problem of unknown genetic mode of inheritance. However, existing joint analysis of combining test statistics from both stages might suffer from a loss of efficiency if the combined test statistics are not sufficient or the weight of the statistic for each stage is not appropriate. In this article, we propose a new strategy for joint analysis by combining the raw data rather than the test statistics across stages and construct a robust MAX3-based test for two-staged GWASs, which can make full use of the information of the data from both stages. Our numerical results show that the proposed procedure is more powerful and computationally much faster than the existing joint analysis procedures. An application to a type 2 diabetes dataset is used to illustrate the proposed approach.     

Robust Poisson likelihood estimation for frailty Cox models: A simulation study

Frailty models can be fit as mixed-effects Poisson models after transforming time-to-event data to the Poisson model framework. We assess, through simulations, the robustness of Poisson likelihood estimation for Cox proportional hazards models with log-normal frailties under misspecified frailty distribution. The log-gamma and Laplace distributions were used as true distributions for frailties on a natural log scale. Factors such as the magnitude of heterogeneity, censoring rate, number and sizes of groups were explored. In the simulations, the Poisson modeling approach that assumes log-normally distributed frailties provided accurate estimates of within- and between-group fixed effects even under a misspecified frailty distribution. Non-robust estimation of variance components was observed in the situations of substantial heterogeneity, large event rates, or high data dimensions.     

The Additive Genetic Gamma Frailty Model

In this paper the additive genetic gamma frailty model is defined. Individual frailties are correlated as a result of an additive genetic model. An algorithm to construct additive genetic gamma frailties for any pedigree is given so that the variance–covariance structure among individual frailties equals the numerator relationship matrix times a variance. The EM algorithm can be used to estimate the parameters in the model. Calculations are similar using the EM algorithm in the shared frailty model, however the E step is not correspondingly simple. This is illustrated re-analysing data, analysed by the shared frailty model in Nielsen et al . (1992), from the Danish adoptive register. Goodness of fit of the additive genetic gamma frailty model can be tested after analysing data with the correlated frailty model. Doing so, a "defect" in the often used and otherwise well behaving likelihood was found     

Testing for the Presence of a Cure Fraction in Clustered Interval‐Censored Survival Data

Clustered interval‐censored survival data are often encountered in clinical and epidemiological studies due to geographic exposures and periodic visits of patients. When a nonnegligible cured proportion exists in the population, several authors in recent years have proposed to use mixture cure models incorporating random effects or frailties to analyze such complex data. However, the implementation of the mixture cure modeling approaches may be cumbersome. Interest then lies in determining whether or not it is necessary to adjust the cured proportion prior to the mixture cure analysis. This paper mainly focuses on the development of a score for testing the presence of cured subjects in clustered and interval‐censored survival data. Through simulation, we evaluate the sampling distribution and power behaviour of the score test. A bootstrap approach is further developed, leading to more accurate significance levels and greater power in small sample situations. We illustrate applications of the test using data sets from a smoking cessation study and a retrospective study of early breast cancer patients.     

A PROBLEM IN ASCERTAINMENT

In studies of disease inheritance, it is more convenient to collect family data by first locating an affected individual and then enquiring about the status of his or her relatives. Although the different categories of children classified by disease, sex, and other covariates may have a particular multinomial distribution among families of a given size, the numbers as ascertained do not have the same distribution because of unequal probabilities of selection of families. The introduction of weighted distributions to correct for ascertainment bias in the estimation of parameters in the classical segregation model can be traced to Fisher in 1934. This theory was presented in a general formulation by C. R. Rao at the First International Symposium on Classical and Contagious Distributions in 1963. Further expansion on the topic was given by C. R. Rao in the ISI Centenary Volume published in 1985. The effects of different two-phase sampling designs on the estimation of parameters in the classical segregation model are examined. An approximation to the classical segregation likelihood model is found to produce results close to those of the exact likelihood function in Monte Carlo simulations for a balanced two-phase design. This has implications for more complex models in which the computation of the exact likelihood is prohibitive, such as for the enhancement of a typical survey sampling plan designed initially for linkage analysis but then used retroactively for a combined segregation and linkage analysis.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

A frailty modeling approach for parental effects in animal breeding

Survival models involving frailties are commonly applied in studies where correlated event time data arise due to natural or artificial clustering. In this paper we present an application of such models in the animal breeding field. Specifically, a mixed survival model with a multivariate correlated frailty term is proposed for the analysis of data from over 3611 Brazilian Nellore cattle. The primary aim is to evaluate parental genetic effects on the trait length in days that their progeny need to gain a commercially specified standard weight gain. This trait is not measured directly but can be estimated from growth data. Results point to the importance of genetic effects and suggest that these models constitute a valuable data analysis tool for beef cattle breeding.     

Asymptotic test of mixture model and its applications to QTL interval mapping

Quantitative trait loci (QTL) mapping has been a standard means in identifying genetic regions harboring potential genes underlying complex traits. Likelihood ratio test (LRT) has been commonly applied to assess the significance of a genetic locus in a mixture model content. Given the time constraint in commonly used permutation tests to assess the significance of LRT in QTL mapping, we study the behavior of the LRT statistic in mixture model when the proportions of the distributions are unknown. We found that the asymptotic null distribution is stationary Gaussian process after suitable transformation. The result can be applied to one-parameter exponential family mixture model. Under certain condition, such as in a backcross mapping model, the tail probability of the supremum of the process is calculated and the threshold values can be determined by solving the distribution function. Simulation studies were performed to evaluate the asymptotic results.     

The universal validity of the possible triangle constraint for affected sib pairs

The authors consider affected‐sib‐pair analysis, in which genetic marker data are collected from families with at least two sibs affected by a disease under investigation. At any locus not linked to the disease gene, a sib pair shares 0, 1 and 2 alleles identical by descent (IBD) with probabilities of 1/4, 1/2 and 1/4, respectively. With linkage, the IBD value increases stochastically. Louis, Payami & Thomson (1987) and Holmans (1993) were the first ones who discovered that the IBD distribution satisfies the “possible triangle constraint” in some situations. Consequently, more powerful statistical procedures can be designed in detecting linkage. It is of statistical and genetical importance to investigate whether the possible triangle constraint remains true under general genetic models. In this paper, the authors introduce a new technique to prove the possible triangle constraint. Their proof is particularly simple for the single disease locus case. The general case is proved by linking IBD distributions between marker loci through a transition probability matrix.     

Maximizing a Family of Optimal Statistics over a Nuisance Parameter with Applications to Genetic Data Analysis

In this article, a simple algorithm is used to maximize a family of optimal statistics for hypothesis testing with a nuisance parameter not defined under the null hypothesis. This arises from genetic linkage and association studies and other hypothesis testing problems. The maximum of optimal statistics over the nuisance parameter space can be used as a robust test in this situation. Here, we use the maximum and minimum statistics to examine the sensitivity of testing results with respect to the unknown nuisance parameter. Examples from genetic linkage analysis using affected sub pairs and a candidate-gene association study in case-parents trio design are studied.     

Using categorical markers as auxiliary variables in log‐rank tests and hazard ratio estimation

Markers, which are prognostic longitudinal variables, can be used to replace some of the information lost due to right censoring. They may also be used to remove or reduce bias due to informative censoring. In this paper, the authors propose novel methods for using markers to increase the efficiency of log‐rank tests and hazard ratio estimation, as well as parametric estimation. They propose a «plug‐in» methodology that consists of writing the test statistic or estimate of interest as a functional of Kaplan–Meier estimators. The latter are then replaced by an efficient estimator of the survival curve that incorporates information from markers. Using simulations, the authors show that the resulting estimators and tests can be up to 30% more efficient than the usual procedures, provided that the marker is highly prognostic and that the frequency of censoring is high.     

Discussion: Why do we test multiple traits in genetic association studies

Zhu and Zhang [Zhu, W., &; Zhang, H. (2009). Why do we test multiple traits in genetic association studies. Journal of the Korean Statistical Society, 38(1), 1–10] publish a paper “Why Do We Test Multiple Traits in Genetic Association Studies?” in this issue. The authors used linear structural equations and acyclic graph as tools to explore the performance of testing multiple traits simultaneously by large-scale simulations for various genetic models. The methods, conclusions and results are of great interest in quantitative genetics. Diseases are caused by dynamic interaction among many genes and many environmental exposures through regulation and metabolism. In the past several decades, researchers have primarily focused on (1) the role of individual genetic variation in determining the diseases and (2) one single trait at a time. Little attention has been paid to determining how the genetic variations and environmental perturbation are integrated into networks which act together to dynamically alter regulations and metabolism leading to the emergence of complex phenotypes and diseases. Pending conceptual and statistical challenges are (1) how to identify networks involved in molecular phenotypes and endpoint clinical phenotypes under perturbation of environments and (2) how to connect DNA variation to disease outcomes through gene regulations and cellular intermediate traits. Structural equations and graphical models of multiple quantitative traits provide a general framework for developing novel analytic strategies for identifying the path from genomic information coupled with the environmental exposures, through gene expressions and other intermediate traits, to the clinical endpoints of complex diseases, to meet the above conceptual and statistical challenges. In this discussion, we use structural equations to analyze multiple intermediate traits of ankylosing spondylitis (AS) as a real example to further demonstrate the importance of network approach to genetic studies of complex traits.     

Marker processes in survival analysis

In the development of many diseases there are often associated variables which continuously measure the progress of an individual towards the final expression of the disease (failure). Such variables are stochastic processes, here called marker processes, and, at a given point in time, they may provide information about the current hazard and subsequently on the remaining time to failure. Here we consider a simple additive model for the relationship between the hazard function at time t and the history of the marker process up until time t. We develop some basic calculations based on this model. Interest is focused on statistical applications for markers related to estimation of the survival distribution of time to failure, including (i) the use of markers as surrogate responses for failure with censored data, and (ii) the use of markers as predictors of the time elapsed since onset of a survival process in prevalent individuals. Particular attention is directed to potential gains in efficiency incurred by using marker process information.