Analysis of familial aggregation in the presence of varying family sizes

Summary.  Family studies are frequently undertaken as the first step in the search for genetic and/or environmental determinants of disease. Significant familial aggregation of disease is suggestive of a genetic aetiology for the disease and may lead to more focused genetic analysis. Of course, it may also be due to shared environmental factors. Many methods have been proposed in the literature for the analysis of family studies. One model that is appealing for the simplicity of its computation and the conditional interpretation of its parameters is the quadratic exponential model. However, a limiting factor in its application is that it is not reproducible , meaning that all families must be of the same size. To increase the applicability of this model, we propose a hybrid approach in which analysis is based on the assumption of the quadratic exponential model for a selected family size and combines a missing data approach for smaller families with a marginalization approach for larger families. We apply our approach to a family study of colorectal cancer that was sponsored by the Cancer Genetics Network of the National Institutes of Health. We investigate the properties of our approach in simulation studies. Our approach applies more generally to clustered binary data.     

Modeling binary familial data using Gaussian copula

Modeling binary familial data has been a challenging task due to the dependence among family members and the constraints imposed on the joint probability distribution of the binary responses. This paper investigates some useful familial dependence structures and proposes analyzing binary familial data using Gaussian copula model. Advantages of this approach are discussed as well as some computational details. An numerical example is also presented with an aim to show the capability of Gaussian copula model in more sophisticated data analysis.     

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.     

Adaptive designs for dose-finding based on efficacy–toxicity response

We propose a new adaptive procedure for dose-finding in clinical trials when both efficacy and toxicity responses are available. We model the distribution of this bivariate binary endpoint using either Gumbel bivariate logistic regression or Cox bivariate binary model. In both cases, the analytic formulae for the Fisher information matrix are obtained, that form the basis for derivation of the locally optimal and adaptive designs.     

A multivariate regression model for continuous genetic traits

Summary.  In many commonly used models for multivariate traits, the likelihood is specified as a mixture of nested sums of products over the unobserved genotypes of all the family members, in which the familial covariance matrices vary in size and structure for different families, and their sizes can be immense for large family units. These issues pose computational difficulties in many applications. Bonney's compound regressive model for univariate traits simplifies the familial covariance structure and reduces the mixture of nested sums only to the parent–offspring level, thus enhancing computation significantly. This model has been extended to the multivariate case in the absence of unobserved genotypes. Here, we further extend this model to incorporate major genes, covariates and multiple loci. As is typical in practice, this causes new computational difficulties. We study the computational issues and explore the behaviour of this extended model.     

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.     

Multivariate association test for rare variants controlling for cryptic and family relatedness

In genetic studies of complex diseases, multiple measures of related phenotypes are often collected. Jointly analyzing these phenotypes may improve statistical power to detect sets of rare variants affecting multiple traits. In this work, we consider association testing between a set of rare variants and multiple phenotypes in family‐based designs. We use a mixed linear model to express the correlations among the phenotypes and between related individuals. Given the many sources of correlations in this situation, deriving an appropriate test statistic is not straightforward. We derive a vector of score statistics, whose joint distribution is approximated using a copula. This allows us to have closed‐form expressions for the p‐values of several test statistics. A comprehensive simulation study and an application to Genetic Analysis Workshop 18 data highlight the gains associated with joint testing over univariate approaches, especially in the presence of pleiotropy or highly correlated phenotypes. The Canadian Journal of Statistics 47: 90–107; 2019 © 2018 Statistical Society of Canada     

Estimation of a semiparametric recursive bivariate probit model in the presence of endogeneity

The classic recursive bivariate probit model is of particular interest to researchers since it allows for the estimation of the treatment effect that a binary endogenous variable has on a binary outcome in the presence of unobservables. In this article, the authors consider the semiparametric version of this model and introduce a model fitting procedure which permits to estimate reliably the parameters of a system of two binary outcomes with a binary endogenous regressor and smooth functions of continuous covariates. They illustrate the empirical validity of the proposal through an extensive simulation study. The approach is applied to data from a survey, conducted in Botswana, on the impact of education on women's fertility. Some studies suggest that the estimated effect could have been biased by the possible endogeneity arising because unobservable confounders (e.g., ability and motivation) are associated with both fertility and education. The Canadian Journal of Statistics 39: 259–279; 2011 © 2011 Statistical Society of Canada     

CAUSAL GRAPHICAL MODELS IN SYSTEMS GENETICS: A UNIFIED FRAMEWORK FOR JOINT INFERENCE OF CAUSAL NETWORK AND GENETIC ARCHITECTURE FOR CORRELATED PHENOTYPES

Causal inference approaches in systems genetics exploit quantitative trait loci (QTL) genotypes to infer causal relationships among phenotypes. The genetic architecture of each phenotype may be complex, and poorly estimated genetic architectures may compromise the inference of causal relationships among phenotypes. Existing methods assume QTLs are known or inferred without regard to the phenotype network structure. In this paper we develop a QTL-driven phenotype network method (QTLnet) to jointly infer a causal phenotype network and associated genetic architecture for sets of correlated phenotypes. Randomization of alleles during meiosis and the unidirectional influence of genotype on phenotype allow the inference of QTLs causal to phenotypes. Causal relationships among phenotypes can be inferred using these QTL nodes, enabling us to distinguish among phenotype networks that would otherwise be distribution equivalent. We jointly model phenotypes and QTLs using homogeneous conditional Gaussian regression models, and we derive a graphical criterion for distribution equivalence. We validate the QTLnet approach in a simulation study. Finally, we illustrate with simulated data and a real example how QTLnet can be used to infer both direct and indirect effects of QTLs and phenotypes that co-map to a genomic region.     

Estimation in Regressive Logistic Regression Analyses of Familial Data with Missing Outcomes

This paper examines a number of methods of handling missing outcomes in regressive logistic regression modelling of familial binary data, and compares them with an EM algorithm approach via a simulation study. The results indicate that a strategy based on imputation of missing values leads to biased estimates, and that a strategy of excluding incomplete families has a substantial effect on the variability of the parameter estimates. Recommendations are made which depend, amongst other factors, on the amount of missing data and on the availability of software.     

Generating Bivariate Uniform Data with a Full Range of Correlations and Connections to Bivariate Binary Data

In this article, random number generation algorithms for generating bivariate uniform data based on a known class of symmetric bivariate uniform distributions that allow the entire correlation range are given, and its previously unrecognized connection with bivariate binary data is established via matching the cumulative distribution functions.     

Extended Bayesian model averaging for heritability in twin studies

Family studies are often conducted to examine the existence of familial aggregation. Particularly, twin studies can model separately the genetic and environmental contribution. Here we estimate the heritability of quantitative traits via variance components of random-effects in linear mixed models (LMMs). The motivating example was a myopia twin study containing complex nesting data structures: twins and siblings in the same family and observations on both eyes for each individual. Three models are considered for this nesting structure. Our proposal takes into account the model uncertainty in both covariates and model structures via an extended Bayesian model averaging (EBMA) procedure. We estimate the heritability using EBMA under three suggested model structures. When compared with the results under the model with the highest posterior model probability, the EBMA estimate has smaller variation and is slightly conservative. Simulation studies are conducted to evaluate the performance of variance-components estimates, as well as the selections of risk factors, under the correct or incorrect structure. The results indicate that EBMA, with consideration of uncertainties in both covariates and model structures, is robust in model misspecification than the usual Bayesian model averaging (BMA) that considers only uncertainty in covariates selection.     

A class of absolutely continuous bivariate distributions

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. Extensive work has been done on the Block and Basu bivariate exponential model over the past several decades. Interestingly it is observed that the Block and Basu bivariate exponential model can be extended to the Weibull model also. We call this new model as the Block and Basu bivariate Weibull model. We consider different properties of the Block and Basu bivariate Weibull model. The Block and Basu bivariate Weibull model has four unknown parameters and the maximum likelihood estimators cannot be obtained in closed form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. We propose to use the EM algorithm for computing the maximum likelihood estimators of the unknown parameters. The proposed EM algorithm can be carried out by solving one non-linear equation at each EM step. Our method can be also used to compute the maximum likelihood estimators for the Block and Basu bivariate exponential model. One data analysis has been preformed for illustrative purpose.     

Nonparametric Covariate-Adjusted Association Tests Based on the Generalized Kendall's Tau()

Identifying the risk factors for comorbidity is important in psychiatric research. Empirically, studies have shown that testing multiple, correlated traits simultaneously is more powerful than testing a single trait at a time in association analysis. Furthermore, for complex diseases, especially mental illnesses and behavioral disorders, the traits are often recorded in different scales such as dichotomous, ordinal and quantitative. In the absence of covariates, nonparametric association tests have been developed for multiple complex traits to study comorbidity. However, genetic studies generally contain measurements of some covariates that may affect the relationship between the risk factors of major interest (such as genes) and the outcomes. While it is relatively easy to adjust these covariates in a parametric model for quantitative traits, it is challenging for multiple complex traits with possibly different scales. In this article, we propose a nonparametric test for multiple complex traits that can adjust for covariate effects. The test aims to achieve an optimal scheme of adjustment by using a maximum statistic calculated from multiple adjusted test statistics. We derive the asymptotic null distribution of the maximum test statistic, and also propose a resampling approach, both of which can be used to assess the significance of our test. Simulations are conducted to compare the type I error and power of the nonparametric adjusted test to the unadjusted test and other existing adjusted tests. The empirical results suggest that our proposed test increases the power through adjustment for covariates when there exist environmental effects, and is more robust to model misspecifications than some existing parametric adjusted tests. We further demonstrate the advantage of our test by analyzing a data set on genetics of alcoholism.     

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.     

Estimation of a Semiparametric Recursive Bivariate Probit Model with Nonparametric Mixing

We consider an extension of the recursive bivariate probit model for estimating the effect of a binary variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate effects and overdispersion. Specifically, the model consists of a system of two binary outcomes with a binary endogenous regressor which includes smooth functions of covariates, hence allowing for flexible functional dependence of the responses on the continuous regressors, and arbitrary random intercepts to deal with overdispersion arising from correlated observations on clusters or from the omission of non‐confounding covariates. We fit the model by maximizing a penalized likelihood using an Expectation‐Maximisation algorithm. The issues of automatic multiple smoothing parameter selection and inference are also addressed. The empirical properties of the proposed algorithm are examined in a simulation study. The method is then illustrated using data from a survey on health, aging and wealth.     

Weibull extension of bivariate exponential regression model with different frailty distributions

We propose bivariate Weibull regression model with frailty in which dependence is generated by a gamma or positive stable or power variance function distribution. We assume that the bivariate survival data follows bivariate Weibull of Hanagal (Econ Qual Control 19:83–90, 2004; Econ Qual Control 20:143–150, 2005a; Stat Pap 47:137–148, 2006a; Stat Methods, 2006b). There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows known frailty distribution. These are the situations which motivate to study this particular model. David D. Hanagal is on leave from Department of Statistics, University of Pune, Pune 411007, India.     

Modeling heterogeneity for bivariate survival data by the Weibull distribution

We propose bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by Weibull distribution. We assume that the bivariate survival data follow bivariate Weibull of Hanagal (Econ Qual Control 19:83–90, 2004). There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate to study this particular model. We propose two-stage maximum likelihood estimation for hierarchical likelihood in the proposed model. We present a small simulation study to compare these estimates with the true value of the parameters and it is observed that these estimates are very close to the true values of the parameters.