Specification of Generalized Linear Mixed Models for Family Data using Markov Chain Monte Carlo Methods

Abstract

Statistical models imposed on family data can be used to partition phenotypic variation into components due to sharing of both genetic and environmental risk factors for disease. Generalized linear mixed models (GLMMs) are useful tools for the analysis of family data, but it is not always clear how to specify individual-level regression equations so that the resulting within-family variance-covariance matrix of the phenotype reflects the correlation implied by the relatedness of individuals within families. This is particularly challenging when families are of varying sizes and compositions. In this paper we propose a general approach to specifying GLMMs for family data that uses a decomposition of the within-family variance-covariance matrix of the phenotype to set up a series of regression equations with fixed and random effects that corresponds to an appropriate genetic model. This "mechanistic" specification is particularly suited to estimation and evaluation of models within a Markov chain Monte Carlo (MCMC) framework. The proposed approach was assessed with simulated data to demonstrate the accuracy of estimation of the variance components. We analyzed data from the Victorian Family Heart Study (families with two generations over-sampled for those with monozygotic and dizygotic twins) and for a binary phenotype (hypertension) that resulted in substantially reduced computation time in the MCMC framework (via WinBUGS) compared with a maximum likelihood approach (via Stata). The proposed approach to model specification in this paper is easily implemented using standard software and can accommodate prior information on the magnitude of fixed or random effects. J Biomet Derivation of the Model SpecificationOur models for data have as their basis Fisher's polygenic model [10], where the genetic contribution to the phenotype is the combined effect of possibly a large number of separate locations or loci on the human genome. At each of these loci an individual inherits one allele (of possibly many) from each parent. The effect of these alleles is assumed to be additive -the presence of each additional allele of the same type changes the phenotype by the same amount (so the effect of two identical alleles is twice the effect of one allele acting alone), and the effect of two different alleles is the sum of the effect of each allele separately. The result of this assumption is a phenotypic model where covariation between individuals depends only on their level of relatedness. If the underlying additive model holds at a sufficient proportion of the active loci, then the polygenic model will be a reasonable approximation to the true but unknown effect of the genetic factors on the phenotype. Common environmental factors might also contribute to similarity of outcomes within families, with the additional assumption that the correlation between phenotypes does not depend of the degree of relatedness of the individuals.We can formalise the above by specifying a model for individual participant data:where Y ij is...

Non‐proportional odds multivariate logistic regression of ordinal family data

“…β or β k ), then gologit2 Stata function can be used to check the proportional odds assumption for either independent or correlated ordinal outcomes (Ellis et al, 2007). However, if inferences regarding both non-proportional covariate effects and the residual correlation structure are of interest, then such analyses are difficult to implement in standard software packages; especially for data comprising clusters of variable sizes and when the correlation between outcomes varies depending on the relationship between subjects in the population, as is the case for family data (Rabe-Hesketh et al, 2008;Jamsen et al, 2012).…”

“…The CLM may be extended to correlated data via the introduction of individual specific random effects to give the cumulative link mixed model (CLMM), and the proportional and non-proportional versions of this model can be fitted to clustered/multilevel data using standard software packages, such as MIXOR (Hedeker and Mermelstein, 1998), SAS PROC NLMIXED (Pinheiro and Bates, 1995;SAS Institute Inc., 2013) and GLLAMM (Rabe-Hesketh et al, 2001). The correlation structure of family data is more complex than that for other types of clustered data, and many random effects are required to model unmeasured genetic and shared environmental sources of variation (Scurrah et al, 2000;Rabe-Hesketh et al, 2008;Jamsen et al, 2012). This makes fitting the CLMM to family data computationally intensive because numerical integration of the complex likelihood function followed by maximisation to yield maximum-likelihood (ML) estimates is required.…”

Herdabilidade e Estudo de Associação Genômica Ampla (GWAS) de atividade física autorreportada: uma investigação de famílias brasileiras

“…Considerando dados de famílias, os modelos empregados neste trabalho para as estimativas de herdabilidade de fenótipos quantitativos e associação com marcadores moleculares seguem os modelos lineares mistos , os quais têm sido amplamente aplicados na medicina, pecuária, oncologia, dentre outras áreas [Almasy eBlangero, 2010, 1998, Chu e Huang, 2017, de Andrade et al, 1999, Kachman e Stroup, 1994.Esses modelos são bastante flexíveis na modelagem de dados tanto da média como dos componentes de (co)variância de variáveis normais sob delineamentos com indivíduos relacionados e têm sido usados em estudos de herdabilidade de fenótipos quantitativos e de mapeamento de genes, mostrando resultados promissores[de Oliveira et al, 2008, Diego et al, 2015, Kris M et al, 2013.Pertencente à classe dos modelos lineares mistos, o modelo mais recomendado e utilizado para explicar a herança de fenótipos foi inicialmente proposto por Fisher em 1918, denominado modelo de componentes de variância e sendo mais tarde chamado modelo linear misto poligênico. Esse modelo assume que as características fenotípicas são influenciadas por, potencialmente, um número grande de loci no genoma, assumindo-se que eles exercem seus efeitos no fenótipo de forma aditiva, ou seja, de forma independente.…”