title: A Bayesian generalized random regression model for estimating heritability using overdispersed count data.
abstract: BACKGROUND
  Faecal egg counts are a common indicator of nematode infection and since it is a heritable trait, it provides a marker for selective breeding. However, since resistance to disease changes as the adaptive immune system develops, quantifying temporal changes in heritability could help improve selective breeding programs. Faecal egg counts can be extremely skewed and difficult to handle statistically. Therefore, previous heritability analyses have log transformed faecal egg counts to estimate heritability on a latent scale. However, such transformations may not always be appropriate. In addition, analyses of faecal egg counts have typically used univariate rather than multivariate analyses such as random regression that are appropriate when traits are correlated. We present a method for estimating the heritability of untransformed faecal egg counts over the grazing season using random regression.
  
  
  RESULTS
  Replicating standard univariate analyses, we showed the dependence of heritability estimates on choice of transformation. Then, using a multitrait model, we exposed temporal correlations, highlighting the need for a random regression approach. Since random regression can sometimes involve the estimation of more parameters than observations or result in computationally intractable problems, we chose to investigate reduced rank random regression. Using standard software (WOMBAT), we discuss the estimation of variance components for log transformed data using both full and reduced rank analyses. Then, we modelled the untransformed data assuming it to be negative binomially distributed and used Metropolis Hastings to fit a generalized reduced rank random regression model with an additive genetic, permanent environmental and maternal effect. These three variance components explained more than 80 % of the total phenotypic variation, whereas the variance components for the log transformed data accounted for considerably less. The heritability, on a link scale, increased from around 0.25 at the beginning of the grazing season to around 0.4 at the end.
  
  
  CONCLUSIONS
  Random regressions are a useful tool for quantifying sources of variation across time. Our MCMC (Markov chain Monte Carlo) algorithm provides a flexible approach to fitting random regression models to non-normal data. Here we applied the algorithm to negative binomially distributed faecal egg count data, but this method is readily applicable to other types of overdispersed data.
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Colette.Mair@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Colette.Mair@glasgow.ac.uk.
authors.forename: Colette
authors.initials: C
authors.lastname: Mair
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Michael.Stear@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Michael.Stear@glasgow.ac.uk.
authors.forename: Michael
authors.initials: M
authors.lastname: Stear
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Paul.Johnson@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Paul.Johnson@glasgow.ac.uk.
authors.forename: Paul
authors.initials: P
authors.lastname: Johnson
authors.affiliation: School of Veterinary Medicine, Bearsden Road, Glasgow, G61 1QH, UK. Matthew.Denwood@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Matthew.Denwood@glasgow.ac.uk.
authors.forename: Matthew
authors.initials: M
authors.lastname: Denwood
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Joaquin.Prada@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Joaquin.Prada@glasgow.ac.uk.
authors.forename: Joaquin Prada
authors.initials: JP
authors.lastname: Jimenez de Cisneros
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Thorsten.Stefan@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Thorsten.Stefan@glasgow.ac.uk.
authors.forename: Thorsten
authors.initials: T
authors.lastname: Stefan
authors.affiliation: Institute of Biodiversity, Animal Health and Comparative Medicine, Bearsden Road, Glasgow, G611QH, UK. Louise.Matthews@glasgow.ac.uk.|The Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow, G12 8QQ, UK. Louise.Matthews@glasgow.ac.uk.
authors.forename: Louise
authors.initials: L
authors.lastname: Matthews
mesh_terms: D000465:Algorithms; D000818:Animals; D001499:Bayes Theorem; D005243:Feces / Q000469:parasitology; D015233:Models, Statistical; D010270:Parasite Egg Count / Q000662:veterinary; D019655:Quantitative Trait, Heritable; D012044:Regression Analysis; D012756:Sheep; D012757:Sheep Diseases / Q000235:genetics* / Q000469:parasitology*