title: Genetic heteroscedastic models for ordinal traits: application to sheep litter size.
abstract: BACKGROUND
  Classical genetic canalization models, which accommodate the mean and variance of a trait separately, provide a flexible approach to take heteroscedasticity for continuous traits into account. However, this model is not appropriate for discrete traits. The aim of this work was to propose heteroscedastic threshold models suitable for the genetic analysis of ordinal data.
  
  
  METHODS
  In order to first fit the mean and variance of ordinal traits separately, we extended the classical threshold model (TM) for discrete data by introducing non-genetic and genetic factors of heterogeneity on the variance of its underlying variable, which leads to a homothetic threshold model HTM and its alternative parameterization HTM' in which the thresholds of different individuals are linked by a homothetic-translation. Relaxing the constraint between the thresholds led us to propose an independent threshold model ITM that was more flexible than HTM' but required the estimation of more parameters. TM, HTM and ITM models were applied to study 19,671 records on litter size in Romane sheep.
  
  
  RESULTS
  Both HTM and ITM were able to disentangle the link between the mean and variance that holds in the classical homoscedastic threshold model. The results obtained for the litter size of Romane ewes showed that the data was best fitted with HTM compared to ITM and TM. The correlations between the observed and predicted variances were equal to 0.6 and 0.2 for HTM and TM, respectively. These analyses showed the existence of a genetic component for the heterogeneity of litter size in sheep that was taken into account in HTM.
  
  
  CONCLUSIONS
  HTM is the most suitable model to study the variability of litter size in sheep. It accommodates both the mean and variance separately while requiring the estimation of only a few parameters.
authors.affiliation: GenPhySE, Université de Toulouse, INRA, INPT, INP-ENVT, Castanet-Tolosan, France.
authors.forename: Samira
authors.initials: S
authors.lastname: Fathallah
authors.affiliation: GenPhySE, Université de Toulouse, INRA, INPT, INP-ENVT, Castanet-Tolosan, France. loys.bodin@toulouse.inra.fr.
authors.forename: Loys
authors.initials: L
authors.lastname: Bodin
authors.affiliation: GenPhySE, Université de Toulouse, INRA, INPT, INP-ENVT, Castanet-Tolosan, France.
authors.forename: Ingrid
authors.initials: I
authors.lastname: David
mesh_terms: D000818:Animals; D001947:Breeding / Q000706:statistics & numerical data*; D005260:Female; D008098:Litter Size / Q000235:genetics*; D015233:Models, Statistical; D010641:Phenotype; D034561:Sheep, Domestic / Q000235:genetics*