Abstract
Population-averaged and subject-specific models are available to evaluate count data when repeated observations per subject are present. The latter are also known in the literature as generalised linear mixed models (GLMM). In GLMM repeated measures are taken into account explicitly through random animal effects in the linear predictor. In this paper the relevant GLMMs are presented based on conditional Poisson or negative binomial distribution of the response variable for given random animal effects. Equations for the repeatability of count data are derived assuming normal distribution and logarithmic gamma distribution for the random animal effects. Using count data on aggressive behaviour events of pigs (barrows, sows and boars) in mixed-sex housing, we demonstrate the use of the Poisson »log-gamma intercept«, the Poisson »normal intercept« and the »normal intercept« model with negative binomial distribution. Since not all count data can definitely be seen as Poisson or negative-binomially distributed, questions of model selection and model checking are examined. Emanating from the example, we also interpret the least squares means, estimated on the link as well as the response scale. Options provided by the SAS procedure NLMIXED for estimating model parameters and for estimating marginal expected values are presented.
Highlights
Count data arise when the trait outcome is determined through a count process, whereby a count variable denotes the number of discrete occurrences of an event of interest
If we consider Yi = (Yi1, ..., Yini ), and μij =, it can be shown that Yi satisfies a multivariate negative binomial (MNB) distribution whose density function can be given as follows: ni
Estimating the parameters in the Poisson normal (PoiN), in the Poisson log-gamma (PoiLG) and in the negative binomial normal (NBinN) model occurs through the use of the maximum likelihood (ML) method, i.e. through setting up and maximising the log-likelihood function
Summary
Count data arise when the trait outcome is determined through a count process, whereby a count variable denotes the number of discrete occurrences of an event of interest. A further differentiation of the subject-specific model is given by making distributional assumptions for the random effects in the linear predictor. On the response scale the GLMMs provide conditional expected values for the count trait, for example for an animal whose effect corresponds to the average of the herd or population. In this paper only subject-specific models, that are models with random effects in the linear predictor, are used to evaluate the count data. On particular it is shown how, for count data, the covariance between repeated observations per subject can be calculated on the response scale. Questions on model selection and checking are extensively illustrated for the analysis of correlated count data
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