Analysis of correlated count data using generalised linear mixed models exemplified by field data on aggressive behaviour of boars

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.

Analysis of correlated count data using generalised linear mixed models exemplified by field data on aggressive behaviour of boars

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.

Distribution patterns of Macrodasyceras hirsutum (Hymenoptera: Torymidae) eggs among Ilex integra (Aquifoliaceae) seeds and berries

Preference-performance hypothesis in host selection by herbivorous insects predicts selective oviposition on plant on which the offspring maximise fitness. The seed parasitoid wasp, Macrodasyceras hirsutum Kamijo (Hymenoptera: Torymidae) selectively lays the eggs into the fertilised seeds of Ilex integra Thunberg (Aquifoliaceae). Only one larva develops in a seed. Therefore, the hypothesis predicts a uniform distribution pattern of wasp eggs among fertilised seeds. Dissection of 531 berries showed that M. hirsutum deposited one to five eggs into a fertilised seed. Iwao's patchiness regression suggested a uniform distribution pattern of M. hirsutum eggs among fertilised seeds at the scales of tree and berry and their random distribution pattern among berries at the tree scale. Destroying the connection of seeds within berries revealed that female wasps randomly selected berries for oviposition in most trees. Generalised linear mixed models showed that the number of fertilised seeds in a berry could not explain the number of eggs in a seed but could explain the number of eggs in a berry. Therefore, this study shows that the wasp females do not distinguish between berries with different numbers of fertilised seeds but distinguish between seeds harbouring different numbers of eggs, which supports the hypothesis.