Fitting Bayesian multiple random effects models

SUMMARYNutrient pollution in rivers is a common problem. It can provoke algae blooms which are related to increased fish mortality. To restore the water status, the regulator recently has promulgated more restrictive regulations. In Flanders for instance, the government has introduced several manure decrees (MDs) to restrict nutrient pollution. Environmental regulations are commonly expressed in terms of threshold levels. This provides a binary response to the decision maker. To handle such data, we propose the use of marginalised generalised linear mixed models. They provide valid inference on trends in the exceedance frequency. The spatio-temporal dependence of the river monitoring network is incorporated by the use of a latent variable. The temporal dependence is assumed to be AR(1) and the spatial dependence is derived from the river topology. The mean model contains a term for the trend and corrects for seasonal variation. The model formulation allows an assessment on the level of individual sampling locations and on a more regional scale. The methodology is applied to a case study on the river Yzer (Flanders). It assesses the impact of the MDs on the violation probability of the nitrate standard. A trend change is detected after the introduction of the second MD.

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“…This is a well-known problem when mixed models are used in a Bayesian context (e.g. Vines et al, 1996).…”

Section: Discussionmentioning

confidence: 99%

Testing for trends in the violation frequency of an environmental threshold in rivers

Clement

Thas

2008

Environmetrics

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“…If only single-site sampling is possible (due to computational limitations), the parameters can be randomly updated in each MCMC step and not in the same sequential order, as suggested by Levine and Casella (2006). An additional alternative to improve mixing is to apply transformation of the location parameters in the model (Vines et al 1996;Waldmann et al 2008).…”

Section: Discussionmentioning

confidence: 99%

“…To our knowledge, this procedure of decomposing the prior distribution has not been utilized before, in the context of parameter estimation in genetics. The decomposion approach was put forward and excecuted successfully in WinBUGS by Vines et al (1996); they utilized a random effect model to analyze clinical data in the context of epidemiology. However, they did not include covariance between random effects, which makes the decomposition procedure more complex.…”

Section: Discussionmentioning

confidence: 99%

Bayesian Inference of Genetic Parameters Based on Conditional Decompositions of Multivariate Normal Distributions

Hallander

Waldmann²,

Wang

et al. 2010

Genetics

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It is widely recognized that the mixed linear model is an important tool for parameter estimation in the analysis of complex pedigrees, which includes both pedigree and genomic information, and where mutually dependent genetic factors are often assumed to follow multivariate normal distributions of high dimension. We have developed a Bayesian statistical method based on the decomposition of the multivariate normal prior distribution into products of conditional univariate distributions. This procedure permits computationally demanding genetic evaluations of complex pedigrees, within the user-friendly computer package WinBUGS. To demonstrate and evaluate the flexibility of the method, we analyzed two example pedigrees: a large noninbred pedigree of Scots pine (Pinus sylvestris L.) that includes additive and dominance polygenic relationships and a simulated pedigree where genomic relationships have been calculated on the basis of a dense marker map. The analysis showed that our method was fast and provided accurate estimates and that it should therefore be a helpful tool for estimating genetic parameters of complex pedigrees quickly and reliably.

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“…After converting the 24‐h day to angles in radians to match the projected normal density definition in , the fractions

F_{ijk}^{M} (τ)

are model‐based and can be written as

F_{ijk}^{M} (τ) = \int_{false(τ - 0.5 false) π / 12}^{false(τ + 0.5 false) π / 12} f (θ ∣ μ_{italicijk}) d θ,

with the two‐dimensional mean vectors

{boldμ}_{ijk} = μ + {boldm}_{k} + {bolds}_{i} + {boldw}_{j} + {boldsw}_{ij}

an overall mean plus mode, state, wave, and state–wave interaction effects. Sweeping (Vines, Gilks, & Wild, ) and centering Gelfand, Sahu, & Carlin () parameterizations are common in linear models to improve mixing and identifiability of parameters, and are also applied here. We restrict the fixed effects to sum to zero

\sum {boldm}_{k} = 0, \sum {bolds}_{i} = 0, \sum {boldw}_{j} = 0,

and use the parameterization

{boldη}_{ij} = {bolds}_{i} + {boldw}_{j} + {boldsw}_{ij}

.…”

Section: Sae Of Chts Departure Timesmentioning

confidence: 99%

Hierarchical Bayesian small area estimation for circular data

Hernandez-Stumpfhauser

Breidt

Opsomer

2016

Can J Statistics

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We consider small area estimation for the departure times of recreational anglers along the Atlantic and Gulf coasts of the United States. A Bayesian area‐level Fay–Herriot model is considered to obtain estimates of the departure time distribution functions. The departure distribution functions are modelled as circular distributions plus area‐specific errors. The circular distributions are modelled as projected normal, and a regression model is specified to borrow information across domains. Estimation is conducted through the use of a Hamiltonian Monte Carlo sampler and a projective approach onto the probability simplex. The Canadian Journal of Statistics 44: 416–430; 2016 © 2016 Statistical Society of Canada

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Fitting Bayesian multiple random effects models

Cited by 29 publications

References 32 publications

Testing for trends in the violation frequency of an environmental threshold in rivers

Testing for trends in the violation frequency of an environmental threshold in rivers

Bayesian Inference of Genetic Parameters Based on Conditional Decompositions of Multivariate Normal Distributions

Hierarchical Bayesian small area estimation for circular data

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