The coupling method was investigated as a method to assess the convergence of the Gibbs sampler when drawing marginal inferences with an animal model. This method is based on the output of two Markov chains with different starting values but the same conditional deviates. The coupling method shows that the Gibbs sampler has an exponential convergence when the variance components are assumed to be known. All the variables in the model have the same rate of convergence, and it is closely related with the largest eigenvalue of a matrix derived from the coefficient matrix of the mixed model equations. Models including the variance components as unknowns showed a rate of convergence equal for all the variables, and this rate of convergence was approximately exponential. The coupling method provides an estimation of the convergence at the current iteration, and it does not require a post-Gibbs analysis as do the single chain-based methods. The coupling method is less computationally demanding than multiple chain methods, because only two chains are required to assess convergence.
MoYA. 2006. Analysis of polygenic traits of Microcystis aeruginosa (Cyanobacteria) strains by restricted maximum likelihood (REML) procedures: 2. Microcystin net production, photosynthesis and respiration. Phycologia Phycologia 45: 243-248.Phenotypic expression of traits is the result of complex interactions between genotype and environment. A combination of experiments and statistics based on restricted maximum likelihood (REML) procedures was utilised for estimating the proportion of genetic variability in a series of phenotypic characters (microcystin net production, cell size, dark respiration rate, photosynthetic capacity, maximum quantum yield and growth rate), as well as their genetic and nongenetie correlations, measured on 18 Microcystis aeruginosa (Cyanobacteria) strains. Results indicated that the phenotypic variability found in these traits were principally due to genetic differences among strains. Thus, the estimated genetic variability ranged from 50% for dark respiration rate to > 90% in cell size or maximum quantum yield. These results support the conclusions that (1) genetic factors are the main cause of the spatial-temporal heterogeneity observed in quantum yield, respiration and toxin production, and (2) light harvesting and toxin production are not under intense natural selection, both having a low adaptive value. The high variance in toxin production resulting from genetic effects introduces a source of unpredictability in water-supply toxic early warnings. In addition, no significant genetic correlations were found between quantum yield and either maximal growth rate or mean cell diameter.
M. Rico, M. ALTAMIRANO, V. LOPEZ-RODAS AND E. COSTAS. 2006. Analysis of polygenic traits of Microcy.Kti.s aerugino.'ia (Cyanobacteria) strains by Restricted Maximum Likelihood (REML) procedures: 1. Size and shape of colonies and cells.Traditionally, morphometrical traits are used to classify cyanobacteria. However, neither the importance of genetic factors in morphological variability nor genetic correlations between different morphometrical traits are known. Six morphometrical traits of individual cells and six morphometrical traits of colonies have been measured in 21 strains of Microcystis aeruginosa. A fixed model and a mixed model allowing genetic relationships between traits were both used to estimate genetic and residual correlations, and genetic variance/total phenotypic variance ratios were used to estimate the 12 morphometrical traits. Results showed that (1) colony size has a low genetic correlation with traits of individual cells; (2) traits measured on individual cells present a higher genetic variance/total phenotypic variance ratio than the corresponding trait in colonies; and (3) genetic correlations between directly measured traits (maximal diameter and axis, minimal diameter and axis, perimeter, and surface) are so high that they may be considered different expressions of the same underlying trait, namely size, so any one of them could be used without significant loss of information, instead of all of them. Fixed and mixed models showed the same conclusions, although mixed models preferably should be used because they take into account the common random genetic basis of traits. Maximal cell diameter was used as a discriminator trait for studying morphological differences between strains from different origins. Results suggest that environment may modulate strain mixtures in Microcystis populations.
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