Bayesian analysis of the linear reaction norm model with unknown covariates1

Abstract

The reaction norm model is becoming a popular approach for the analysis of genotype x environment interactions. In a classical reaction norm model, the expression of a genotype in different environments is described as a linear function (a reaction norm) of an environmental gradient or value. An environmental value is typically defined as the mean performance of all genotypes in the environment, which is usually unknown. One approximation is to estimate the mean phenotypic performance in each environment and then treat these estimates as known covariates in the model. However, a more satisfactory alternative is to infer environmental values simultaneously with the other parameters of the model. This study describes a method and its Bayesian Markov Chain Monte Carlo implementation that makes this possible. Frequentist properties of the proposed method are tested in a simulation study. Estimates of parameters of interest agree well with the true values. Further, inferences about genetic parameters from the proposed method are similar to those derived from a reaction norm model using true environmental values. On the other hand, using phenotypic means as proxies for environmental values results in poor inferences.

Caracterização da interação genótipo-ambiente e comparação entre modelos para ajuste do ganho pós-demama de bovinos Devon via normas de reação

“…Os Os resultados deste estudo permitem concluir, de acordo com diversos autores (Kolmodin et al, 2002;Calus & Veerkamp, 2003;Calus et al, 2004;Pollott & Greeff, 2004;Cardoso et al, 2005a;Su et al, 2006), que há interação genótipo-ambiente na população Devon estudada essencialmente em virtude do efeito de escala, pois, quando consideradas as normas de reação, os valores genéticos apresentaram maior variação nos valores positivos do gradiente ambiental em comparação aos níveis ambientais mais baixos. Ou seja, à medida que o ambiente se torna mais favorável, aumenta a diferença entre os valores genéticos dos indivíduos, uma vez que esses indivíduos expressam melhor seu potencial.…”

Genotype by environment interaction for litter size in pigs as quantified by reaction norms analysis

“…Data analyses were carried out with the DMU program (Madsen and Jensen, 2004) as extended by Su et al (2006). This approach performs a Markov Chain Monte Carlo analysis that allows for inferring the environmental covariate and the other parameters in the model simultaneously, and for accommodating heterogeneous residual variances.…”

Phenotypic plasticity of composite beef cattle performance using reaction norms model with unknown covariate

“…The maternal heritability (h 2 m ) and the proportion of the total phenotypic variance due to maternal permanent environment effect (c 2 ) were obtained in a similar manner. More details about the hierarchical reaction norms model with unknown covariate are reported by Su et al (2006).…”

“…The hierarchical reaction norms model with unknown covariate (Su et al, 2006) was used to estimate genetic parameters and study the G 3 E. In the present study, the CG was chosen as the environmental descriptor, that is, the environmental covariate of the reaction norms. This covariate was jointly estimated with the reaction norms of the animals.…”

“…Traditionally, one approximation proposed is to compute the mean phenotypic of the trait studied in each environment and then use these estimates as known covariates in the model (Kolmodin et al, 2002 and. According to Su et al (2006), this approach can lead to bias in the estimation of genetic parameters and breeding values. To overcome this problem, these authors proposed the simultaneous estimation of environmental values and other parameters of the model (reaction norm model with unknown covariate), which provided more accurate estimates.…”