REML approach for adjusting the Fusarium head blight rating to a phenological date in inoculated selection experiments of wheat

Abstract

Fusarium head blight is one of the most important wheat diseases causing grain yield and quality losses as well as mycotoxin contamination all over the world. Since Fusarium cannot be reliably controlled with fungicides, breeding has become a favorable tool to decrease the infection severity. In most cases, selection for Fusarium resistance is done by artificial infection in the field. However, there is a risk in preferring late heading genotypes, because heading of wheat is negatively correlated to head blight severity. Because an indirect selection for late maturity is not intended, we considered a statistical approach to avoid this problem. In this paper, we propose a mixed model to analyze extensive Fusarium head blight rating in resistance breeding experiments of wheat. The objective of the analysis was to select for Fusarium resistance, while at the same time ensuring that late heading genotypes, which show less head blight over the shorter vegetation period, are not preferred. Thus, selection was to be done such that genetic variability for heading date was retained. Therefore, the statistical model contained a covariate to adjust for differences in the heading date. The use of covariate adjustment is an easily handled alternative to a bivariate analysis. Covariate adjustment will in practice often work almost equally well as bivariate analysis. Any statistical software with powerful mixed model analysis tools can be used for this type of analysis. We propose an ad hoc method to obtain heritability estimates and a form of LSD (least significance difference) as a measure of accuracy on the basis of the proposed model and under special consideration of the experimental design. The ad hoc LSD was used as a rough measure to judge rankings of genotypic means (BLUPs). Friedman’s super smoother was used to compare smoothed rank estimates for adjusted and unadjusted genotypes against increasing smoothed heading dates. Traits were transformed to meet the model assumptions, especially homogeneity of errors and normality, and back-transformation of means and standard errors was conducted by using the delta method.

Appendix

Back-transformation of means and standard errors

A genotype mean $\bar x_{\text{i}}$ on the logit scale with standard error se _i is back-transformed in the original scale by

$$ h^{ - 1} (\bar x_i ) \cong \frac{{\exp (\bar x_i )}}{{\left[ {1 + \exp (\bar x_i )} \right]}} + \frac{1}{2} \times {\text{Var}}(\bar x_i )\left[ {\frac{{\partial ^2 }}{{\partial {{\bar x_i}^2} }}h^{ - 1} (\bar x_i )} \right] $$

and

$$ {\text{Var(}}h^{ - 1} (\bar x_i )) \cong {\text{se}}_{i}^2 \left[ {\frac{\partial }{{\partial {\bar x_i}}}h^{ - 1} (\bar x_i )} \right]^2 . $$

In case of differences of genotype means (i.e., pairwise comparisons of genotype means) $d = \bar x_1 - \bar x_2$ we back-transformed the expectations and variances by

$$ E(h^{ - 1} (d)) = h^{ - 1} (\bar x_1 ) - h^{ - 1} (\bar x_2 ) $$

and

$$ {\text{Var(}}h^{ - 1} (d)) = {\text{Var}}(\bar x_1 )\left[ {\frac{\partial }{{\partial \bar x_1 }}h^{ - 1} (\bar x_1 )} \right]^2 + {\text{Var}}(\bar x_2 )\left[ {\frac{\partial }{{\partial \bar x_2 }}h^{ - 1} (\bar x_2 )} \right]^2 - 2\;{\text{Cov}}\;{\text{(}}\bar x_1 ,\bar x_2 {\text{)}}\left[ {\frac{\partial }{{\partial \bar x_1 }}h^{ - 1} (\bar x_1 )} \right]\;\left[ {\frac{\partial }{{\partial \bar x_2 }}h^{ - 1} (\bar x_2 )} \right]. $$

${\text{Var}}(\bar x_1 ),\,{\text{Var}}(\bar x_2 )$ and ${\text{Cov}}(\bar x_1 ,\bar x_2 )$ are the variances and covariances of genotype means and covariances of genotype means on the transformed scale.

For back-transformation from the logit scale, the formulas for first and second-order derivatives are:

$$ \frac{\partial }{{\partial {\bar x}}}h^{ - 1} (\bar x) = \frac{{\exp (\bar x)}}{{[1 + \exp (\bar x)]^2 }}\quad {\text{and}}\quad \frac{{\partial ^2 }}{{\partial \bar x^2 }}h^{ - 1} (\bar x) = \frac{{\exp (\bar x)\;(1 - \exp (\bar x))}}{{[1 + \exp (\bar x)]^3 }}. $$