Browsing by Subject "Gemischte Modelle"

Now showing 1 - 3 of 3

Biometrical approaches for analysing gene bank evaluation data on barley (Hordeum spec.)
(2007) Hartung, Karin; Piepho, Hans-Peter
This thesis explored methods to statistically analyse phenotypic data of gene banks. Traits of the barley data (Hordeum spp.) of the gene bank of the IPK-Gatersleben were evaluated. The data of years 1948-2002 were available. Within this period the ordinal scale changed from a 0-5 to a 1-9 scale after 1993. At most gene banks reproduction of accessions is currently done without any experimental design. With data of a single year only rarely do accessions have replications and there are only few replications of a single check for winter and summer barley. The data of 2002 were analysed separately for winter and summer barley using geostatistical methods. For the traits analysed four types of variogram model (linear, spherical, exponential and Gaussian) were fitted to the empirical variogram using non-linear regression. The spatial parameters obtained by non-linear regression for every variogram model then were implemented in a mixed model analysis and the four model fits compared using Akaike's Information Criterion (AIC). The approach to estimate the genetical parameter by Kriging can not be recommended. The first points of the empirical variogram should be explained well by the fitted theoretical variogram, as these represent most of the pairwise distances between plots and are most crucial for neighbour adjustments. The most common well-fitting geostatistical models were the spherical and the exponential model. A nugget effect was needed for nearly all traits. The small number of check plots for the available data made it difficult to accurately dissect the genetical effect from environmental effects. The threshold model allows for joint analysis of multi-year data from different rating scales, assuming a common latent scale for the different rating systems. The analysis suggests that a mixed model analysis which treats ordinal scores as metric data will yield meaningful results, but that the gain in efficiency is higher when using a threshold model. The threshold model may also be used when there is a metric scale underlying the observed ratings. The Laplace approximation as a numerical method to integrate the log-likelihood for random effects worked well, but it is recommended to increase the number of quadrature points until the change in parameter estimates becomes negligible. Three rating methods (1%, 5%, 9-point rating) were assessed by persons untrained (A) and experienced (B) in rating. Every person had to rate several pictograms of diseased leaves. The highest accuracy was found with Group B using the 1%-scale and with Group A using the 5%-scale. With a percentage scale Group A tended to use values that are multiples of 5%. For the time needed per leaf assessment the Group B was fastest when using the 5% rating scale. From a statistical point of view both percent ratings performed better than the ordinal rating scale and the possible error made by the rater is calculable and usually smaller than with ratings by rougher methods. So directly rating percentages whenever possible leads to smaller overall estimation errors, and with proper training accuracy and precision can be further improved. For gene banks augmented designs as proposed by Federer and by Lin et al. offer themselves, so an overview is given. The augmented designs proposed by Federer have the advantage of an unbiased error estimate. But the random allocation of checks is a problem. The augmented design by Lin et al. always places checks in the centre plot of every whole plot. But none of the methods is based on an explicit statistical model, so there is no well-founded decision criterion to select between them. Spatial analysis can be used to find an optimal field layout for an augmented design, i.e. a layout that yields small least significant differences. The average variance of a difference and the average squared LSD were used to compare competing designs, using a theoretical approach based on variations of two anisotropic models and different rotations of anisotropy axes towards field reference axes. Based on theoretical calculations, up to five checks per block are recommended. The nearly isotropic combinations led to designs with large quadratic blocks. With strongly anisotropic combinations the optimal design depends on degree of anisotropy and rotation of anisotropy axes: without rotation small elongated blocks are preferred; the closer the rotation is to 45° the more squarish blocks and the more checks are appropriate. The results presented in this thesis may be summarised as follows: Cultivation for regeneration of accessions should be based on a meaningful and statistically analysable experimental field design. The design needs to include checks and a random sample of accessions from the gene pool held at the gene bank. It is advisable to utilise metric or percentage rating scales. It can be expected that using a threshold model increases the quality of multivariate analysis and association mapping studies based on phenotypic gene bank data.
Estimating heritability in plant breeding programs
(2019) Schmidt, Paul; Piepho, Hans-Peter
Heritability is an important notion in, e.g., human genetics, animal breeding and plant breeding, since the focus of these fields lies on the relationship between phenotypes and genotypes. A phenotype is the composite of an organism’s observable traits, which is determined by its underlying genotype, by environmental factors and by genotype-environment interactions. For a set of genotypes, the notion of heritability expresses the proportion of the phenotypic variance that is attributable to the genotypic variance. Furthermore, as it is an intraclass correlation, heritability can also be interpreted as, e.g., the squared correlation between phenotypic and genotypic values. It is important to note that heritability was originally proposed in the context of animal breeding where it is the individual animal that represents the basic unit of observation. This stands in contrast to plant breeding, where multiple observations for the same genotype are obtained in replicated trials. Furthermore, trials are usually conducted as multi-environment trials (MET), where an environment denotes a year × location combination and represents a random sample from a target population of environments. Hence, the observations for each genotype first need to be aggregated in order to obtain a single phenotypic value, which is usually done by obtaining some sort of mean value across trials and replicates. As a consequence, heritability in the context of plant breeding is referred to as heritability on an entry-mean basis and its standard estimation method is a linear combination of variances and trial dimensions. Ultimately, I find that there are two main uses for heritability in plant breeding: The first is to predict the response to selection and the second is as a descriptive measure for the usefulness and precision of cultivar trials. Heritability on an entry-mean basis is suited for both purposes as long as three main assumptions hold: (i) the trial design is completely balanced/orthogonal, (ii) genotypic effects are independent and (iii) variances and covariances are constant. In the last decades, however, many advancements in the methodology of experimental design for and statistical analysis of plant breeding trials took place. As a consequence it is seldom the case that all three of above mentioned assumptions are met. Instead, the application of linear mixed models enables the breeder to straightforwardly analyze unbalanced data with complex variance structures. Chapter 2 exemplarily demonstrates some of the flexibility and benefit of the mixed model framework for typically unbalanced MET by using a bivariate mixed model analyses to jointly analyze two MET for cultivar evaluation, which differ in multiple crucial aspects such as plot size, trial design and general purpose. Such an approach can lead to higher accuracy and precision of the analysis and thus more efficient and successful breeding programs. It is not clear, however, how to define and estimate a generalized heritability on an entry-mean basis for such settings. Therefore, multiple alternative methods for the estimation of heritability on an entry-mean basis have been proposed. In Chapter 3, six alternative methods are applied to four typically unbalanced MET for cultivar evaluation and compared to the standard method. The outcome suggests that the standard method over-estimates heritability, while all of the alternative methods show similar, lower estimates and thus seem able to handle this kind of unbalanced data. Finally, it is argued in Chapter 4 that heritability in plant breeding is not actually based on or aiming at entry-means, but on the differences between them. Moreover, an estimation method for this new proposal of heritability on an entry-difference basis (H_Delta^2/h_Delta^2) is derived and discussed, as well as exemplified and compared to other methods via analyzing four different datasets for cultivar evaluation which differ in their complexity. I argue that regarding the use of heritability as a descriptive measure, H_Delta^2/h_Delta^2, can on the one hand give a more detailed and meaningful insight than all other heritability methods and on the other hand reduces to other methods under certain circumstances. When it comes to the use of heritability as a means to predict the response to selection, the outcome of this work discourages this as a whole. Instead, response to selection should be simulated directly and thus without using any ad hoc heritability measure.
Statistical methods for analysis of multienvironment trials in plant breeding : accuracy and precision
(2021) Buntaran, Harimurti; Piepho, Hans-Peter
Multienvironment trials (MET) are carried out every year in different environmental conditions to evaluate a vast number of cultivars, i.e., yield, because different cultivars perform differently in various environmental conditions, known as genotype×environment interactions. MET aim to provide accurate information on cultivar performance so that a recommendation of which cultivar performs the best in a growers’ field condition can be available. MET data is often analysed via mixed models, which allow the cultivar effect to be random. The random effect of cultivar enables genetic correlation to be exploited across zones and considering the trials’ heterogeneity. A zone can be viewed as a larger target of population environments. The accuracy and precision of the cultivar predictions are crucial to be evaluated. The prediction accuracy can be evaluated via a cross-validation (CV) study, and the model selection can be done based on the lowest mean squared error prediction (MSEP). Also, since the trials’ locations hardly coincide with growers’ field, the precision of predictions needs to be evaluated via standard errors of predictions of cultivar values (SEPV) and standard errors of the predictions of pairwise differences of cultivar values (SEPD). The central objective of this thesis is to assess the model performance and conduct model selection via a CV study for zone-based cultivar predictions. Chapter 2 assessed the performance between empirical best linear unbiased estimations (EBLUE) and empirical best linear unbiased predictions (EBLUP) for zone-based prediction. Different CV schemes were done for the single-year and multi-year datasets to mimic the practice. A complex covariance structure such as factor-analytic (FA) was imposed to account for the heterogeneity of cultivar×zone (CZ) effect. The MSEP showed that the EBLUP models outperformed the EBLUE models. The zonation was necessary since it improved the accuracy and was preferable to make cultivar recommendations. The FA structure did not improve the accuracy compared to the simpler covariance structure, and so the EBLUP model with a simple covariance structure is sufficient for the single and multi-year datasets. Chapter 3 assessed the single-stage and stagewise analyses. The three weighting methods were compared in the stagewise analysis: two diagonal approximation methods and the fully efficient method with the unweighted analysis. The assessment was based on the MSEP instead of Pearson’s and Spearman’s correlation coefficients since the correlation coefficients are often very close between the compared models. The MSEP showed that the single-stage EBLUP and the stagewise weighting EBLUP strategy were very similar. Thus, the loss of information due to diagonal approximation is minor. In fact, the MSEP showed a more apparent distinction between the single-stage and the stagewise weighting analyses with the unweighted EBLUE compared to the correlation coefficients. The simple compound-symmetric covariance structure was sufficient for the CZ effect than the more complex structures. The choice between the single-stage and stagewise weighting analysis, thus, depends on the computational resources and the practicality of data handling. Chapter 4 assessed the accuracy and precision of the predictions for the new locations. The environmental covariates were combined with the EBLUP in the random coefficient (RC) models since the covariates provide more information for the new locations. The MSEP showed that the RC models were not the model with the smallest MSEP, but the RC models had the lowest SEPV and SEPD. Thus, the model selection can be done by joint consideration of the MSEP, SEPV, and SEPD. The models with EBLUE and covariate interaction effects performed poorly regarding the MSEP. The EBLUP models without RC performed best, but the SEPV and SEPD were large, considered unreliable. The covariate scale and selection are essential to obtain a positive definite covariance matrix. Employing unstructured covariance int the RC is crucial to maintaining the RC models’ invariance feature. The RC framework is suitable to be implemented with GIS data to provide an accurate and precise projection of cultivar performance for the new locations or environments. To conclude, the EBLUP model for zoned-based predictions should be preferred to obtain the predictions and rankings closer to the true values and rankings. The stagewise weighting analysis can be recommended due to its practicality and its computational efficiency. Furthermore, projecting cultivar performances to the new locations should be done to provide more targeted information for growers. The available environmental covariates can be utilised to improve the predictions’ accuracy and precision in the new locations in the RC model framework. Such information is certainly more valuable for growers and breeders than just providing means across a whole target population of environments.