Best linear unbiased prediction (BLUP) is a standard method for estimating random effects of a mixed model. This method was originally developed in animal breeding for estimation of breeding values and is now widely used in many areas of research. It does not, however, seem to have gained the same popularity in plant breeding and variety testing as it has in animal breeding. In plants, application of mixed models with random genetic effects has up until recently been mainly restricted to the estimation of genetic and nongenetic components of variance, whereas estimation of genotypic values is mostly based on a model with fixed effects. This paper reviews recent developments in the application of BLUP in plant breeding and variety testing. These include the use of pedigree information to model and exploit genetic correlation among relatives and the use of flexible variance-covariance structures for genotype-by-environment interaction. We demonstrate that BLUP has good predictive accuracy compared to other procedures. While pedigree information is often included via the so-called numerator relationship matrix ðAÞ, we stress that it is frequently straightforward to exploit the same information by a simple mixed model without explicit reference to the A-matrix.

Recent review articles in this journal have compared the relative merits of two prominent statistical models for analyzing yield‐trial data: Additive main effects and multiplicative interaction (AMMI) and genotype main effects and genotype × environment interaction (GGE). This review addresses more than 20 issues that require clarification after controversial statements and contrasting conclusions have appeared in those recent reviews. The AMMI2 mega‐environment display incorporates more of the genotype main effect and captures more of the genotype × environment (GE) interaction than does GGE2, thereby displaying the which‐won‐where pattern more accurately for complex datasets. When the GE interaction is captured well by one principal component, the AMMI1 display of genotype nominal yields describes winning genotypes and adaptive responses more simply and clearly than the GGE2 biplot. For genotype evaluation within a single mega‐environment, a simple scatterplot of mean and stability is more straightforward than the mean vs. stability view of a GGE2 biplot. Diagnosing the most predictively accurate member of a model family is vital for either AMMI or GGE, both for gaining accuracy and delineating mega‐environments.

Heritability is often used by plant breeders and geneticists as a measure of precision of a trial or a series of trials. Its main use is for computing the response to selection. Most formulas proposed for calculating heritability implicitly assume balanced data and independent genotypic effects. Both of these assumptions are often violated in plant breeding trials. This article proposes a simulation-based approach to tackle the problem. The key idea is to directly simulate the quantity of interest, e.g., response to selection, rather than trying to approximate it using some ad hoc measure of heritability. The approach is illustrated by three examples.

There is growing evidence of escalating wildlife losses worldwide. Extreme wildlife losses have recently been documented for large parts of Africa, including western, Central and Eastern Africa. Here, we report extreme declines in wildlife and contemporaneous increase in livestock numbers in Kenya rangelands between 1977 and 2016. Our analysis uses systematic aerial monitoring survey data collected in rangelands that collectively cover 88% of Kenya’s land surface. Our results show that wildlife numbers declined on average by 68% between 1977 and 2016. The magnitude of decline varied among species but was most extreme (72–88%) and now severely threatens the population viability and persistence of warthog, lesser kudu, Thomson’s gazelle, eland, oryx, topi, hartebeest, impala, Grevy’s zebra and waterbuck in Kenya’s rangelands. The declines were widespread and occurred in most of the 21 rangeland counties. Likewise to wildlife, cattle numbers decreased (25.2%) but numbers of sheep and goats (76.3%), camels (13.1%) and donkeys (6.7%) evidently increased in the same period. As a result, livestock biomass was 8.1 times greater than that of wildlife in 2011–2013 compared to 3.5 times in 1977–1980. Most of Kenya’s wildlife (ca. 30%) occurred in Narok County alone. The proportion of the total “national” wildlife population found in each county increased between 1977 and 2016 substantially only in Taita Taveta and Laikipia but marginally in Garissa and Wajir counties, largely reflecting greater wildlife losses elsewhere. The declines raise very grave concerns about the future of wildlife, the effectiveness of wildlife conservation policies, strategies and practices in Kenya. Causes of the wildlife declines include exponential human population growth, increasing livestock numbers, declining rainfall and a striking rise in temperatures but the fundamental cause seems to be policy, institutional and market failures. Accordingly, we thoroughly evaluate wildlife conservation policy in Kenya. We suggest policy, institutional and management interventions likely to succeed in reducing the declines and restoring rangeland health, most notably through strengthening and investing in community and private wildlife conservancies in the rangelands.

This paper reviews properties of ridge regression for genomewide (genomic) selection and establishes close relationships with other methods to model genetic correlation among relatives, including use of a kinship matrix and the simple matching coefficient as computed from marker data. A number of alternative models are then proposed exploiting ties between genetic correlation based on marker data and geostatistical concepts. A simple method for automatic marker selection is proposed. The methods are exemplified using a series of experiments with test‐cross hybrids of maize (Zea mays L.) conducted in five environments. Results underline the need to appropriately model genotype–environment interaction and to employ an independent estimate of error. It is also shown that accounting for genetic effects not captured by markers may be important.

Designed experiments conducted by crop scientists often give rise to several random sources of variation. Pertinent examples are split-plot designs, series of experiments and repeated measurements taken on the same field plot. Data arising from such experiments may be conveniently analysed by mixed models. While the mixed model framework is by now very well developed theoretically, and good software is readily available, the technology is still underutilized. The purpose of the present paper is, therefore, to encourage more widespread use of mixed models. We outline basic principles, which help in setting up mixed models appropriate in a given situation, the main task required from users of mixed model software. Several examples are considered to demonstrate key issues. The theoretical underpinnings are briefly sketched in so far as they are practically relevant for making informed use of mixed-model computer packages. Finally, a brief review is given of some recent methodological developments, which are of interest to the plant sciences. A German version of this paper is available from the corresponding author upon request.

Association-mapping methods promise to overcome the limitations of linkage-mapping methods. The main objectives of this study were to (i) evaluate various methods for association mapping in the autogamous species wheat using an empirical data set, (ii) determine a marker-based kinship matrix using a restricted maximum-likelihood (REML) estimate of the probability of two alleles at the same locus being identical in state but not identical by descent, and (iii) compare the results of association-mapping approaches based on adjusted entry means (two-step approaches) with the results of approaches in which the phenotypic data analysis and the association analysis were performed in one step (one-step approaches). On the basis of the phenotypic and genotypic data of 303 soft winter wheat (Triticum aestivum L.) inbreds, various association-mapping methods were evaluated. Spearman's rank correlation between P-values calculated on the basis of one-and two-stage association-mapping methods ranged from 0.63 to 0.93. The mixedmodel association-mapping approaches using a kinship matrix estimated by REML are more appropriate for association mapping than the recently proposed QK method with respect to (i) the adherence to the nominal a-level and (ii) the adjusted power for detection of quantitative trait loci. Furthermore, we showed that our data set could be analyzed by using two-step approaches of the proposed association-mapping method without substantially increasing the empirical type I error rate in comparison to the corresponding one-step approaches.

All-pairwise comparisons among a set of t treatments or groups are one of the most frequent tasks in applied statistics. Users of statistical software are accustomed to the familiar lines display, in which treatments that do not differ significantly, are connected by a common line or letter. Availability of the lines display is restricted mainly to the balanced analysis of variance set-up. This limited availability is at stark variance with the diversity of statistical methods and models, which call for multiple comparisons. The present paper describes a general method for graphically representing any set of t(t-1)/2 all-pairwise significance statements (p-values) for t treatments by a familiar letter display, which is applicable regardless of the underlying data structure or the statistical method used for comparisons. The method reproduces the familiar lines display in case of the balanced analysis of variance. Its broad applicability is demonstrated using data from an international multi-environment wheat yield trial and from a fish catching survey.