Efficiency of augmented p-rep designs in multi-environmental trials

Key message

The paper shows that unreplicated designs in multi-environmental trials are most efficient. If replication per environment is needed then augmented p-rep designs outperform augmented and replicated designs in triticale and maize.

Abstract

In plant breeding, augmented designs with unreplicated entries are frequently used for early generation testing. With limited amount of seed, this design allows to use a maximum number of environments in multi-environmental trials (METs). Check plots enable the estimation of block effects, error variances and a connection of otherwise unconnected trials in METs. Cullis et al. (J Agri Biol Environ Stat 11:381–393, 2006) propose to replace check plots from a grid-plot design by plots of replicated entries leading to partially replicated (p-rep) designs. Williams et al. (Biom J 53:19–27, 2011) apply this idea to augmented designs (augmented p-rep designs). While p-rep designs are increasingly used in METs, a comparison of the efficiency of augmented p-rep designs and augmented designs in the range between replicated and unreplicated designs in METs is lacking. We simulated genetic effects and allocated them according to these four designs to plot yields of a triticale and a maize uniformity trial. The designs varied in the number of environments, but have a fixed number of entries and total plots. The error model and the assumption of fixed or random entry effects were varied in simulations. We extended our simulation for the triticale data by including correlated entry effects which are common in genomic selection. Results show an advantage of unreplicated and augmented p-rep designs and a preference for using random entry effects, especially in case of correlated effects reflecting relationships among entries. Spatial error models had minor advantages compared to purely randomization-based models.     

A note on the Design of Unreplicated Trials

One approach to the design of early generation trials is to have a number of replicated check varieties and a larger number of unreplicated or new varieties. For such trials we define the class of α–α‐designs as an extension to the augmented lattice square designs of Federer (2002). Efficient latinized α–α‐designs can be conveniently constructed using the design generation package CycDesigN.     

Optimal complete block designs to adjust for interplot competition with a covariance analysis

Optimal complete block designs are given for variety trials with interplot competition. In the paper, competition is modeled using height differences between neighboring varieties. When variety heights are known at the design stage, a simulated annealing algorithm is proposed. The optimal designs tend to reduce height differences between neighboring varieties. When variety heights are not known in advance, neighbor-balanced designs at distances 1 and 2 are shown to have optimality properties.     

The design and analysis of breeders' trials of sugar beet (Beta vulgaris) using two-dimensional blocking structures

A method for generating two-dimensional blocking designs to fit any shape and size of experimental area is presented. The method takes an alpha design appropriate to the experimental area and imposes additional blocking perpendicular to the alpha blocks. Improvements to the design are then made by repositioning entries within the alpha blocks. Although this method of construction is less sophisticated than for other two-dimensional designs, the designs are particularly suited to large scale breeders trials where no alternative two-dimensional design may exist. The designs have been used for 3 yr in a sugar beet breeding programme, and have given improvements in efficiency over one-dimensional alpha designs.     

Construction of resolvable spatial row-column designs

Resolvable row-column designs are widely used in field trials to control variation and improve the precision of treatment comparisons. Further gains can often be made by using a spatial model or a combination of spatial and incomplete blocking components. Martin, Eccleston, and Gleeson presented some general principles for the construction of robust spatial block designs which were addressed by spatial designs based on the linear variance (LV) model. In this article we define the two-dimensional form of the LV model and investigate extensions of the Martin et al. principles for the construction of resolvable spatial row-column designs. The computer construction of efficient spatial designs is discussed and some comparisons made with designs constructed assuming an autoregressive variance structure.     

A comparison of methods for constructing confidence intervals after phase II/III clinical trials

Recently, in order to accelerate drug development, trials that use adaptive seamless designs such as phase II/III clinical trials have been proposed. Phase II/III clinical trials combine traditional phases II and III into a single trial that is conducted in two stages. Using stage 1 data, an interim analysis is performed to answer phase II objectives and after collection of stage 2 data, a final confirmatory analysis is performed to answer phase III objectives. In this paper we consider phase II/III clinical trials in which, at stage 1, several experimental treatments are compared to a control and the apparently most effective experimental treatment is selected to continue to stage 2. Although these trials are attractive because the confirmatory analysis includes phase II data from stage 1, the inference methods used for trials that compare a single experimental treatment to a control and do not have an interim analysis are no longer appropriate. Several methods for analysing phase II/III clinical trials have been developed. These methods are recent and so there is little literature on extensive comparisons of their characteristics. In this paper we review and compare the various methods available for constructing confidence intervals after phase II/III clinical trials.     

Analysis of yield and oil from a series of canola breeding trials. Part I. Fitting factor analytic mixed models with pedigree information

In this paper multiplicative mixed models have been used for the analysis of multi-environment trial (MET) data for canola oil and grain yield. Information on pedigrees has been included to allow for the modelling of additive and nonadditive genetic effects. The MET data set included a total of 19 trials (synonymous with sites or environments), which were sown across southern Australia in 2007 and 2008. Each trial was designed as a p-rep design using DiGGeR with the default prespecified spatial model. Lines in their first year of testing were unreplicated, whereas there were two or three replications of advanced lines or varieties. Pedigree information on a total of 578 entries was available, and there were 69 entries that had unknown pedigrees. The degree of inbreeding varied from 0 (55 entries) to nearly fully inbred (337 entries). Subsamples of 2 g harvested grain were taken from each plot for determination of seed oil percentage by near infrared reflectance spectroscopy. The MET analysis for both yield and oil modelled genetic effects in different trials using factor analytic models and the residual plot effects for each trial were modelled using spatial techniques. Models in which pedigree information was included provided significantly better fits to both yield and oil data.     

Optimal design of experiments with anticipated pattern of missing observations

We propose a general method of designing an experiment when there are potentially failing trials. We use polynomial models and the Michaelis-Menten model as examples and construct different types of optimal designs under a broad class of response probability functions. We show that the usual optimal designs, that assume all observations are available at the end of the experiment, can be quite inefficient if the anticipated missingness pattern is not accounted for at the design stage. We also investigate robustness properties of the proposed designs to specification of their nominal values and the response probability functions.     

A comparison of experimental designs for selection in breeding trials with nested treatment structure

Plant breeders frequently evaluate large numbers of entries in field trials for selection. Generally, the tested entries are related by pedigree. The simplest case is a nested treatment structure, where entries fall into groups or families such that entries within groups are more closely related than between groups. We found that some plant breeders prefer to plant close relatives next to each other in the field. This contrasts with common experimental designs such as the α-design, where entries are fully randomized. A third design option is to randomize in such a way that entries of the same group are separated as much as possible. The present paper compares these design options by simulation. Another important consideration is the type of model used for analysis. Most of the common experimental designs were optimized assuming that the model used for analysis has fixed treatment effects. With many entries that are related by pedigree, analysis based on a model with random treatment effects becomes a competitive alternative. In simulations, we therefore study the properties of best linear unbiased predictions (BLUP) of genetic effects based on a nested treatment structure under these design options for a range of genetic parameters. It is concluded that BLUP provides efficient estimates of genetic effects and that resolvable incomplete block designs such as the α-design with restricted or unrestricted randomization can be recommended.     

Robustness and Efficiency of D‐optimal Experimental Designs in a Growth Problem

To assess tree growth, for example in diameter, a forester typically measures the trees at regular time points. We call such designs equidistant. In this paper we look at the robustness and efficiency of several experimental designs, using the D‐optimality criterion, in a case study of diameter growth in cork oaks. We compare D‐optimal designs (unrestricted and replication‐free) with equidistant designs. We further compare designs in different experimental regions. Results indicate that the experimental region should be adequate to the problem, and that D‐optimal designs are substantially more efficient than equidistant designs, even under parameter mis‐specification.     

Inter-plot competition in yield trials of potatoes (Solanum tuberosum L.) with single-drill plots

An experiment was carried out to assess the significance of inter-plot competition in a yield trial of potato cultivars. Seventeen cultivars were deliberately chosen and assessed for yield in single-drill and four-drill plots. Inter-plot competition for fresh-weight yield was a significant factor in the single-drill plots. It was modelled using a common competition coefficient with a covariate based on neighbour fresh-weight yields. In contrast, there was no statistically significant inter-plot competition for specific gravity. After adjustment for inter-plot competition, varietal ranking in estimated monoculture yield differed little from that based on unadjusted means. However, there was a reduction in the range of yield estimates, and a closer agreement with the observed pure-stand yields from the inner two drills of the four-drill plots. The adjustment for monoculture performance was most pronounced for the higher and lower yielding varieties, as expected from the assumption that the performance of high yielding varieties was enhanced in a competitive environment at the expense of low yielding ones. A general and flexible method of estimating competition coefficients in variety trials, together with a suitable algorithm, was developed and is explained in an appendix. It was used to check for inter-plot competition in a number of potato trials with single-drill plots and a large number of entries. Competition was found in some trials but not in others. Thus, where potato tubers are grown in single-drill plots for assessment of fresh-weight yield, adjustment should be made for inter-plot competition when evidence of inter-drill competition is found.     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Internal pilots for observational studies

Study planning often involves selecting an appropriate sample size. Power calculations require specifying an effect size and estimating “nuisance” parameters, e.g. the overall incidence of the outcome. For observational studies, an additional source of randomness must be estimated: the rate of the exposure. A poor estimate of any of these parameters will produce an erroneous sample size. Internal pilot (IP) designs reduce the risk of this error ‐ leading to better resource utilization ‐ by using revised estimates of the nuisance parameters at an interim stage to adjust the final sample size. In the clinical trials setting, where allocation to treatment groups is pre‐determined, IP designs have been shown to achieve the targeted power without introducing substantial inflation of the type I error rate. It has not been demonstrated whether the same general conclusions hold in observational studies, where exposure‐group membership cannot be controlled by the investigator. We extend the IP to observational settings. We demonstrate through simulations that implementing an IP, in which prevalence of the exposure can be re‐estimated at an interim stage, helps ensure optimal power for observational research with little inflation of the type I error associated with the final data analysis.     

Non-Iterative Least Squares Estimation of Missing Values in Replicated Latin Square Designs

In this paper an attempt has been made to estimate several missing values in replicated latin square designs. The explicit computable expressions for the non-iterative least squares estimates of the missing values are presented for particular patterns of missing values.     

Improving the Flexibility and Efficiency of Phase II Designs for Oncology Trials

Summary Phase II trials in oncology are usually conducted as single-arm two-stage designs with binary endpoints. Currently available adaptive design methods are tailored to comparative studies with continuous test statistics. Direct transfer of these methods to discrete test statistics results in conservative procedures and, therefore, in a loss in power. We propose a method based on the conditional error function principle that directly accounts for the discreteness of the outcome. It is shown how application of the method can be used to construct new phase II designs that are more efficient as compared to currently applied designs and that allow flexible mid-course design modifications. The proposed method is illustrated with a variety of frequently used phase II designs.     

Drop-the-losers design: normal case

Drop-the-losers designs are statistical designs which have two stages of a trial separated by a data based decision. In the first stage k experimental treatments and a control are administered. During a transition period, the empirically best experimental treatment is selected for continuation into the second phase, along with the control. At the study's end, inference focuses on the comparison of the selected treatment with the control using both stages' data. Traditional methods used to make inferences based on both stages' data can yield tests with higher than advertised levels of significance and confidence intervals with lower than advertised confidence. For normally distributed data, methods are provided to correct these deficiencies, providing confidence intervals with accurate levels of confidence. Drop-the-losers designs are particularly applicable to biopharmaceutical clinical trials where they can allow Phase II and Phase III clinical trials to be conducted under a single protocol with the use of all available data.     

A New Class of Matrices with Application in Experimental Designs

A very general class of balanced matrices is defined in this paper. These matrices are generalisations of the balanced orthogonal designs of Rao (1970) and generalised balanced matrices of Dey and Midha (1976). Some methods of construction of these balanced matrices are discussed. An application of these matrices in experimental designs is also given.     

Random and fixed effects in plant genetics

Summary A general model for any type of genetic entry is developed which takes into account both the factorial model of gene effects and the ancestral sources, whether inbred lines or outbred varieties, of the genes.Utilizing the model, various genetic designs of fixed entries are explored for the estimation of genetic effects and the testing of genetic hypotheses. These designs consisted of generation means — parents, crosses, various types of backcrosses, and so on — stemming from one or more pairs of parents, and of hybrid combinations from factorial mating designs. Limitations, from the standpoint of genetic effects that can be estimated and genetic hypotheses that can be tested, are developed in considerable detail.When entries from the factorial mating designs are considered to be random, attention is focused on the estimation of genetic variances, rather than effects, and on the concomitant changes in the tests of genetic hypotheses. While there is considerable improvement over fixed entries in the number of types of genetic variances that can be estimated, and of genetic hypotheses that can be tested, they are still very limited in contrast to what would be most desirable.Paper No. 6018 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, North Carolina. This investigation was supported in part by NIH Research Grant No. GM 11546 from the National Institute of General Medical Sciences.Preliminary: This paper was presented at the 7th International Biometric Conference. Since then it has come to various people's attention and I have been encouraged to give it a wider distribution. Except for editing, the paper is essentially as originally written.     

A nested-intensity design for surveying plant diversity

Managers of natural landscapes need cost-efficient, accurate, and precise systems to inventory plant diversity. We investigated a nested-intensity sampling design to assess local and landscape-scale heterogeneity of plant species richness in aspen stands in southern Colorado, USA. The nested-intensity design used three vegetation sampling techniques: the Modified-Whittaker, a 1000-m² multiple-scale plot (n = 8); a 100-m² multiple-scale Intensive plot (n = 15); and a 100-m² single-scale Extensive plot (n = 28). The large Modified-Whittaker plot (1000 m²) recorded greater species richness per plot than the other two sampling techniques (P < 0.001), estimated cover of a greater number of species in 1-m² subplots (P < 0.018), and captured 32 species missed by the smaller, more numerous 100-m² plots of the other designs. The Intensive plots extended the environmental gradient sampled, capturing 17 species missed by the other techniques, and improved species–area calculations. The greater number of Extensive plots further expanded the gradient sampled, and captured 18 additional species. The multi-scale Modified-Whittaker and Intensive designs allowed quantification of the slopes of species–area curves in the single-scale Extensive plots. Multiple linear regressions were able to predict the slope of species–area curves (adj R ² = 0.64, P < 0.001) at each Extensive plot, allowing comparison of species richness at each sample location. Comparison of species–accumulation curves generated with each technique suggested that small, single-scale plot techniques might be very misleading because they underestimate species richness by missing locally rare species at every site. A combination of large and small multi-scale and single-scale plots greatly improves our understanding of native and exotic plant diversity patterns.     

A genome-wide search for wild-species alleles that increase horticultural yield of processing tomatoes

To identify QTLs associated with horticultural yield it is necessary to conduct replicated plot trials of the tested genotypes. The first step in the utilization of an introgression-line (IL) population of Lycopersicon pennellii in a processing-tomato variety (M82) for mapping such QTLs was to screen 51 ILs in a non-replicated plot trial. The results of this survey were compared to those obtained in a replicated trial of the same genotypes grown as single plants at wide spacing. Fruit characteristics were similar between the two stands, but yield was generally different. Eight lines that outperformed the control in the plot survey were subjected to detailed analysis in the following year. The effects of these introgressions, measured on single plants, were reproducible relative to the previous year's results. In a replicated plot trial of these ILs and their hybrids involving two genetic backgrounds, the product of yield and total soluble solids (horticultural yield) in seven of the eight hybrids was 7–13% higher than that of their nearly isogenic controls. The results revealed a consistent trend in the interaction between introgression effects and genetic background. Combining the two introgressions with the largest contribution to horticultural yield in plots resulted in a 20% increase relative to the control in the third year. This research highlights the potential of wild germ plasm for yield improvement and the ability of nearly isogenic populations to achieve this goal.