Keywords

1 Introduction

Breeding programmes deal with large number of activities , including evaluation of hundreds or thousands of genotypes and selection of the best individuals to comprise the next generation of individuals or to be released as new cultivars. Genetic testing is an expensive task that constitutes the largest activity performed in any breeding programme. Phenotyping of genotypes is particularly demanding on small breeding programmes, such is the case of most tropical crops, and for this reason, all activities that aim to maximise (or optimise) the use and quality of the information generated from genetic tests are critical. The basis for this evaluation and selection originates from data and information generated from field and greenhouse experiments, so these need to be carefully planned and analysed.

Genetic tests can be optimised through three different ways: (1) design of experiments, (2) implementation and measurement of trials and (3) statistical analysis. Appropriate selection of the experimental design, their implementation and then their statistical analyses can yield considerable benefits resulting in greater precision of estimates of genetic parameters leading to increased genetic gains from successful selections, and better operational decisions that depend on information obtained from genetic tests , such as heritability , genotype-by-environment interactions , trait-to-trait correlations , etc. Such optimisation can be classified into ‘a priori’ and ‘a posteriori’: the former related to actions that are implemented before the experiment is established (i.e. at the design stage), while ‘a posteriori’ are those actions that are critical to implement once the experiment is established and often relate to tools to be used in for statistical analysis.

There is a plethora of classical and modern literature on the ideal characteristics of a wide array of experimental designs. However, no single design will suit all experimental objectives and environmental conditions found in field tests around the world. Hence, the choice of the ‘best’ design must be made carefully. Statistical and computational tools can be used to generate experimental layouts with great efficiency, where, as always, the principles of replication, randomisation and blocking are critical (for more details about these principles, see Welham et al. (2014)).

Randomised complete block (RCB) designs are the most frequently used in plant breeding. Blocking is important to minimise variability, and this design is effective when within-replicate (or block) variability is relatively small. Where there is large site heterogeneity, or when there are many genotypes to be evaluated, other experimental designs can be more efficient. For example, incomplete block (IB) designs allow for a better control of site heterogeneity by specifying smaller compartments that include a (planned) subset of the genotypes to be tested. IB designs are often generated by implementing an alpha design, a particular class of IB design where the number of genotypes (or entries) is a multiple of block size (John and Williams 1995). Another efficient alternative is the use of row-column (RC) designs that consider both row and columns within a replicate as complete or incomplete blocks. Both of these designs provide greater control of site heterogeneity and can be generated using an array of public and commercial software. For more details about the use and analysis of these designs in the context of plant breeding, we recommend Williams et al. (2002). Other efficient design options include the use of restricted randomisation such as latinisation , nested structures and spatial designs (Whitaker et al. 2002), which can increase the efficiency of the experiments.

For early generation variety trials where large numbers of genotypes are often tested, there may be insufficient planting material to replicate all genotypes. One of the most widely used designs is the use of grid plots where checks (or control genotypes) are repeated several times arranged in a block or incomplete block, depending on the experimental design implemented. Test genotypes are unreplicated and allocated at random to the remaining plots. Examples of this are the various augmented block designs developed by Federer (1956) and Federer and Raghavarao (1975). In an alternative approach, Cullis et al. (2006) proposed the use of partially replicated (p-rep) designs in which a subset of the test genotypes are replicated two or more times, and these are arranged in a resolvable spatial design. Then, the unreplicated test genotypes are randomly allocated to the remaining plots. For a fixed amount of resources, Cullis et al. (2006) found that p-rep designs result in a greater genetic gain than augmented designs.

The second optimisation of genetic testing focuses on the implementation of and measurement within a field design. Here, it is important to observe carefully all operational aspects of field testing, including documentation, labelling, site preparation and crop maintenance. One aspect that is critical here refers to preparing the site in such a way that environmental heterogeneity is minimised. This applies to all soil selection and preparation before planting and its management while the trial is active. In addition, to ensure the best quality of the phenotypic data originating from these trials, adequate definitions of response variables and clarity and consistency on measurement protocols are critical. Any actions that decrease experimental noise will increase the precision of the estimation of genetic parameters and, therefore, increase heritability estimates.

It is also important to collect the most accurate and reliable data that will be used to make decisions on which genotypes are rejected, advanced or ultimately commercially released. For example, Australian sugarcane breeders evaluate genotypes on the basis of their relative economic genetic value for traits of commercial importance – how much value would a genotype add to industry profitability if grown commercially (Wei et al. 2006).

Having collected the data, the genetic tests can be optimised through statistical analysis. This has been an area that has had several important advances over the last few decades. Of special interest for plant breeding is the use of linear mixed models (LMM) that combine estimation procedures such as residual maximum likelihood (REML) to estimate variance components and to predict random effects (or best linear unbiased predictions, BLUP ). LMMs are an extension to the traditional linear models (LM) that allow for more flexible assumptions such as correlations among experimental units (e.g. temporal correlation) and among effects (e.g. by considering the numerator relationship matrix of genetic effects or BLUP ) and heterogeneity of variances (e.g. different error variances for each block or site).

Modern analysis of complex and unbalanced data to obtain parameters, such as site-to-site and trait-to-trait genetic correlations, is possible by fitting LMMs that estimate variance components. Spatial analysis (Gilmour et al. 1997) of field experiments is a useful tool that incorporates the co-ordinates of the experimental units (plots or plants) into the LMM to account for physical proximity by modelling the error structure (i.e. correlations among observations), something that can be extended easily to also model competition among neighbouring plants. This is also particularly important with augmented and p-rep designs where spatial analysis allows for extracting better genetic information from the unreplicated test genotypes.

The greatest benefit of LMMs is that it is possible to combine data from many sources, with different levels of unbalance, into a complex model that will maximise the use of this information to estimate genetic parameters. For example, multi-environment trials (MET) use information from several trials, where not all genotypes are present in all sites, and for each site, there might be different numbers of replicates and precision and therefore heritabilities . These trials are evaluated together into a single LMM to estimate overall breeding values and genetic correlations among sites.

Many statistical tools can be implemented ‘a posteriori’ given a field dataset. One of these is post hoc blocking, where, for a given experimental layout (say a RCB design), a new blocking structure is superimposed on top of the original, and a new linear model is fitted as if the superimposed blocking structure belonged to the original design (Gezan et al. 2006). This tool increases the precision of estimates of heritability and of the predicted genetic values at little extra cost by only marginally increasing the complexity of the analysis.

In the next sections, some of these modern statistical approaches, such as interplot competition and spatial analysis, will be first defined and then illustrated in more detail.

2 Accounting for Interplot Competition

In early-stage selection trials , most plant improvement programmes face the challenge of finding a few incrementally superior individuals from among a large number of lines produced by cross-pollination (Stringer et al. 2011). Due to limitations on planting material and space for field testing, genotypes are often planted in trials in small, partly replicated, single-row plots. Such trials are subject to variation arising from spatial variability and interplot competition, which makes the identification of elite genotypes problematic. Unless accounted for, spatial variability and interplot competition may seriously affect the estimates of genetic merit and, hence, reduce genetic progress.

Interplot competition (also known as interference ) arises when a treatment or response on one experimental or measurement unit may affect the response on neighbouring units (Martin and Eccleston 2004) and is caused by both genetic and environmental sources (Magnussen 1989). It is difficult to quantify and there is no universal method to account for the competitive interactions among genotypes. As resource limitations generally preclude the use of multi-row plots to account for interplot competition, statistical approaches have been developed to adjust for competition in the design and analysis of field trials.

One alternative is to use appropriate experimental layouts, such as the neighbour-balanced (NB) designs suggested by Williams (1952), Street and Street (1987) and Azaïs et al. (1993). However, these designs are not practical where large numbers of genotypes are to be screened, due to the number of replicates required to achieve balance between neighbouring genotypes (Kempton 1982).

In regard to statistical analyses, Besag and Kempton (1986) presented two approaches to estimate interplot competition . Building on earlier work by Kempton (1982), they developed the phenotypic interference model , which is a simultaneous autoregressive approach where competition is assumed to be directly related to yields of neighbouring plots. This has been applied successfully to a wide range of crops including sugar beet (Kempton 1982; Durban et al. 2001), potatoes (Connolley et al. 1993), swedes (Bradshaw 1989) and trees (Resende et al. 2005).

The second model developed by Besag and Kempton (1986) is the treatment or genotypic interference model and was originally proposed by Pearce (1957). In this model, competition effects are associated with genotype differences in characteristics such as plant height, tillering ability, date to maturity and canopy size (Kempton and Lockwood 1984; Talbot et al. 1995). Here, competition effects are associated with the average genotypic value of the nearest neighbouring genotypes rather than the phenotypic response (Stringer et al. 2011). In addition, each treatment is assumed to have a direct effect and a neighbour effect on adjacent plots.

3 Incorporating Spatial Variation

In early-stage field trials, which are typically large, growing conditions may be quite variable across the trial area, leading to the phenomenon known as spatial variability. One of the oldest techniques available to minimise the effect of this variability is the method of check (or control) plots (Wiancko 1914) in which replicated plots are distributed over the trial site as checks and are used as a benchmark to assess the yields of test plots. It is assumed that the checks and test varieties show the same general pattern of response to soil fertility over a trial as the test varieties. If this is not true, then the method of check plots will actually increase the error of assessment (Kempton 1984a; Besag and Kempton 1986). An alternative approach, which may be more useful for dealing with small-scale variation , is spatial or nearest neighbour (NN) analysis where a plot parameter is adjusted by using information from immediate neighbours. Although Papadakis (1937) proposed the earliest NN method , it lacked efficiency.

Spatial methods were largely neglected by statisticians until Wilkinson et al. (1983) developed the smooth trend plus independent error model on which most spatial models have since been based (Stringer et al. 2012). Since then, there have been many alternative approaches, including the one-dimensional models of Gleeson and Cullis (1987) and the two-dimensional approaches of Cullis and Gleeson (1991). Spatial analysis has been successfully applied to early generation trials by Cullis et al. (1992), who found that the response to selection for the spatial method was greater than for check plot method proposed by Wiancko (1914). In all of these models, the covariance structure of the plot errors was modelled as a single component. These techniques were later extended by Gilmour et al. (1997), who demonstrated that modelling plot errors alone as a single process may not be appropriate in most cases, requiring the spatial variation to be partitioned into three components. This approach is currently used to analyse over 1000 cereal variety trials in Australia annually (Stringer et al. 2012) and has resulted in increased accuracy and precision in the estimates of genotype effects in a wide range of crops (Apiolaza et al. 2000; Dutkowski et al. 2002; Gilmour et al. 1997; Grondona et al. 1996; Qiao et al. 2000; Sarker et al. 2001; Silva et al. 2001; Singh et al. 2003).

The methods developed by Gilmour et al. (1997), and later refined by Stefanova et al. (2009), partition spatial variation into three additive components. Atkin (2012) defines these as :

  • Local trend , reflecting small smooth changes due to parameters such as fertility, soil moisture and light

  • Nonstationary global trend which is usually aligned with the columns and rows of a field trial and associated with large-scale changes across the trial, for example, large-scale moisture or fertility gradients

  • Extraneous variation , which usually arises from management practices or experimental procedures that have recurrent patterns, such as spraying operations or serpentine harvesting (harvesting columns of rows in alternating directions)

Previous approaches at removing global trend involved first or second differencing of the data (Gleeson and Cullis 1987), but this often overcomplicated the model. Gilmour et al. (1997) recommend directly fitting nonstationary global trends through the use of polynomial or spline functions (Verbyla et al. 1999) to the row and column co-ordinates. The modelling approach developed by Gilmour et al. (1997) is a sequential approach and commences by including design factors such as replicates that reflect the trial design. Next, modelling local trend is undertaken using a first-order separable autoregressive process in the row and column directions. During the modelling procedure, diagnostic tools, such as the sample variogram and trellis plots , play a large role in determining what effects should be included in a model. This is always followed by formal assessment by using the Wald test for fixed effects and REML likelihood ratio test for random effects.

4 Modelling Competition and Spatial Variation

As indicated earlier, interplot competition arises when a treatment or response on one unit affects the response on neighbouring units (Martin and Eccleston 2004). For example, in sugarcane, estimates of cane yield are affected more by interplot competition than are estimates of sugar content when genotypes are evaluated in single-row plots (Fig. 1.1), because plants in adjoining plots compete for resources such as water, fertiliser and sunlight (Jackson and McRae 2001; McRae and Jackson 1998; Skinner 1961; Stringer and Cullis 2002). This often results in a negative correlation between neighbouring plots, biasing estimates of cane yield.

Fig. 1.1
figure 1

A typical sugarcane family trial layout in a rectangular array of plots

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Although there are many approaches in the literature that individually model spatial variability or interplot competition , there are only a few studies that jointly account for both sources of bias. Durbán Reguera (1998) and Durban et al. (2001) presented one such approach. They used cubic smoothing splines to model spatial global trend together with the phenotypic interference model for competition (Stringer et al. 2011). Genotype effects were considered fixed and adjusted profile likelihood was used for parameter estimation (McCullagh and Tibshirani 1990). This model was limited by not considering genotypes to be random nor incorporating a spatial process to model local trend. In a small simulation study based on the Rothamsted downy mildew data, Durbán Reguera (1998) found that the profile likelihood gave biased estimates of the variance components and in some cases the competition parameter was also biased. However, when using McCullagh and Tibshirini’s adjustment to the profile likelihood, bias in the parameters of interest was small. Matassa (2003) developed a method combining both models from Besag and Kempton (1986) for interplot competition together with the methods of Gilmour et al. (1997) for spatial variability. Matassa’s approach was similar to Durbán Reguera (1998) and Durban et al. (2001) in that genotype effects were fixed. However, Matassa (2003) used marginal likelihood and profile likelihood for parameter estimation. On comparing the estimation procedures in a simulation study, Matassa (2003) found that the preferred method depended on what terms were included in the design matrix and also on the sign of the trend parameter.

Stringer et al. (2011) developed an alternative approach to jointly model spatial variability and interplot competition. They partitioned spatial variability into global trend and extraneous variation (Gilmour et al. 1997) and allowed for both genotypic (Besag and Kempton 1986) and residual level competition. Genotype effects were considered to be random, as recommended by Smith et al. (2001), and REML was used for parameter estimation. Stringer et al. (2011) presented two simultaneous autoregressive processes to model competition at the residual level. They recommended an equal-roots second-order autoregressive model for trials where competition is dominant and an equal-roots third-order autoregressive model where both competition and spatial variability exist.

In sorghum breeding trials in Australia, parental lines are evaluated in single-row plots where both interplot competition and spatial variability are present (Hunt et al. 2013). Hunt et al. (2013) extended the methods of Stringer et al. (2011) used for sugarcane clonal trials, by incorporating pedigree information into a LMM. This allowed Hunt et al. (2013) to partition total genetic effects into additive and nonadditive components for parent evaluation in the presence of both competition and spatial effects. The methods developed by Stringer et al. (2011) and Hunt et al. (2013) are routinely applied to sugarcane clonal and full-sib family (produced from biparental cross-pollination) trials from Queensland, Australia. In such trials, clones and families are evaluated in single-row plots and large spatial trends and interplot competition are regularly present. The presence of spatial variation and interplot competition effects in sugarcane family trials will probably have a similar effect on estimating additive genetic effects of sugarcane parents as with estimating total genetic or clonal effects and lead to biased estimates of breeding values (BV) for cane yield (in the presence of both spatial variation and interplot competition) and sugar content (in the presence of spatial variation only). In turn, this will bias the ranking of parents and so impact the outcomes of parental selection.

The two statistical models explained below (a basic RCB design model with and without modelling spatial variation) can be used to estimate additive genetic effects of sugarcane parents from family trials for parent selection (Atkin 2012) by applying some of the techniques used by Stringer et al. (2011) and Hunt et al. (2013).

4.1 Base Model: RCB Without Modelling Spatial Variation

Consider an experiment consisting of p trials that contains a total of m genotypes (families). Each trial is laid out in a rectangular array of r rows and c columns (n = r × c) (Fig. 1.1). Where the data are ordered as rows within columns, the mixed linear model for y (n×1) combined across trials is

$$ \mathbf{y}=\mathbf{Xb}+{\mathbf{Z}}_g\mathbf{g}+{\mathbf{Z}}_u\mathbf{u}+\mathbf{e} $$

where b (b×1) is a vector of fixed effects with the associated design matrix X (n×b); g (mp×1) contains the random genotype and genotype by environment effects of m entities in each of p trials with indicator matrix Z g (n×mp); u (d×1) contains the random replicate effects with associated design matrix Z u (n×d); and e (n×1) is a vector of plot error effects combined across trials. Vector b contains only an overall mean effect for each trial or more complex design structures.

Here, vector g is the random genotypic effect of unique parents (for each trial), where a sugarcane parent can be used as either a male or a female, or both. Using a biparental model (or a reduced animal model) (Mrode 2005; Quaas and Pollak 1980), vector g is then further partitioned into additive and nonadditive genetic effects as per Costa e Silva et al. (2004). The prediction of BVs described here is also applicable to the next model described below: the spatial model.

4.2 Spatial Model: RCB Plus Modelling Spatial Variation

The above model can be extended to include the partitioning of spatial variability, where vector b contains an overall mean for each trial, as well as trial-specific modelling due to global trend (Stefanova et al. 2009). Global trend is accommodated in the model by using design factors such as linear row and/or linear column effects or by fitting spline functions to the row and column co-ordinates (Verbyla et al. 1999). Vector u includes effects associated with the modelling of extraneous variation due to experimental procedures and blocking design factors specific to each trial or sub-trial (in cases where a trial comprised of two or more sub-trials). For each trial, vector e is further partitioned into a vector that represents a spatially dependent process and a vector of residual errors (Gilmour et al. 1997).

Local spatial trend is modelled using a first-order separable autoregressive (AR) process in the row (AR(1)) and column directions (AR(1)), as recommended by Cullis et al.(1998), Gilmour et al. (1997) and Grondona et al. (1996). After fitting the local trend, diagnostic tools such as the sample variogram and trellis plots (Gilmour et al. 1997) can be used to determine if global spatial trend and/or extraneous variation needed to be included in the model. An example of a theoretical variogram for an AR(1) × AR(1) process is given in Fig. 1.2. This variogram has a smooth appearance and an exponential increase in the row and column directions reaching a plateau giving it a ‘tabletop’ appearance. Departure from this smooth appearance indicates the presence of extraneous variation; similarly, if the sample variogram fails to reach a plateau in the row and/or column direction, this indicates the presence of a global trend that needs to be incorporated into the LMM (Stringer and Cullis 2002). An example of the presence of a global trend is given in Fig. 1.3 for which a linear row and column effect would then be fitted. The inclusion of these fixed effects is based on visual inspection of the sample variogram followed by a formal assessment using the Wald test (Agresti 1990). An example of extraneous variation is given in Fig. 1.4 for which a random row effect would be fitted. The inclusion of random effects is also based on visual inspection for the sample variogram , followed by the use of the likelihood ratio test to ascertain if the change in REML log-likelihood for random effects is significant.

Fig. 1.2
figure 2

Example of a theoretical variogram for an AR(1) × AR(1) process in the absence of both global and extraneous trend (From Stringer and Cullis (2002) – used with permission)

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Fig. 1.3
figure 3

Example of a sample variogram for an AR(1) × AR(1) process indicating the presence of a global trend in the row and column direction (From Stringer and Cullis (2002) – used with permission)

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Fig. 1.4
figure 4

Example of a sample variogram for an AR(1) × AR(1) process indicating the presence of extraneous variation (From Stringer and Cullis (2002) – used with permission)

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These types of analyses are routinely performed in the Australian sugarcane breeding programme using the statistical package ASReml (Gilmour et al. 2006) and could be extended to other tropical crops where competition and spatial effects are often experienced in field trials.

5 Further Approaches That Incorporate Genotype-by-Environment Interactions

Mixed model analyses of data from multi-environment trials (METs) can be used to partition the total variation into sources such as trial, genotype and genotype-by-environment (G×E) interactions . Although this provides an estimate of the magnitude of G×E, it does not provide any insight into the nature of G×E effects (Kempton 1984b). Multiplicative methods are particularly useful at describing G×E interactions and have been widely used in a fixed-effects setting. The earliest of these was the regression on mean model, where either the phenotypic values or interaction is regressed on environmental indices. This was first suggested by Yates and Cochran (1938) and enhanced by Finlay and Wilkinson (1963); however, this approach assumes genotypes respond linearly to environmental change (Flores et al. 1998). Freeman (1973) suggested the use of multiplicative methods in genetic analyses, and of these, the additive main effects and multiplicative interaction (AMMI) has been used very widely (Gauch and Zobel 1988). AMMI combines the additive analysis of variance for main effects with the multiplicative principal component analysis for the interaction. However, AMMI requires data to be balanced and, hence, it can be too restrictive for the analysis of MET data for most crops (Smith et al. 2001).

A key issue often neglected in G×E studies is the need to model plot-level residuals (Smith et al. 2001). Individual trials routinely exhibit spatial variability (Atkin 2012) as a correlation in residuals among neighbouring plots, and it is common for residual variances to differ among trials. Estimates of genotype main effects and G×E interactions may be biased if this is not accounted for (Cullis et al. 1998). An approach which overcomes these limitations was developed by Smith et al. (2001), in which spatial variability within a trial is partitioned into local and global trends and extraneous variation using the methods of Gilmour et al. (1997) (described previously), and the heterogeneity of residual variance among trials is accounted for. This is called the factor analytic model (FA) and implies that genetic effects are correlated between trials; hence, it allows the genetic variation at each environment to differ and allows for different covariances between pairs of environments (Smith et al. 2001). This flexibility requires a large number of variance components to be estimated. However, the particular FA model proposed by Smith et al. (2001) uses the algorithms of Thompson et al. (2003) by providing a parsimonious fit for parameter estimation. Therefore, the multiplicative mixed model with random genotype effects, i.e. the FA model (as used by Chapman et al. 2004), is currently considered the most appropriate approach for the analysis of G×E interactions in breeding programmes for many crops including sugarcane.

6 Final Remarks

The array of statistical tools available in quantitative genetics for the analysis of messy and complex data that originates from breeding trials is diverse, noisy and continuously evolving. The emergence of powerful statistical software that can deal with this data has allowed breeders to extract more information from each experiment, including aspects such as competition and spatial correlations. In addition, the availability of large quantities of molecular data to be incorporated into the linear mixed models , for example, by calculating observed relationships among genotypes based on molecular markers (VanRaden 2008), has widened the options to improve and optimise the design and analysis of genetic experiments. Here we have presented some tools, but these modern tools will constitute, in the near future, a daily part of all statistical analysis performed by many breeding programmes across the world.