Agricultural Statistics & Data Science · Design and analysis of experiments · also Pooled Analysis, Stability Analysis, MET analysis
Combining trials across sites and seasons: testing homogeneity of error variances, pooled analysis, genotype by environment interaction, the classical stability models and AMMI and GGE biplots that decide variety release in national coordinated trials.
In the exams: AICE-JRF Ag Stat Units 3-4 'groups of experiments', 'stability parameters' (heavy); AICE-JRF Agronomy Unit 6 'pooled analysis' (heavy); ASRB NET Ag Stat (present)
No single trial can tell whether a variety or practice will work across a region. This subject covers the analysis of experiments repeated over locations and years: groups of experiments and the homogeneity of error variances, the pooled analysis and its error terms, genotype x environment interaction and its partitioning, the Finlay-Wilkinson and Eberhart-Russell stability models, other stability parameters (Wricke's ecovalence, Shukla's stability variance, Perkins-Jinks), AMMI models, GGE biplots and mega-environments, mixed-model analysis of multi-environment trials with factor-analytic models, and the analysis of perennial and repeated-season experiments.
It builds on the basic and factorial design subjects (the individual trials are RCBDs, split plots or incomplete block designs), on the linear models subject (expected mean squares and fixed versus random effects) and especially on the mixed-models subject, whose REML, BLUP, heterogeneous variances and covariance structures are used throughout chapter 8 and chapter 9. The genetic interpretation of stability and its use in breeding are developed further in the biometrical techniques subject.
In examinations, ICAR AICE-JRF/SRF Agricultural Statistics names groups of experiments in Unit 4 and genotype x environment interaction and stability parameters in Unit 3, and the statistics unit of AICE-JRF Agronomy names pooled analysis. Typical questions test the combined ANOVA and its error terms, Bartlett's test, the interpretation of the regression coefficient and deviation variance, ecovalence, AMMI degrees of freedom and biplot reading, GGE which-won-where and mega-environments, and heritability over environments.
A variety or practice recommended for a region must perform well across the locations and seasons of that region, not just at one research station in one year. Agricultural experiments are therefore repeated as groups of experiments: the same set of treatments tested at several locations, in several years, or both. National coordinated trials, international nurseries and regional variety trials are all groups of experiments. Analysing them together answers questions no single trial can: whether treatment differences are consistent, how large genotype by environment interaction is, which entries are broadly or specifically adapted, and how much replication over environments a recommendation needs.
Before pooling, the error variances of the individual experiments are compared, because the combined analysis uses a pooled error and assumes it is common. With s experiments, error mean squares s_i squared on f_i degrees of freedom, Bartlett's test compares them: the statistic, a function of the logarithms of the error mean squares and their pooled value, is approximately chi-square with s - 1 degrees of freedom. When all experiments have the same error degrees of freedom, a quick check is the ratio of the largest to the smallest error mean square; many texts advise pooling directly when this ratio is small (a commonly quoted rule of thumb is below about 3 or 4) and caution otherwise. Bartlett's test is sensitive to non-normality, and Levene-type tests are alternatives. If error variances are heterogeneous, the options are a weighted analysis (each experiment's data or treatment means weighted by the reciprocal of its error variance), grouping experiments with similar variances, a transformation that stabilises variances, or a mixed model that allows a separate residual variance for each environment, which is now the usual choice.
Suppose six locations each ran an RCBD with the same error degrees of freedom, and the error mean squares were 0.20, 0.25, 0.22, 0.30, 0.18 and 0.27 (t/ha squared). The ratio of the largest to the smallest is 0.30/0.18 = 1.67, small enough for most texts to accept pooling; Bartlett's test would confirm homogeneity. If one location had an error mean square of 0.90, the ratio would be 5, and pooling would give that noisy site the same weight as the precise ones. The analyst would then examine that trial for problems (lost plots, damage, poor management), and either analyse with site-specific error variances or weight sites by the reciprocal of their error variances.
The structure of a trial series should be planned: the target population of environments (the region, management levels and seasons for which recommendations are wanted), the number of sites and years, the number of replicates per site, and the common set of entries and checks. Sites chosen for convenience near research stations may not represent farmers' conditions; deliberately including stress-prone and on-farm sites improves the relevance of conclusions. A balanced core set of entries tested everywhere simplifies analysis, but modern mixed models tolerate unbalanced series in which entries are tested for different numbers of years.
In a weighted combined analysis, each environment's treatment means are weighted by w_j = r / s_j squared, the reciprocal of the variance of a mean in that environment, so that precise trials count more. A variety's weighted mean over environments is the sum of w_j times its mean divided by the sum of w_j. With three sites whose error mean squares are 0.2, 0.2 and 0.8 and equal replication, the third site receives a quarter of the weight of each of the others. Weighting is the forerunner of the mixed-model approach in which each environment has its own residual variance.
For g genotypes tested in e environments (location-year combinations), each as an RCBD with r replicates, the combined ANOVA has: environments (e - 1 degrees of freedom); replicates within environments (e(r - 1)); genotypes (g - 1); genotype x environment interaction ((g - 1)(e - 1)); and pooled error (e(r - 1)(g - 1)), the sum of the individual errors. With locations and years as separate factors, the environment and interaction lines split into locations, years, locations x years, G x L, G x Y and G x L x Y. Which mean square tests which effect depends on whether environments are fixed or random. If environments are regarded as a random sample of the target region, which is the usual view for recommendation purposes, genotypes are tested against the G x E mean square, not against the pooled error: a genotype difference must be large relative to its inconsistency across environments to be trusted. The G x E interaction itself is tested against the pooled error. Testing genotypes against the pooled error answers only whether genotypes differed in these particular environments.
With genotypes fixed and environments random, E(MS genotypes) = sigma squared + r sigma_ge squared + re theta_g, where theta_g is the sum of squared genotype effects over (g - 1), and E(MS G x E) = sigma squared + r sigma_ge squared. Their ratio tests genotypes on (g - 1) and (g - 1)(e - 1) degrees of freedom. The variance of a genotype mean over the trial series is (sigma_ge squared + sigma squared / r) / e, which shows that adding environments reduces it more effectively than adding replicates when G x E is substantial; this is the statistical basis for testing over many sites with few replicates.
Take g = 8 varieties tested at e = 5 locations, each an RCBD with r = 3 replicates. The degrees of freedom are 4 for locations, 10 for replicates within locations, 7 for varieties, 28 for varieties x locations and 70 for pooled error, 119 in total. If the variety mean square is 2.4, the variety x location mean square 0.6 and the pooled error 0.2, then with locations random the variety F is 2.4/0.6 = 4.0 on 7 and 28 degrees of freedom, and the interaction F is 0.6/0.2 = 3.0 on 28 and 70 degrees of freedom; both would be significant at 5%. Testing varieties against the pooled error would give F = 12 and exaggerate the evidence for consistent differences. The variance component for G x E is (0.6 - 0.2)/3 = 0.133.
With locations and years crossed, G x L measures repeatable site-specific differences (useful for zoning recommendations), G x Y measures season-to-season inconsistency common to all sites, and G x L x Y measures unrepeatable interaction. If G x L is large and G x Y small, recommendations can be targeted to locations; if G x Y and G x L x Y dominate, interaction is unpredictable and the remedy is wide testing and selection for stability. In practice G x L x Y is often the largest interaction component, which is why multi-year testing is required before variety release.
If the environments are of interest in themselves (for example three specific irrigation commands or two named seasons that will recur), they may be treated as fixed, and genotype effects are then tested against the pooled error, with conclusions confined to those environments. In variety testing, however, the sites and years are nearly always a sample of the future conditions in which varieties will be grown, so the random-environment view and the G x E error term are appropriate. Being explicit about the target population of environments resolves most disputes about error terms.
With environments random, the standard error of the difference between two variety means over e environments is the square root of 2 MS(G x E) / (re) in the balanced case. In the worked example above, this is the square root of 2 x 0.6 / 15 = 0.28 t/ha, compared with 0.16 t/ha if the pooled error were wrongly used. The difference illustrates why the choice of error term matters for recommendations.
Genotype x environment (G x E) interaction occurs when genotypes respond differently to environments. Two kinds matter. Non-crossover (quantitative) interaction changes the size of genotype differences but not their ranking: one variety is always best, but by more in good environments. Crossover (qualitative) interaction changes rankings: variety A is best at some sites and variety B at others. Crossover interaction is what complicates recommendations and motivates the search for mega-environments, groups of sites within which rankings are consistent. G x E arises from differences among environments in rainfall, temperature, soil, pests and diseases, sowing date and management, and from genotypic differences in phenology, stress tolerance and disease resistance. Understanding its causes, by relating interaction patterns to environmental covariates and genotypic traits, is more useful than merely testing its significance.
The G x E sum of squares, with (g - 1)(e - 1) degrees of freedom, can be partitioned in several ways. Regression on an environmental index (chapter 4) extracts a heterogeneity-of-regressions component with g - 1 degrees of freedom and leaves deviations. Tukey's one-degree-of-freedom component and Mandel's row and column regressions are related multiplicative partitions. Principal component analysis of the interaction matrix, as in AMMI (chapter 6), extracts successive multiplicative terms. Genotype-wise contributions (Wricke's ecovalence, chapter 5) apportion the interaction among genotypes. Partitioning by environmental covariates (factorial regression) relates the interaction to measured variables such as rainfall or temperature.
Two varieties tested at a wet site and a dry site yield 5.0 and 4.0 t/ha (variety A and B) at the wet site and 2.0 and 2.6 t/ha at the dry site. The variety means over sites are 3.5 and 3.3, so A has the higher mean, but B wins at the dry site: a crossover interaction. Recommending A everywhere would lose 0.6 t/ha at dry sites. If the dry-site pattern were repeatable over years, the region would contain two mega-environments with different recommended varieties.
Factorial regression relates G x E to measured environmental variables. For each genotype, a regression slope on a covariate such as rainfall during flowering or mean maximum temperature is estimated, and the genotype x covariate interaction sum of squares is a part of the G x E sum of squares with g - 1 degrees of freedom per covariate. Such analyses convert statistical interaction into agronomic explanation (variety B tolerates drought) and allow prediction for new environments whose covariates are known. Crop-model outputs and soil data can serve as covariates too.
In large trial series the G x E interaction is almost always statistically significant, because the pooled error has many degrees of freedom. The useful questions are its size relative to genotype variance (the ratio sigma_ge squared / sigma_g squared), whether it involves crossover changes in ranking among the leading entries, and whether it is repeatable across years. A significant but small, non-crossover interaction does not change recommendations; a large, repeatable crossover interaction does.
Yates and Cochran (1938) suggested regressing each genotype's yields on the mean yield of all genotypes in each environment. Finlay and Wilkinson (1963), working with barley varieties, used this approach on a logarithmic scale and popularised the regression coefficient as a measure of stability: a coefficient of 1 means average stability (the genotype responds to environments like the average genotype), below 1 means above-average stability (less response to environmental improvement, relatively better in poor environments), and above 1 means below-average stability (strong response, suited to favourable environments). Combined with mean yield, the coefficient classifies genotypes as generally adapted (high mean, slope near 1), specifically adapted to favourable environments (slope above 1) or to poor ones (slope below 1).
Eberhart and Russell (1966) wrote the model as y_ij = mu_i + b_i I_j + delta_ij, where mu_i is the mean of genotype i, I_j the environmental index (the mean of all genotypes in environment j minus the grand mean, so that the indices sum to zero), b_i the regression coefficient and delta_ij the deviation from regression. The coefficient is estimated as b_i = sum over j of y_ij I_j / sum of I_j squared, and the deviation variance S squared d_i is estimated from the residual sum of squares of each genotype's regression, corrected for the pooled error. A stable genotype in their sense has a high mean, b_i close to 1 and S squared d_i close to zero, the last indicating that its response is predictable. For example, with environmental indices -2, -1, 1 and 2, a genotype yielding 2, 4, 6 and 8 has b = (2 x -2 + 4 x -1 + 6 x 1 + 8 x 2) / (4 + 1 + 1 + 4) = 14/10 = 1.4, a responsive genotype. The analysis partitions environments plus G x E into environments (linear), genotype x environment (linear) with g - 1 degrees of freedom, and pooled deviations.
Joint regression has well-known weaknesses. The environmental index is computed from the same data as the genotype yields, so it is not independent of them; with few genotypes this biases the coefficients. Regression on the mean captures only one pattern of interaction and often explains a small part of it. The coefficient's interpretation depends on the range of environments sampled. Despite this, the method remains widely used and examined because of its simplicity and its agronomic interpretation.
Suppose three genotypes have means 4.2, 4.6 and 3.9 t/ha, regression coefficients 0.95, 1.40 and 0.70, and deviation variances near zero, significantly positive and near zero respectively. The first is generally adapted and predictable: a good candidate for wide recommendation. The second has the highest mean and responds strongly to favourable conditions, but its significant deviations make its performance less predictable; it may suit high-input environments. The third is relatively better in poor environments but has a low mean, which may make it valuable only where conditions are consistently harsh. The judgement combines all three parameters with knowledge of the target environments.
Whether b_i differs from 1 is tested by t = (b_i - 1)/SE(b_i), with the standard error from the deviations of that genotype's regression. Whether the deviation variance is zero is tested by comparing the genotype's deviation mean square with the pooled error. In the combined analysis, the significance of the G x environment (linear) line, tested against pooled deviations, tells whether genotypes differ in their regression slopes at all; if not, the slope classification is not supported by the data.
Finlay and Wilkinson analysed log yields, which turns proportional differences into additive ones; on the original scale, a genotype that is a constant percentage better than another has a regression coefficient above 1 simply because absolute differences grow with the environmental mean. Whether stability is judged on the original or the log scale therefore changes the conclusions for high-yielding genotypes, and the scale should be chosen with that in mind and reported.
Wricke (1962) defined the ecovalence of genotype i as its contribution to the G x E sum of squares: W_i = sum over j of (y_ij - y_i. - y_.j + y_..) squared, where y_i. is the genotype mean, y_.j the environment mean and y_.. the grand mean (computed on means over replicates). The W_i sum to the G x E sum of squares of the table of means. A genotype with low ecovalence contributes little interaction and is stable in the sense of following the environment means closely; ecovalence measures stability in the dynamic or agronomic sense of consistent relative performance, not constant yield.
Shukla (1972) proposed estimating a separate interaction-plus-error variance for each genotype, the stability variance sigma squared_i, which is a linear function of the ecovalences: genotypes with large sigma squared_i are unstable. Shukla's statistic can be tested for heterogeneity among genotypes and extended by removing the part of each genotype's interaction explained by a covariate. In mixed-model terms, it corresponds to allowing heterogeneous G x E variances for genotypes. Ecovalence and Shukla's variance rank genotypes identically.
Perkins and Jinks (1968) used a model equivalent to joint regression but with the regression term written as beta_i = b_i - 1, so that a genotype with average response has beta_i = 0; it emphasised the biometrical genetic interpretation of the interaction. Other measures include the coefficient of variation of a genotype across environments (static stability, which favours unresponsive genotypes), Lin and Binns' superiority measure (the mean squared distance of a genotype from the best genotype in each environment), Kang's rank-sum combining yield and stability, and non-parametric measures based on ranks. Static stability (constant yield regardless of environment) is useful for traits like quality, but for yield, farmers usually prefer dynamic stability with high mean yield.
For a genotype with interaction residuals (y_ij - y_i. - y_.j + y_..) of 0.3, -0.2, 0.1 and -0.2 in four environments, the ecovalence is 0.09 + 0.04 + 0.01 + 0.04 = 0.18. A genotype with residuals 0.6, -0.5, 0.4 and -0.5 has ecovalence 0.36 + 0.25 + 0.16 + 0.25 = 1.02 and contributes far more to the interaction. Because ecovalences sum to the interaction sum of squares of the means table, they can be expressed as percentages of it, which makes them easy to compare between trials of different scales.
Different measures emphasise different concepts and often rank genotypes differently. Static measures (variance or CV across environments) favour genotypes that do not respond to improved conditions, which farmers rarely want for yield. Dynamic measures (ecovalence, Shukla's variance, deviations from regression) favour genotypes that track the environment average. Measures combining yield and stability (superiority, rank-sum indices, AMMI or GGE distances, FA-based overall performance and stability) are usually the most useful for recommendations. Whatever is used, it should be computed over a representative set of environments and years.
Many classical stability parameters have mixed-model counterparts. Shukla's variance corresponds to genotype-specific interaction variances; the Eberhart-Russell model to a random regression of genotype effects on an environmental index; AMMI and GGE to factor-analytic models. Fitting them as mixed models handles unbalanced data and heterogeneous error variances and gives standard errors that account for the whole variance structure, which is why modern variety-testing systems compute stability from FA models rather than from classical formulas.
The additive main effects and multiplicative interaction (AMMI) model, developed by Gauch, Zobel and colleagues (Zobel, Wright and Gauch, 1988), fits genotype and environment main effects by ordinary ANOVA and then applies a principal component analysis (singular value decomposition) to the matrix of interaction residuals: y_ij = mu + g_i + e_j + sum over k of lambda_k alpha_ik gamma_jk + residual, where lambda_k is the k-th singular value and alpha_ik and gamma_jk are the genotype and environment scores on the k-th interaction principal component axis (IPCA). Each axis captures a pattern of interaction; the first few axes usually capture most of the interaction 'signal', and the rest is treated as noise.
Gollob's method assigns degrees of freedom g + e - 1 - 2k to the k-th IPCA, for g genotypes and e environments, and tests each axis against the pooled error by an F test; for g = 10 and e = 6, IPCA1 has 13 degrees of freedom. Cornelius proposed alternative F tests, and cross-validation (predicting held-out replicates) chooses the number of axes that best predicts new data; Gauch showed that a parsimonious AMMI model can predict better than the raw cell means, because it removes noise. Typically AMMI1 or AMMI2 (one or two axes) is retained.
The AMMI1 biplot plots genotype and environment main effects (or means) on the horizontal axis against IPCA1 scores on the vertical axis. Genotypes with IPCA1 scores near zero have little interaction and are broadly adapted; a genotype and an environment with large scores of the same sign interact positively (the genotype does especially well there). The AMMI2 biplot plots IPCA1 against IPCA2. AMMI stability values combining several axes have been proposed. AMMI is a fixed-effects descriptive and predictive tool; it does not by itself account for unequal error variances or unbalanced data, for which mixed-model factor-analytic models (chapter 8) are the counterpart.
For g = 10 genotypes in e = 6 environments, the interaction has 9 x 5 = 45 degrees of freedom. Using Gollob's rule, IPCA1 has 10 + 6 - 1 - 2 = 13, IPCA2 has 11, IPCA3 has 9, and the residual after three axes has 45 - 33 = 12. If IPCA1 and IPCA2 are significant against the pooled error and IPCA3 is not, an AMMI2 model is retained. Cornelius's tests are more conservative and often retain fewer axes; cross-validation may retain fewer still, emphasising prediction over description.
Each cell mean of a genotype x environment table contains the true value plus noise from the replicates. The additive main effects capture most of the signal, and the first few interaction axes capture structured interaction; the remaining axes are dominated by noise. Discarding them gives smoothed cell estimates that, as Gauch demonstrated with cross-validation, can be as accurate as cell means based on several more replicates. The same shrinkage idea underlies BLUP and FA mixed models.
AMMI is usually computed on the table of genotype x environment means, with the interaction residuals obtained by double-centring (removing genotype and environment means). Results depend on the set of genotypes and environments included; adding or removing an outlying environment can change the axes considerably. Checking the robustness of the main conclusions by refitting without unusual environments or genotypes is good practice before drawing recommendations from a biplot.
Yan and colleagues (2000) argued that for variety evaluation the genotype main effect (G) and the G x E interaction should be considered together, since both matter for which genotype is best in an environment. The GGE biplot is obtained by environment-centring the genotype x environment table of means (subtracting each environment's mean, which removes the environment main effect but keeps G + GE), optionally scaling, and plotting the first two principal components from a singular value decomposition. Different scalings (singular value partitioning) emphasise genotype comparisons or environment comparisons.
Several views answer different questions. The which-won-where view draws a polygon connecting the genotypes farthest from the origin and lines from the origin perpendicular to its sides; these divide the biplot into sectors, and the vertex genotype of each sector is the best in the environments falling in it. Environments in different sectors with different winning genotypes indicate crossover interaction and possibly different mega-environments. The mean-versus-stability view uses the average-environment coordinate (AEC) axis, a line through the origin and the average environment: genotypes are ranked by their projections onto it (mean performance), and their distance from it indicates instability. In the discriminativeness-versus-representativeness view, environments with long vectors discriminate well among genotypes, and environments with small angles to the AEC axis are representative of the average environment; the cosine of the angle between two environment vectors approximates the correlation between them, so acute angles indicate positively correlated environments and obtuse angles negatively correlated ones.
A mega-environment is a group of environments that share the same best genotypes, with crossover interaction between groups but not within them. Identifying mega-environments over several years (single-year patterns can be accidental) allows breeding and recommendation to be targeted, with G x E within a mega-environment treated as unpredictable noise to be handled by testing over many sites. The AMMI versus GGE debate in the literature concerns which decomposition is more informative; both describe the same data, and the practical conclusions often agree when interpreted carefully.
Imagine a GGE biplot in which the polygon has five vertex genotypes, and the perpendicular lines divide the plane into five sectors. Six environments fall in the sector of genotype G3, two in the sector of G7, and the other sectors contain no environments. The data suggest two candidate mega-environments, with G3 best in the first and G7 in the second; vertex genotypes in empty sectors are not best anywhere and may be poor everywhere. If the same split appears in several years, and the two groups of environments differ in a recognisable way (for example altitude or rainfall), the mega-environment structure is credible.
Genotypes are ranked along the average-environment coordinate (AEC) abscissa, from below-average to above-average mean performance. Their perpendicular distance from it measures their contribution to interaction: a genotype far along the abscissa and close to it combines high mean and stability. This view is the GGE counterpart of plotting mean against a stability parameter, but it is derived from the same decomposition as the rest of the biplot. Interpreting it requires the appropriate scaling (genotype-focused) and a biplot that explains a large share of G + GE variation.
In singular value partitioning, the singular values can be allocated to genotype scores (genotype-focused scaling), to environment scores (environment-focused scaling) or split symmetrically. Genotype-focused scaling is appropriate for comparing genotypes and for the mean-versus-stability view; environment-focused scaling for comparing environments and judging discriminativeness and representativeness. Environments can also be standardised by their standard deviations before decomposition, which gives each environment equal weight regardless of its variability. A biplot should state which scaling and standardisation were used.
A biplot displays only the first two components; if they explain a small share of the variation, patterns in the plot may be misleading. Biplots of a single year can suggest mega-environments that do not repeat. They also show no standard errors, so differences between nearby genotypes may not be significant. Biplots are best used as exploratory summaries alongside formal analyses and multi-year evidence.
Real multi-environment trial (MET) data are rarely balanced: entries enter and leave over years, sites differ in design and number of replicates, error variances differ greatly between sites, and spatial variation needs modelling within each trial. Mixed models handle all of this, building on the mixed-models subject. The simplest MET mixed model has fixed environments and random genotype and G x E effects, with a separate residual variance for each environment. Genotype main-effect BLUPs then estimate broad adaptation, and heritability or reliability can be computed for the series.
More realistic models let the genetic variance differ between environments and the genetic correlation between environments differ between pairs. The fully unstructured model estimates e variances and e(e - 1)/2 covariances, e(e + 1)/2 parameters, 55 for 10 environments, which is usually too many to estimate reliably. Factor-analytic (FA) models, introduced for METs by Smith, Cullis and Thompson (2001), approximate the genetic covariance matrix by a few multiplicative factors plus environment-specific variances: an FA model with k factors has e(k + 1) - k(k - 1)/2 parameters, 20 for FA1 with 10 environments. FA models are the mixed-model counterpart of AMMI and GGE, giving genotype scores and environment loadings, and they are now standard in many national variety testing systems. The broad-sense heritability of genotype means over a balanced series of e environments with r replicates is H squared = sigma_g squared / (sigma_g squared + sigma_ge squared / e + sigma squared / (re)). With sigma_g squared = 4, sigma_ge squared = 2, sigma squared = 6, e = 3 and r = 2, H squared = 4 / (4 + 0.67 + 1) = 0.71.
As in the mixed-models subject, genotypes are random for selection (BLUPs exploit genetic correlations among environments, predicting performance in environments where a genotype was not tested) and fixed when a few varieties must be compared without shrinkage. Two-stage analyses (first each trial with its own spatial model, then the adjusted means with weights in the MET model) are common for very large data sets, while one-stage analyses fit everything together and are more efficient when feasible.
With 10 environments, the unstructured genetic covariance matrix has 55 parameters, the FA1 model 20 (10 loadings and 10 specific variances), and FA2 29 (20 loadings, minus 1 constraint, plus 10 specific variances). Selection among them uses REML likelihoods and information criteria. The proportion of genetic variance explained by the factors, environment by environment, shows how well each site is represented by the common factors; sites with large specific variances behave idiosyncratically. Heritability in the example of the main text, H squared = 4/(4 + 2/3 + 6/6) = 0.71, would rise to 4/(4 + 2/6 + 6/12) = 0.83 with six environments and two replicates, showing the value of more environments when G x E is present.
Because FA and unstructured models estimate genetic correlations between environments, a genotype's BLUP in an environment where it was not tested borrows from its performance in correlated environments. This is valuable in sparse testing, where each entry is tested at only some sites, a strategy increasingly used in breeding programmes to test more entries with the same resources. Its success depends on adequate connectivity: enough entries in common between sites to estimate the correlations.
In a two-stage analysis, each trial is first analysed with its own design and spatial model to produce adjusted genotype means and their variance-covariance matrix; the second stage fits the MET model to these means, weighting by the inverse of their variances (or the full covariance matrix). Two-stage analysis is convenient for large networks and close to one-stage efficiency when the weights are used properly; ignoring the weights treats precise and imprecise trials alike and discards information.
Perennial crops (fruit trees, tea, coffee, sugarcane with ratoons, forage grasses and legumes, plantation crops) give repeated harvests from the same plots over seasons or years. Unlike annual multi-environment trials, the seasons are not independent environments: the same plants, plots and treatment effects persist, carry-over effects occur (a heavy crop one year can reduce the next, as in alternate bearing in fruit trees), and the correlation between seasons tends to decline as they get further apart. Treating seasons as replicates, or as independent environments, is therefore wrong.
The classical approach is a split plot in time, with treatments in main plots and seasons as a sub-plot factor; as discussed in the factorial and mixed-models subjects, its F tests assume sphericity. Better approaches model the covariance between seasons explicitly in a mixed model (AR(1), heterogeneous AR(1), antedependence, factor-analytic or unstructured structures across harvests), with plots as subjects. Totals or means over seasons (cumulative yield) give a simple, valid analysis of overall performance; separate analyses per season show how effects develop; and regression of yield on time per plot, followed by analysis of the slopes, summarises trends. In long-term fertility and cropping-system experiments, the same considerations apply over decades, and repeatability (the correlation between repeated records of the same unit) is a useful summary of consistency.
Perennial variety trials at several sites combine both structures: sites as environments and harvests as repeated measures within sites. Mixed models with a site x harvest structure for genotype effects (for example a factor-analytic model across site-harvest combinations, or separable site and harvest covariance models) are used in breeding programmes for forages, sugarcane and tree crops. The guiding principle is the same as for annual METs: model the variance and covariance structure that the data actually have, so that genotype predictions and their standard errors are honest.
A forage trial with six varieties in four replicates is cut five times a year for three years. A naive analysis treating the 15 cuts as replicates would grossly overstate precision. A valid simple analysis uses total annual yield per plot, analysed as a split plot with years as the sub-plot factor, or better, a mixed model with plots as subjects and an AR(1) or unstructured covariance across years. A finer analysis models all 15 cuts with a structure separating season within year from year, testing whether varieties differ in seasonal distribution of yield as well as in total.
In tree crops such as mango, apple, olive and citrus, a heavy crop often reduces the next year's crop, producing negative correlation between successive years within trees. A mixed model with an AR(1) structure would estimate a negative correlation; treatments that reduce alternate bearing (pruning, thinning, nutrient management) can be identified by their effect on the year-to-year variance and correlation as well as on mean yield. Analysing only the cumulative yield would miss these effects.
In perennial breeding, the genetic correlation between early and later harvests determines how early selection can be made: if first-year yield is highly genetically correlated with yield over the life of the stand, selection after one year is efficient. These correlations are estimated from mixed models with an unstructured or factor-analytic genetic covariance across harvests and are central to the economics of breeding long-lived crops.
Long-term fertility and rotation experiments, some running for many decades, are repeated-season experiments on a grand scale. Their analysis must allow for slow trends in soil properties, changes in varieties and management over time, and correlation between years on the same plots. Methods include mixed models with temporal correlation, analyses of trends in treatment differences, and sustainability indices based on yield level and variability. The Rothamsted classical experiments and many national long-term fertiliser experiments are the reference examples.
Students in agronomy, breeding and agricultural statistics analyse multi-location data in many theses. In examinations they must write the combined ANOVA with correct error terms, test homogeneity of variances, compute and interpret regression coefficients, ecovalence and heritability over environments, and read AMMI and GGE biplots.
A five-year biometrician in a variety-testing or breeding programme analyses coordinated trials each season: checks error variances and spatial models per site, fits mixed MET models with heterogeneous variances or FA structures, reports genotype BLUPs or BLUEs with stability measures, and prepares biplots for variety identification committees. They also advise on how many sites and years a candidate needs before release.
A senior head of a testing network designs the trial system: number and choice of sites, years of testing, the analysis model, how mega-environments are defined and revised, and how results feed release decisions and recommendations by zone. They decide when to move from joint regression and AMMI to FA mixed models across the network.
A PhD with long experience develops G x E methodology (FA models, environmental covariates, genomic prediction across environments), teaches it and advises national and international programmes. As reviewer, they judge whether stability claims in a release proposal or paper rest on an adequate analysis.
The formula for Bartlett's statistic and the exact estimators of Shukla's variance and Eberhart-Russell's deviation variance are described, not written in full. Rules of thumb for pooling (variance ratios) differ between texts and are given as approximate. Factorial regression with environmental covariates, genomic prediction across environments and Bayesian MET models are mentioned only briefly. Software for AMMI, GGE and FA models is not covered in detail.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed with a script; the values needed are stated in the question.
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