Agricultural Statistics & Data Science · Statistical genetics and biometrics · also Statistical Genetics II, Biometrical Methods in Breeding
The analyses a breeding programme runs every season: path analysis, D-squared divergence, diallel and line x tester analysis for combining ability, generation-mean analysis, North Carolina designs, heterosis, prediction of response to selection and the construction of selection indices.
In the exams: AICE-JRF Ag Stat Unit 3 'path coefficient analysis, heterosis, GCA and SCA, diallel, line x tester, response to selection, selection index' (heavy); AICE-JRF Genetics & Plant Breeding (heavy); ASRB NET Ag Stat (heavy)
Biometrical techniques are the statistical tools plant and animal breeders use to understand genetic variation and to choose parents, crosses and selections. This subject covers variability parameters, genotypic and phenotypic correlations and path analysis; genetic divergence by Mahalanobis D squared and clustering; heterosis and inbreeding depression; diallel analysis by Griffing's and Hayman's methods; line x tester analysis; generation mean analysis and scaling tests; North Carolina designs and the triple test cross; response to selection for individual, family and combined selection and correlated response; the Smith-Hazel selection index; sire evaluation and breeding-value estimation in animals; and stability parameters as used in breeding.
It builds on the statistical genetics subject (genetic variance components, heritability and correlation between relatives), uses the linear and mixed models of this track for estimation (REML, BLUP), applies multivariate methods for divergence, and refers to the multi-environment trials subject for stability analysis and to the advanced design subject for designs for diallel crosses. Those topics are referred to rather than repeated.
In examinations, ICAR AICE-JRF/SRF Agricultural Statistics Unit 3 names path coefficient analysis, heterosis, general and specific combining abilities, diallel crosses, line x tester analysis, response to individual, family and combined selection and the construction of selection indices, and the Genetics and Plant Breeding and Animal Genetics papers test the same methods. Numerical questions ask for heritability, GCV, genetic advance, path coefficients, heterosis, diallel degrees of freedom, scaling tests, response to selection and index coefficients.
Biometrical analysis of a breeding population starts with the partition of phenotypic variation. From a replicated trial of genotypes (an RCBD with r replicates), the genotypic variance is estimated as sigma squared g = (MS genotypes - MS error) / r and the phenotypic variance as sigma squared p = sigma squared g + sigma squared e, where sigma squared e is the error mean square (on a plot basis). The genotypic and phenotypic coefficients of variation are GCV = (sigma g / mean) x 100 and PCV = (sigma p / mean) x 100; broad-sense heritability is h squared = sigma squared g / sigma squared p; and the expected genetic advance under selection is GA = k h squared sigma p, where k is the selection differential in standard units (2.06 for selecting the best 5%). For example, with sigma squared g = 16, sigma squared e = 9 and a mean of 50, GCV = 8%, PCV = 10%, h squared = 0.64 and GA = 2.06 x 0.64 x 5 = 6.6, or 13.2% of the mean. Covariances between traits are estimated in the same way from the analysis of covariance, and the genotypic correlation r_g = Cov_g(x, y) / (sigma_gx sigma_gy) measures association due to genes (pleiotropy or linkage), while the phenotypic correlation also includes environmental association. Genotypic correlations matter for indirect selection and selection indices. The derivation of these parameters from genetic models, and the estimation of heritability in narrow sense, are covered in the statistical genetics subject; the mixed-models subject gives REML methods that are preferable with unbalanced data.
Correlation shows association but not how it arises. Path analysis, developed by Sewall Wright and applied to plant breeding by Dewey and Lu (1959), partitions the correlation of each component trait with yield into a direct effect (the standardised partial regression coefficient, the path coefficient P) and indirect effects through other components: r_iy = P_iy + sum over j of r_ij P_jy. The path coefficients are obtained by solving these simultaneous equations, and the residual effect, the square root of (1 - sum of P_iy r_iy), measures the influence of factors not included. For two components with r_1y = 0.6, r_2y = 0.5 and r_12 = 0.3, solving P_1 + 0.3 P_2 = 0.6 and 0.3 P_1 + P_2 = 0.5 gives P_1 = 0.495 and P_2 = 0.352; the indirect effect of trait 1 through trait 2 is 0.3 x 0.352 = 0.105, the coefficient of determination is 0.6 x 0.495 + 0.5 x 0.352 = 0.47, and the residual effect is 0.73, showing that much of the yield variation is due to traits not studied. Path analysis assumes a causal structure specified by the analyst; it does not discover causes, and with highly correlated components it suffers from the same instability as multiple regression with collinearity (linear models subject).
High heritability with high genetic advance suggests additive gene action and good prospects for simple selection; high heritability with low genetic advance suggests non-additive action, for which heterosis breeding may be better. These rules of thumb rest on broad-sense heritability, which includes dominance and epistatic variance, and on estimates from a single trial, which include genotype x environment interaction in the genotypic variance. Estimates from several environments, separating sigma squared ge, are more reliable, as discussed in the multi-environment trials subject.
Genotypic correlations are estimated from mean products of the analysis of covariance in the same way as variances: Cov_g = (MP genotypes - MP error)/r. Because they combine several estimated quantities, their standard errors are large unless the number of genotypes is substantial, and estimates outside the range -1 to 1 occasionally occur in small trials, a sign of imprecision. Multivariate REML estimation constrains the genetic covariance matrix to be valid and is preferred for serious work.
Path analysis can include several layers, for example components affecting intermediate traits that affect yield, and is then a special case of structural equation modelling. Sequential path analysis in crops arranges traits in first-, second- and third-order components and estimates paths layer by layer. Whatever the structure, the diagram should reflect biological knowledge of development, and results should be presented with the correlation matrix so readers can check the computations.
Breeders choose parents that are genetically diverse, expecting more heterosis and more transgressive segregants from crosses between divergent parents. Mahalanobis's generalised distance D squared = delta' S^(-1) delta, where delta is the vector of differences between two genotypes' means for p characters and S the pooled within-genotype (error) dispersion matrix, measures divergence while allowing for correlations among characters and their different scales. Rao (1952) set out its computation by transforming the correlated characters into uncorrelated linear combinations (pivotal condensation), after which D squared is the sum of squared differences of the transformed variables. In the simple case of two uncorrelated characters with error variances 4 and 1 and mean differences 2 and 1, D squared = 4/4 + 1/1 = 2. For n genotypes there are n(n - 1)/2 distances, 435 for 30 genotypes. The contribution of each character to divergence is often summarised by the number of times it ranks first in contributing to the transformed differences, a descriptive rather than inferential measure.
Genotypes are grouped by Tocher's method, which adds genotypes to a cluster in order of increasing distance as long as the average intra-cluster distance stays below the maximum of the minimum distances of each genotype to the others, or by hierarchical clustering (for example Ward's method) and visualised in a dendrogram. Average intra- and inter-cluster distances describe the structure; parents are chosen from different, distant clusters with good means for important traits. Canonical (discriminant) analysis displays the genotypes in two or three dimensions. Geographical origin often does not match genetic divergence, which is a frequent finding of such studies. With molecular markers, genetic distances based on allele sharing complement morphological D squared, and multivariate methods from the multivariate analysis subject (principal components, cluster analysis) apply.
D squared estimates have sampling error, and with few replicates or many characters the dispersion matrix is poorly estimated. Using fewer, well-chosen characters, more replicates, or principal components of the characters before computing distances improves stability. Significance of individual D squared values can be tested with a chi-square approximation, but cluster patterns should be judged for robustness rather than over-interpreted.
The expectation that more divergent parents give more heterotic crosses holds only up to a point: very divergent parents may be poorly adapted to each other or carry unfavourable alleles, and heterosis depends on dominance at loci where the parents differ in the relevant direction. Studies relating D squared to heterosis find positive but often weak associations, so divergence is one criterion among several in parent choice.
Heterosis (hybrid vigour) is the superiority of an F1 hybrid over its parents. Mid-parent heterosis (relative heterosis) is (F1 - MP)/MP x 100, where MP is the mean of the two parents; better-parent heterosis (heterobeltiosis) is (F1 - BP)/BP x 100; and standard (economic) heterosis is the superiority over a standard commercial check. With parents yielding 40 and 60 and an F1 of 66, MP = 50, mid-parent heterosis is 32% and better-parent heterosis 10%. Standard heterosis is the measure that matters for release of a hybrid. The significance of heterosis is tested with t tests using the error mean square of the trial: for mid-parent heterosis, the standard error of F1 - MP is the square root of (3 MSE / 2r).
Inbreeding depression is the decline in vigour on selfing or inbreeding: (F1 - F2)/F1 x 100 between generations. In the absence of epistasis, heterosis depends on dominance at loci where the parents differ, and the F2 mean is expected to lie halfway between the F1 and the mid-parent, because half the heterozygosity is lost: in the example, the expected F2 mean is 66 - (66 - 50)/2 = 58, an inbreeding depression of 12.1%. Large departures from this expectation suggest epistasis. The dominance hypothesis (accumulation of favourable dominant alleles) and the overdominance hypothesis (superiority of heterozygotes per se) explain heterosis; most evidence favours dominance with some pseudo-overdominance from repulsion linkage. Generation mean analysis (chapter 6) estimates the components.
Heterosis estimated at one site may not hold elsewhere; hybrids are tested over locations and years like any variety, and the stability of their advantage over the standard check is part of the release decision. Heterosis for yield often reflects heterosis in components (more tillers, larger seeds), which path analysis of hybrid data can explore.
With r replicates and error mean square MSE, the standard error of F1 - BP is the square root of (2 MSE / r), and of F1 - MP the square root of (3 MSE / 2r). For example, with MSE = 8 and r = 3, these are 2.31 and 2.00. The t value is the observed difference divided by the standard error, on the error degrees of freedom. Reporting heterosis without such tests invites over-interpretation of small differences.
A diallel cross mates a set of p parents in all combinations. Griffing (1956) defined four methods according to the entries included: method 1, parents, F1s and reciprocals (p squared entries); method 2, parents and F1s without reciprocals (p(p + 1)/2); method 3, F1s and reciprocals without parents (p(p - 1)); and method 4, F1s only (p(p - 1)/2). He also distinguished model I, with fixed parents (inference about the particular parents), from model II, with random parents (inference about a population, giving variance components). The analysis partitions the entries' variation into general combining ability (GCA, the average performance of a parent in crosses, with p - 1 degrees of freedom), specific combining ability (SCA, the deviation of a particular cross from expectation based on its parents' GCA) and, where reciprocals are present, reciprocal effects. For method 2 with p = 6, there are 21 entries, 5 degrees of freedom for GCA and 15 for SCA; for method 4 with p = 6, 15 crosses, 5 for GCA and 9 for SCA. GCA effects are associated mainly with additive genetic variance and SCA with non-additive (dominance and epistatic) variance; the ratio of GCA to SCA variance (or Baker's ratio 2 sigma squared GCA / (2 sigma squared GCA + sigma squared SCA)) indicates the predominant type of gene action. Parents with high GCA are preferred for pedigree breeding; crosses with high SCA involving at least one high-GCA parent are candidates for hybrids.
Hayman (1954) and Jinks developed an analysis of method 1 or 2 diallels in terms of genetic components. For each parental array r (the crosses with parent r as one parent), Vr is the variance within the array and Wr the covariance between the array and the non-recurrent parents. Under the assumptions (diploid segregation, homozygous parents, no reciprocal differences, no epistasis, no multiple alleles, independent distribution of genes among parents), the regression of Wr on Vr has slope 1. The intercept indicates the average degree of dominance: a positive intercept means partial dominance, a line through the origin complete dominance, and a negative intercept overdominance. The order of parents along the line shows which carry more dominant genes (those near the origin). Components of variation are estimated: D (additive), H1 and H2 (dominance), F (relative frequency of dominant and recessive alleles) and h squared (overall dominance effect). The mean degree of dominance is the square root of H1/D (below 1 partial dominance, above 1 overdominance), and H2/4H1 estimates the proportion of genes with positive and negative effects (0.25 when they are equal). Because the assumptions are strong and often violated, results should be interpreted cautiously and checked with the Wr-Vr regression and tests of uniformity of Wr - Vr.
With p = 7 parents in method 2 (parents and one-way F1s), there are 28 entries; GCA has 6 degrees of freedom and SCA 21. In method 4 (F1s only), there are 21 crosses; GCA has 6 and SCA 14 degrees of freedom (p(p - 3)/2 = 14). In method 1, with reciprocals, there are 49 entries and the reciprocal effects take 21 degrees of freedom. Including parents (methods 1 and 2) allows comparison of crosses with parents and use of Hayman's approach; excluding them (methods 3 and 4) focuses on crosses and is preferred when parents differ in kind from hybrids, for example because of inbreeding depression.
If the parents are elite lines of direct interest, model I (fixed) gives GCA and SCA effects for those parents. If they are a random sample from a population, model II gives variance components and inferences about additive and dominance variance in that population. Many published studies use model I and then interpret the components as if model II applied, which is not strictly valid.
Modern analyses fit diallel data with mixed models in which GCA and SCA are random effects, possibly with the numerator relationship among parents, and with site and site interactions when crosses are tested in several environments. This handles missing crosses, unbalanced replication and multiple sites, and gives BLUPs of GCA for parent selection. The classical Griffing tables remain useful for balanced data and for understanding the structure.
Tests of the adequacy of the additive-dominance model include the regression coefficient of Wr on Vr (should not differ from 1) and the homogeneity of Wr - Vr across arrays (should be constant). Failure of these tests indicates epistasis, non-independent gene distribution or other violations, and the genetic components should then not be interpreted literally. Many applied diallel studies omit these tests, which weakens their conclusions.
Line x tester analysis (Kempthorne, 1957) crosses each of l lines with each of t testers, giving lt crosses, often evaluated with the parents. It allows many lines to be assessed with fewer crosses than a diallel and is widely used to evaluate inbred lines for hybrid breeding. The crosses' variation is partitioned into lines (l - 1 degrees of freedom), testers (t - 1) and lines x testers ((l - 1)(t - 1)). With 10 lines and 3 testers, there are 30 crosses, 9, 2 and 18 degrees of freedom respectively. GCA effects of lines are estimated from line totals over testers, GCA of testers from tester totals over lines, and SCA effects from the line x tester interaction. The proportional contribution of lines, testers and their interaction to the total crosses' sum of squares is commonly reported.
Variance components for GCA and SCA are obtained from the expected mean squares under a random model, and, from covariances of half-sibs and full-sibs, additive and dominance variances are estimated: with non-inbred parents (inbreeding coefficient F = 0), sigma squared A = 4 Cov(HS), and with fully inbred parents (F = 1), sigma squared A = 2 Cov(HS); dominance variance is estimated from the SCA component similarly. The choice of testers matters: a broad-based tester (a population) reveals general combining ability, while an inbred tester from the opposite heterotic group reveals hybrid value directly. Line x tester analysis shares the assumptions of the diallel about absence of epistasis for genetic interpretation.
With 12 lines and 4 testers, there are 48 crosses; the degrees of freedom are 11 for lines, 3 for testers and 33 for lines x testers. If the trial is an RCBD with 3 replicates including the 16 parents, there are 64 entries and 126 error degrees of freedom. The proportional contributions of lines, testers and L x T to the crosses' sum of squares show whether variation among crosses comes mainly from lines' GCA or from specific interactions.
In hybrid maize and other crops, lines are evaluated with testers from the opposite heterotic group so that the GCA measured reflects performance in the type of hybrid that will be produced. Testers should be few, stable and representative; using a tester closely related to the lines inflates or distorts combining-ability estimates.
Generation mean analysis uses the means of related generations from a cross between two inbred parents, typically P1, P2, F1, F2 and the backcrosses B1 (F1 x P1) and B2 (F1 x P2), to estimate gene effects and detect epistasis. Mather's scaling tests check whether the means fit an additive-dominance model: A = 2B1 - P1 - F1, B = 2B2 - P2 - F1, C = 4F2 - 2F1 - P1 - P2 and D = 2F2 - B1 - B2. Each quantity should be zero, within sampling error, if there is no epistasis; their variances follow from the variances of the generation means (for example V(A) = 4 V(B1) + V(P1) + V(F1)), and each is tested by t. Cavalli's joint scaling test fits the three-parameter model (m, d, h) to all generation means by weighted least squares and tests the fit by chi-square. For example, with P1 = 20, P2 = 10, F1 = 18, B1 = 20 and B2 = 13, A = 40 - 20 - 18 = 2 and B = 26 - 10 - 18 = -2; whether these differ significantly from zero depends on their standard errors.
When scaling tests indicate epistasis, Hayman's six-parameter model expresses the generation means in terms of m (mean), d (additive), h (dominance), i (additive x additive), j (additive x dominance) and l (dominance x dominance) effects, estimated from six generations. The type of epistasis is classified by the signs of h and l: opposite signs indicate duplicate epistasis, which reduces heterosis and makes selection in early generations less effective, while the same sign indicates complementary epistasis. Additive and additive x additive effects (d and i) are fixable by selection in self-pollinated crops, whereas h, j and l favour heterosis breeding. Generation means describe net effects summed over loci and can cancel when effects at different loci have opposite signs, which is a limitation shared by all biometrical models.
If the variances of the generation means are V(P1) = 0.5, V(F1) = 0.4 and V(B1) = 0.6, then V(A) = 4 x 0.6 + 0.5 + 0.4 = 3.3 and the standard error of A is 1.82; a value of A = 2 gives t = 1.1, not significant. Backcross and F2 generations need more plants than parents and F1 because they segregate and their means are needed precisely; the numbers of plants should be planned with this in mind.
Generation mean analysis estimates net effects summed over all segregating loci, so it cannot by itself say how many genes are involved. Effective factor estimates (such as the Castle-Wright estimator) give rough lower bounds, while QTL mapping, described in the statistical genomics subject, locates individual loci. The two approaches complement each other.
In Cavalli's joint scaling test, each generation mean is weighted by the reciprocal of its variance, so precisely estimated generations count more. The fitted three-parameter model is tested by a chi-square statistic with degrees of freedom equal to the number of generations minus three; a significant chi-square indicates epistasis or linkage effects and calls for the six-parameter model.
Comstock and Robinson (1948, 1952) introduced three mating designs for estimating additive and dominance variance in random-mating populations. In Design I, each male is mated to a different set of females (females nested within males); the variance among males estimates one quarter of the additive variance when F = 0 (sigma squared m = Cov(HS) = sigma squared A / 4), and the variance among females within males includes additive and dominance variance. In Design II, each of a set of males is mated to each of a set of females (factorial); male and female variances both estimate additive variance and the male x female interaction estimates one quarter of the dominance variance. Design III backcrosses F2 individuals to both parental inbreds, giving the most precise estimate of the average level of dominance.
Kearsey and Jinks (1968) extended Design III by crossing each F2 (or other) individual to P1, P2 and the F1, giving three families per individual with means L1i, L2i and L3i. The test for epistasis uses L1i + L2i - 2L3i, which should be zero for every individual in the absence of epistasis; sums L1i + L2i + L3i estimate additive variance, and differences L1i - L2i estimate dominance variance. Because it tests epistasis directly and does not depend on the assumptions of the diallel, the triple test cross is regarded as a powerful design, though it is labour-intensive.
For a random-mating population such as maize or forage grasses, NC II gives additive and dominance variance from one factorial mating with modest numbers of parents. For a cross between two inbreds, generation means estimate effects quickly, and the triple test cross tests epistasis and estimates variances in the F2. For a set of fixed elite lines, diallel or line x tester analyses estimate combining abilities. The choice follows the population, the breeding objective and the resources for making crosses.
Interpretation of NC designs in terms of additive and dominance variance assumes linkage equilibrium, no epistasis, diploid inheritance and random choice of parents. In F2-derived populations, linkage disequilibrium biases dominance estimates upward when linked loci are in repulsion, which is one reason pseudo-overdominance is often found in early generations and declines after random mating.
North Carolina designs and triple test crosses need many crosses and large families to estimate variance components with useful precision; tens of males and hundreds of progenies are typical, which limits their use to well-resourced programmes and research studies.
The response to selection in one generation is R = h squared S, where S is the selection differential (the mean of the selected parents minus the population mean) and h squared the narrow-sense heritability. Writing S = i sigma p, where i is the intensity of selection (about 2.06 for 5% selected, 1.76 for 10%, 1.40 for 20% and 2.67 for 1%), gives R = i h squared sigma p = i h sigma A. With S = 10 units and h squared = 0.4, R = 4 units per generation. Response per year divides by the generation interval, which is why speed breeding and genomic selection, which shorten the interval, increase gain.
When individual heritability is low, selection on family means is more accurate, because averaging reduces environmental noise; within-family selection is useful when families share large common environments. Lush's combined selection uses an index weighting an individual's own phenotype and its family mean, and is at least as efficient as either alone. The relative efficiency of these methods depends on heritability, family size, the relationship among family members and the intraclass correlation of phenotypes within families, and formulas are given in quantitative genetics texts.
Selection on trait X changes trait Y through the genetic correlation: the correlated response is CR_Y = i h_X h_Y r_g sigma pY. Indirect selection on X is more efficient than direct selection on Y (with equal intensity) when h_X r_g > h_Y. For example, with h squared X = 0.6, h squared Y = 0.2 and r_g = 0.6, the relative efficiency is 0.6 x 0.775 / 0.447 = 1.04, slightly better than direct selection. Indirect selection is valuable when Y is expensive or late to measure, as when selecting for yield through a highly heritable component or a marker.
Suppose grain yield (Y) has heritability 0.2 and phenotypic standard deviation 0.5 t/ha, panicle number (X) has heritability 0.6, and their genetic correlation is 0.6. With 10% selected on X (i = 1.76), the correlated response in yield is 1.76 x 0.775 x 0.447 x 0.6 x 0.5 = 0.18 t/ha per cycle, while direct selection on yield with the same intensity gives 1.76 x 0.2 x 0.5 = 0.18 t/ha. The two are almost equal here, matching the relative efficiency of 1.04; indirect selection would win clearly if X were cheaper or allowed more intense selection or shorter cycles.
The breeder's equation predicts response for one generation; over several generations response can continue near the predicted rate for some time but slows as allele frequencies change, as linkage disequilibrium created by selection (the Bulmer effect) reduces variance, and as favourable alleles approach fixation. Realised heritability, estimated from the observed response divided by the cumulative selection differential, checks predictions in selection experiments.
The tabulated intensities assume an infinite population and a normal distribution. When few candidates are selected from a small population, the expected intensity is slightly lower; tables or simulation give corrected values. Inbreeding from selecting few parents reduces genetic variance and long-term response, which breeders balance against short-term gain by limiting the contribution of each family.
Breeders usually want to improve several traits at once, with different economic values. Smith (1936) for plants and Hazel (1943) for animals proposed the selection index I = b'P, a linear combination of phenotypic values, chosen to maximise the correlation with the aggregate genotype H = a'G, where a contains the economic weights. The index coefficients solve P b = G a, so b = P^(-1) G a, where P is the phenotypic and G the genotypic variance-covariance matrix of the traits. For a single trait, b = h squared. With two uncorrelated traits having phenotypic variances 25 and 16, genotypic variances 10 and 4, and economic weights 1 and 2, b = (10/25 x 1, 4/16 x 2) = (0.4, 0.5). The expected response in the aggregate genotype is i sigma_I, where sigma_I squared = b'P b: here b'P b = 0.16 x 25 + 0.25 x 16 = 8, sigma_I = 2.83, and with i = 2.06 the gain is about 5.8 units of H. The accuracy of the index is the correlation between I and H.
Restricted selection indices (Kempthorne and Nordskog) hold the change in some traits at zero; desired-gains indices achieve specified relative changes without economic weights; base indices use the economic weights themselves as coefficients, which is robust when P and G are poorly estimated. The Smith-Hazel index relies on good estimates of P and G; with small samples, estimated indices can perform worse than simpler ones. Selection indices are the precursor of BLUP-based selection, in which mixed-model predictions of breeding values for several traits are combined with economic weights; the mixed-models subject covers the estimation side.
Economic weights express how much one unit improvement in each trait is worth, and they change with markets. Indices are often compared under several sets of weights to see whether selection decisions are robust. When economic weights are hard to justify, desired-gains indices, which aim at specified ratios of improvement, or base indices are practical alternatives.
When traits are correlated, the index coefficients depend on the full matrices: a trait with no economic value can receive a non-zero coefficient if it is genetically correlated with a valuable trait and is highly heritable, because it carries information about the aggregate genotype. Conversely, a valuable trait may receive a small coefficient if its information is already captured by correlated traits in the index.
A selection index built from an individual's own records and its relatives' records, with known variances, is equivalent to BLUP when fixed effects are known. BLUP generalises the index by estimating fixed effects simultaneously and by handling any pattern of relatives, which is why it replaced classical indices in animal evaluation while the index concept remains central to multi-trait selection objectives.
In dairy cattle, a bull's genetic merit for milk yield cannot be measured on the bull itself, so it is estimated from his daughters' records: progeny testing. A simple regressed estimate of the sire's breeding value is EBV = b x (mean of daughters' deviations from contemporaries), with b = 2n / (n + k), where n is the number of daughters and k = (4 - h squared) / h squared. With h squared = 0.25, k = 15; with 20 daughters, b = 40/35 = 1.14, so a daughter average deviation of +100 kg gives an EBV of +114 kg, and the predicted transmitting ability (half the breeding value) is +57 kg. The accuracy of the estimate is the square root of n / (n + k), 0.76 here; it rises with more daughters. Historical methods include the daughter-dam comparison and the contemporary comparison, which accounted for herd and year effects in simpler ways.
Henderson's best linear unbiased prediction (BLUP), introduced in the mixed-models subject, became the international standard for sire and cow evaluation because it accounts for herd-year-season effects, unequal numbers of daughters, genetic trends and non-random mating simultaneously. The sire model fits sires as random effects; the animal model fits every animal with the numerator relationship matrix, using information on all relatives and giving breeding values for cows as well as bulls. Reliability, the squared correlation between predicted and true breeding values, is reported with each evaluation. Genomic BLUP extends the approach with marker-based relationships, as described in the statistical genomics subject. Interpreting evaluations correctly (comparing animals within the same evaluation base, using reliability) is part of a practitioner's skill.
With heritability 0.25, a sire with 10 daughters has accuracy sqrt(10/25) = 0.63, with 50 daughters 0.88 and with 100 daughters 0.93. Progeny testing programmes therefore aim at tens to hundreds of daughters per bull, distributed across many herds to avoid confounding with herd effects. Genomic evaluation now gives young bulls moderate reliability before they have daughters, shortening generation intervals.
Comparing daughters with herd-mates calving at the same time (contemporaries) removes much of the herd and season effect. BLUP generalises this by fitting herd-year-season as a fixed effect (or random in some models) and using genetic relationships, so that sires whose daughters are concentrated in good herds are not unfairly favoured. Well-defined contemporary groups remain essential for accurate evaluations.
Evaluation methods have moved from daughter averages to BLUP sire models, animal models and multi-trait, test-day and genomic models. Each step used more information and more realistic models, and the accuracy of young animals' evaluations rose accordingly. The statistical core, BLUP in a mixed model with a relationship matrix, has remained the same.
The multi-environment trials subject treats stability analysis in detail: Finlay-Wilkinson and Eberhart-Russell regression, ecovalence, Shukla's variance, AMMI, GGE and factor-analytic mixed models. In breeding programmes these tools are applied at advanced stages, when lines are tested over locations and years, to choose lines combining high mean performance with appropriate adaptation. Lines with high mean and regression coefficient near 1 with small deviations are candidates for wide adaptation; lines with high slope suit favourable environments; specific adaptation may be targeted to mega-environments.
Several approaches combine mean and stability in a single selection criterion: Kang's rank-sum, Lin and Binns' superiority measure, the weighted average of absolute scores from AMMI, and, in mixed models, BLUP-based overall performance and stability from factor-analytic models. The genetic basis of stability (whether it is heritable and correlated with yield) varies by crop and trait, so stability measures should be estimated over enough environments and, ideally, several years before they guide selection. Stability across environments is also a target of robust design thinking, as discussed in the response surface subject.
If a stability measure is to be selected for, it must vary genetically and be measured with reasonable precision. Studies in several crops show that regression coefficients on environmental indices are often heritable to some degree, while deviation variances are estimated imprecisely. Selection for mean performance across a representative set of environments indirectly selects for broad adaptation and is usually the most effective strategy.
In animal breeding, the equivalent concern is genotype x environment interaction across production systems or climates, for example whether bulls rank the same in pasture-based and housed systems. Genetic correlations between the same trait measured in different environments, estimated by multi-trait mixed models, quantify this; a correlation below about 0.8 is often taken to indicate important re-ranking.
Breeding reports should state the number and type of environments, the stability statistic used, its standard error or test, and how mean and stability were combined in selection decisions, so that release committees can judge the evidence.
Students in plant and animal breeding compute these parameters in nearly every thesis: variability, heritability and genetic advance; correlations and path coefficients; D squared clustering; combining ability from diallel or line x tester crosses; generation means. In examinations they must perform the calculations and interpret them correctly, including their assumptions.
A five-year biometrician in a breeding programme analyses crossing designs, estimates genetic parameters with mixed models, builds selection indices with the breeders' economic weights, computes expected responses to plan programmes, and supports hybrid development with combining-ability analyses. In animal programmes, they run or interpret BLUP evaluations and explain reliabilities to breeders.
A senior quantitative geneticist sets breeding strategy: which mating designs and testers to use, how to combine phenotypic, pedigree and genomic information in selection, how to weight traits, and how to measure realised genetic gain. They decide when classical biometrical methods are adequate and when mixed-model and genomic approaches are needed.
A PhD with long experience develops biometrical methodology (models for combining ability with relationships, multi-trait selection theory, stability genetics) and advises national programmes. As reviewer, they check whether published genetic parameters rest on adequate designs, assumptions and estimation methods.
Derivations of expected mean squares for diallel and North Carolina designs, Hayman's component estimators and formulas for relative efficiency of family and combined selection are outlined, not given in full. The genetic theory behind variance components and correlations between relatives is in the statistical genetics subject. Values of selection intensity are standard rounded values; exact values depend on population size. Genomic selection is mentioned only in passing.
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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