Agricultural Statistics & Data Science · Statistical genetics and biometrics · also Statistical Genetics I, Biometrical Genetics
Genetics as a statistical science: segregation and linkage analysis, gene and genotype frequencies, Hardy-Weinberg and the forces that change it, polygenic models, breeding value and dominance, partitioning genetic variance, correlation between relatives and inbreeding. The Veterinary track teaches animal-genetics principles; this goes to the estimation theory.
In the exams: AICE-JRF Ag Stat Unit 3 (heavy: every unit named); ASRB NET Ag Stat (heavy); AICE-JRF Genetics & Plant Breeding (present)
Statistical genetics is the statistical theory of inheritance: how genes are transmitted, how their frequencies change in populations, and how genetic variation in quantitative traits is described and estimated. This subject covers the analysis of segregation ratios; detection and estimation of linkage; gene and genotype frequencies and the Hardy-Weinberg law; linkage disequilibrium and sex-linked genes; selection, mutation, migration and their equilibria; Fisher's fundamental theorem; the polygenic model with breeding values, dominance, average effects and epistasis; partitioning of genetic variance; correlation between relatives; regular systems of inbreeding; and the estimation of heritability, repeatability and genetic correlation.
It provides the theory that the biometrical techniques subject applies (path analysis, combining ability, selection indices, sire evaluation) and that the statistical genomics subject extends to markers. Estimation of variance components uses the methods of the mixed-models subject. Veterinary readers will find animal-genetics principles and breeding systems in the Veterinary track; this subject concentrates on the statistical theory and estimation.
In examinations, ICAR AICE-JRF/SRF Agricultural Statistics Unit 3 names nearly every topic here: statistical analysis of segregation, detection and estimation of linkage, gene and genotypic frequencies, random mating and equilibrium, disequilibrium due to linkage and sex linkage, selection, mutation and migration, Fisher's fundamental theorem, polygenic systems, breeding value, dominance, average effect, epistasis, partitioning of genetic variance, correlation between relatives, regular inbreeding and estimation of heritability, repeatability and genetic correlation. Numerical questions are common.
Mendelian genetics predicts ratios among the progeny of crosses: 3:1 for a dominant single gene in an F2, 1:2:1 with codominance, 1:1 in a test cross, 9:3:3:1 for two independent dominant genes, and modified ratios (such as 9:7, 15:1, 12:3:1 or 9:3:4) when genes interact. Whether observed counts agree with a hypothesised ratio is tested with the chi-square goodness-of-fit statistic, the sum of (observed - expected) squared / expected, on k - 1 degrees of freedom for k classes when the ratio is fully specified. For example, 70 tall and 30 dwarf plants among 100 in an F2 against an expected 75:25 give chi-square = 25/75 + 25/25 = 1.33 on 1 degree of freedom, below the 5% critical value of 3.84, so the data are consistent with a 3:1 ratio. With small expected numbers (below about 5 in a class) Yates' continuity correction or exact binomial and multinomial tests are used. Agreement with a ratio does not prove the genetic hypothesis: several hypotheses may fit the same counts, and sample sizes are often too small to distinguish 9:7 from 3:1 or 15:1 from 3:1 with confidence.
When several families or seasons are tested, the total chi-square (sum over families) is partitioned into a pooled chi-square (computed from the summed counts, 1 degree of freedom for a two-class ratio) and a heterogeneity chi-square (the difference), with degrees of freedom equal to the number of families minus one. A significant heterogeneity chi-square indicates that families differ in their ratios, so pooling is not justified. For a dihybrid F2 with four phenotypic classes, the 3 degrees of freedom of the chi-square for 9:3:3:1 are partitioned into the segregation of gene A (1 degree of freedom, 3:1), the segregation of gene B (1, 3:1) and their joint segregation (1), which tests independence. A significant joint-segregation component, with normal single-gene segregation, is the first evidence of linkage.
When two genes interact, the F2 classes combine into modified ratios: complementary genes give 9:7, duplicate dominant genes 15:1, dominant epistasis 12:3:1 and recessive epistasis 9:3:4. Testing a 9:7 hypothesis with 95 coloured and 65 white plants out of 160 gives expected numbers 90 and 70 and chi-square = 25/90 + 25/70 = 0.63 on 1 degree of freedom, consistent with complementary gene action. The same data tested against 3:1 (expected 120 and 40) give a very large chi-square and are rejected, showing how the choice of hypothesis is informed by biology and by testing alternatives.
Systematic departures from Mendelian ratios at a marker locus (segregation distortion) arise from gametic or zygotic selection, chromosomal rearrangements or genotyping errors. In mapping populations, markers with strong distortion are flagged and examined; distorted regions can bias recombination estimates if not handled carefully.
Distinguishing close ratios needs large samples. To separate 9:7 from 1:1 with reasonable power requires a few hundred plants, and to distinguish 15:1 from 63:1 requires many more. Planning the F2 or test-cross population size from the expected class frequencies, as in any power calculation, avoids inconclusive studies.
Linked genes on the same chromosome are inherited together unless separated by crossing over; the recombination fraction r is the proportion of recombinant gametes, ranging from 0 (complete linkage) to 0.5 (independent assortment). In a test cross of a double heterozygote with the double recessive, gametes are seen directly in the progeny, and r is estimated as the number of recombinant progeny divided by the total, with standard error the square root of r(1 - r)/n. For example, with 172 AB, 168 ab, 32 Ab and 28 aB progeny from a coupling-phase test cross (n = 400), r = 60/400 = 0.15, with standard error 0.018. In an F2, recombinant and parental gametes cannot be distinguished directly; r is estimated by maximum likelihood, by the product-ratio method of Fisher and Balmukand, or by Emerson's method, and coupling and repulsion phases give different formulas and different amounts of information.
The evidence for linkage is summarised by the LOD score, log10 of the ratio of the likelihood of the data at the estimated r to the likelihood under r = 0.5; a LOD score of 3 or more (odds of 1000:1 in favour of linkage) is the conventional threshold for declaring linkage in mapping. Recombination fractions are converted to map distances (in centimorgans) by map functions that allow for multiple crossovers: Haldane's function d = -(1/2) ln(1 - 2r), which assumes no interference, and Kosambi's function d = (1/4) ln[(1 + 2r)/(1 - 2r)], which allows for interference. For r = 0.15, Haldane gives 17.8 cM and Kosambi 15.5 cM. In three-point crosses, the coefficient of coincidence (observed double crossovers divided by the number expected if crossovers were independent) measures interference, which is one minus the coefficient. Linkage map construction with many markers is developed in the statistical genomics subject.
In an F2 from a coupling-phase (AB/ab) double heterozygote, the frequency of the double recessive class is (1 - r) squared / 4, so tightly linked genes produce many double recessives; in repulsion (Ab/aB) it is r squared / 4, so double recessives are rare. Estimates from repulsion F2 data are therefore less precise for small r. Maximum likelihood combines all classes efficiently and gives standard errors from the information function.
In a three-point test cross, the gene order is identified from the rarest classes (double crossovers), recombination fractions for adjacent intervals are estimated, and the coefficient of coincidence compares observed double crossovers with the product of the two single recombination fractions times n. For example, with r = 0.10 and 0.15 in adjacent intervals and 1000 progeny, 15 double crossovers are expected; observing 9 gives a coefficient of coincidence of 0.6 and interference of 0.4.
Molecular markers made linkage analysis routine: thousands of markers are scored on mapping populations (F2, backcross, recombinant inbred lines, doubled haploids), pairwise recombination fractions and LOD scores are computed, markers are grouped into linkage groups and ordered by algorithms that minimise total map length or maximise likelihood. Polyploid species complicate segregation patterns and require specialised models. These methods are developed in the statistical genomics subject.
In a population, the frequency of allele A is p and of allele a is q = 1 - p. The Hardy-Weinberg law states that in a large random-mating population, without selection, mutation or migration, genotype frequencies after one generation of random mating are p squared (AA), 2pq (Aa) and q squared (aa), and remain constant. Allele frequencies are estimated by counting alleles: with 400 AA, 400 Aa and 200 aa individuals (n = 1000), p = (2 x 400 + 400)/2000 = 0.6. The expected genotype numbers under Hardy-Weinberg are 360, 480 and 160, and chi-square = 1600/360 + 6400/480 + 1600/160 = 27.8 on 1 degree of freedom (three classes, minus one for the total, minus one for estimating p), which rejects equilibrium: there is an excess of homozygotes, as might arise from inbreeding or population mixture. When the recessive class is distinguishable but heterozygotes are not, q is estimated as the square root of the frequency of recessive homozygotes, assuming equilibrium: if 4% of animals show a recessive trait, q = 0.2 and the frequency of carriers 2pq = 0.32.
With several alleles at frequencies p, q, r, ..., equilibrium genotype frequencies are the terms of (p + q + r + ...) squared: homozygote frequencies are squares and heterozygote frequencies twice the products, as in the ABO blood groups. For a sex-linked (X-linked) gene, males carry one copy, so male genotype frequencies equal allele frequencies; females follow Hardy-Weinberg proportions. This is why X-linked recessive traits are much commoner in males: with q = 0.1, 10% of males but only 1% of females are affected. The approach to equilibrium for sex-linked genes is discussed in chapter 4.
Hardy-Weinberg proportions require random mating with respect to the locus, a large population, no selection, mutation or migration, and equal allele frequencies in the sexes for autosomal loci. Departures in a sample may also reflect genotyping errors (for example heterozygotes scored as homozygotes), population structure (the Wahlund effect, an excess of homozygotes when subpopulations with different frequencies are pooled) or selection at that locus. In genomic data, Hardy-Weinberg tests are a standard quality check.
Positive assortative mating (similar phenotypes mating together) increases homozygosity for the genes affecting the trait and increases the genetic variance in the population, while negative assortative mating does the reverse. Unlike inbreeding, assortative mating affects only the loci related to the trait used in mating decisions.
When heterozygotes cannot be distinguished, frequencies can be estimated only by assuming equilibrium, and the equilibrium cannot then be tested with the same data. Codominant markers (most DNA markers) avoid this problem, which is one reason they replaced phenotypic markers in population studies.
For k alleles there are k(k + 1)/2 genotypes, and a chi-square test of Hardy-Weinberg proportions has k(k + 1)/2 - k degrees of freedom, that is k(k - 1)/2. With many alleles and small samples, expected counts are small, and exact tests by permutation are used instead of the chi-square approximation.
For two loci, linkage disequilibrium is the non-random association of alleles in gametes, measured by D = p_AB - p_A p_B (equivalently D = p_AB p_ab - p_Ab p_aB). Unlike single-locus Hardy-Weinberg equilibrium, which is reached in one generation of random mating, gametic equilibrium is approached gradually: D decays by a factor (1 - r) each generation, so D_t = (1 - r)^t D_0. With r = 0.1 and D_0 = 0.2, after 5 generations D = 0.2 x 0.9^5 = 0.118; for unlinked loci (r = 0.5), D halves every generation. Linkage disequilibrium is created by population admixture, selection, drift in small populations and new mutations, and its decay with distance underlies association mapping and genomic selection (statistical genomics subject). Standardised measures D' (D divided by its maximum possible value) and r squared (the squared correlation between allelic states) are used to compare loci with different allele frequencies.
For an X-linked gene, sons receive their X chromosome from their mothers, and daughters one X from each parent. If allele frequencies differ between the sexes, the frequency in males in one generation equals that in females in the previous generation, and the frequency in females equals the average of both sexes' frequencies in the previous generation. The difference between the sexes therefore halves each generation and changes sign, so frequencies approach equilibrium in a damped oscillation. The equilibrium frequency is the weighted average p = (2/3) p_f + (1/3) p_m, reflecting that females carry two-thirds of the population's X chromosomes.
After a cross between two inbred lines, linkage disequilibrium is maximal in the F2 and declines with each generation of random mating at a rate set by recombination. Populations with extensive linkage disequilibrium allow genes to be mapped with few markers but with low resolution; populations with little disequilibrium (old, large, outbred) need many markers but give fine resolution. This trade-off guides the choice between bi-parental mapping populations and diverse association panels.
Suppose in one generation the frequency of an X-linked allele is 0.6 in females and 0.3 in males. In the next generation, males take the females' previous frequency, 0.6, and females take the average, (0.6 + 0.3)/2 = 0.45. The difference between sexes has changed from +0.3 to -0.15, halving and reversing. The equilibrium is (2/3)(0.6) + (1/3)(0.3) = 0.5.
In small populations, drift generates linkage disequilibrium even between unlinked loci, and its expected level is roughly inversely related to effective population size. Estimates of effective population size from linkage disequilibrium among markers are now widely used in livestock and wildlife genetics.
For two biallelic loci, D can be tested with a chi-square on 1 degree of freedom computed as n r squared, where r squared is the squared correlation between allelic states and n the number of gametes or haplotypes. With genotype rather than haplotype data, haplotype frequencies are estimated by the EM algorithm before computing disequilibrium.
Selection changes allele frequencies through differences in fitness among genotypes. With selection against a recessive phenotype, fitnesses 1, 1 and 1 - s for AA, Aa and aa, the change in the frequency of a per generation is delta q = -s q squared (1 - q) / (1 - s q squared). Selection is slow when the recessive allele is rare, because most copies are hidden in heterozygotes. With complete selection against the recessive (s = 1), q_t = q_0 / (1 + t q_0), so the number of generations needed to reduce q from q_0 to q_t is t = 1/q_t - 1/q_0: from 0.2 to 0.1 takes 10 - 5 = 5 generations, but from 0.01 to 0.005 takes 100. Selection against dominants is fast. When the heterozygote is fittest (overdominance), with fitnesses 1 - s1, 1 and 1 - s2 for AA, Aa and aa, a stable equilibrium exists at q = s1/(s1 + s2); with s1 = 0.1 and s2 = 0.3, q = 0.25.
Recurrent mutation from A to a at rate u and back at rate v leads to an equilibrium q = u/(u + v). Because mutation rates are small, mutation changes frequencies very slowly on its own. Balanced against selection, mutation maintains deleterious alleles at low frequencies: for a recessive deleterious allele, q is approximately the square root of (u/s); for a dominant one, p is approximately u/s. Migration at rate m from a population with frequency q_m changes frequency by delta q = m(q_m - q) per generation, pulling the population towards the migrants' frequency; migration is the most powerful of these forces in homogenising populations. Random genetic drift in finite populations, discussed with inbreeding in chapter 10, adds random change and eventual fixation or loss.
With q = 0.4 and selection against the recessive homozygote with s = 0.5, delta q = -0.5 x 0.16 x 0.6 / (1 - 0.5 x 0.16) = -0.048/0.92 = -0.052, so q falls to about 0.348 in one generation. As q falls, the change per generation shrinks, which is why deleterious recessives persist in populations at low frequencies and why carrier testing (now by DNA tests) is needed to eliminate them quickly.
In small populations, allele frequencies change randomly from generation to generation because of sampling of gametes. The variance of the change is pq/(2N_e) per generation, and alleles are eventually fixed or lost, the probability of fixation of a neutral allele equalling its current frequency. Drift and inbreeding are two aspects of the same process, as chapter 10 shows.
An equilibrium is stable if frequencies return to it after small disturbances. The overdominance equilibrium is stable; with underdominance (heterozygote least fit), the equilibrium is unstable and the population moves towards fixation of one allele depending on the starting frequency. Mutation-selection balance is stable and explains the persistence of rare genetic defects.
In crops, gene flow by pollen between fields and by seed exchange among farmers acts as migration. In seed production, isolation distances limit unwanted gene flow to maintain varietal purity, and in conservation of landraces, some gene flow maintains diversity while too much erodes local adaptation. The migration formula shows that even modest rates of gene flow, sustained over generations, bring populations close to the migrants' frequencies.
R. A. Fisher (1930) stated that 'the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time'. In modern terms, the part of the change in mean fitness due to changes in allele frequencies under natural selection equals the additive genetic variance in fitness divided by mean fitness. The theorem connects population genetics with quantitative genetics: selection acts on additive variance, the variance that parents transmit to offspring through average effects of alleles.
Because additive variance is non-negative, the theorem was long read as saying that mean fitness always increases. Price (1972) and later Ewens and Edwards clarified that it refers only to the 'partial' change due to natural selection acting on allele frequencies, holding the environment (including the genetic background) constant; total mean fitness can fall because of environmental deterioration, frequency dependence and epistasis. A corollary is that traits closely related to fitness tend to have low heritability, because selection exhausts additive variance in fitness while non-additive and environmental variance remain, which is observed for fertility and survival traits in livestock and crops.
Robertson's secondary theorem extends Fisher's idea: the change in the mean of a trait due to selection equals the additive genetic covariance between the trait and relative fitness. This links natural selection to the breeder's equation of the biometrical techniques subject, where artificial selection defines fitness through the breeder's choice of parents.
Low heritability of fitness traits such as fertility, longevity and disease resistance does not mean genes are unimportant; it means additive variance is small relative to total variance. Improving such traits needs large progeny groups, accurate recording and genomic or BLUP methods that use information from many relatives.
Fisher's theorem was one of the foundations of the modern synthesis, alongside Wright's work on drift and population structure and Haldane's mathematical theory of selection. Together they established population genetics as a statistical science, and Fisher's work on the analysis of variance grew partly out of his attempt to partition variation in quantitative traits among relatives.
Quantitative traits are controlled by many genes, each with small effect, plus environment. For a single locus, assign genotypic values a, d and -a to AA, Aa and aa, measured from the mid-point of the homozygotes; d = 0 means no dominance, d = a complete dominance and d > a overdominance. With allele frequencies p (A) and q (a), the population mean is M = a(p - q) + 2dpq. The average effect of substituting A for a is alpha = a + d(q - p); it depends on allele frequencies as well as on a and d, which is why the same gene has different effects in different populations.
The breeding value of an individual is the value judged by the mean of its progeny, twice the expected deviation of its progeny from the population mean, and equals the sum of the average effects of its alleles: 2q alpha for AA, (q - p) alpha for Aa and -2p alpha for aa. The difference between genotypic value and breeding value is the dominance deviation. The additive genetic variance is the variance of breeding values, sigma squared A = 2pq alpha squared, and the dominance variance sigma squared D = (2pqd) squared. For p = q = 0.5, a = 1 and d = 0.5, alpha = 1, sigma squared A = 0.5 and sigma squared D = 0.0625. Even with considerable dominance, much of the genetic variance is usually additive at intermediate allele frequencies.
When genotypic values at one locus depend on genotypes at another, interaction (epistatic) deviations arise, and their variance is partitioned into additive x additive, additive x dominance and dominance x dominance components. Epistatic variance is usually small relative to additive variance in estimates from random-mating populations, though epistasis may be important for specific crosses, as generation mean analysis (biometrical techniques subject) can show.
With a = 1, d = 0.5 and p = 0.8, q = 0.2, the average effect is alpha = 1 + 0.5(0.2 - 0.8) = 0.7, and the population mean is M = 1(0.6) + 2(0.5)(0.8)(0.2) = 0.76. The breeding values are 2(0.2)(0.7) = 0.28 for AA, (0.2 - 0.8)(0.7) = -0.42 for Aa and -2(0.8)(0.7) = -1.12 for aa. The additive variance is 2(0.8)(0.2)(0.49) = 0.157 and the dominance variance (2 x 0.8 x 0.2 x 0.5) squared = 0.0256. Comparing with the p = q = 0.5 example shows how variance components change with allele frequency.
A substitution of A for a changes the genotype from aa to Aa in some individuals and from Aa to AA in others; the average change, weighted by the frequencies with which each occurs, is alpha. When A is rare, most substitutions convert aa to Aa, gaining a + d; when A is common, most convert Aa to AA, gaining a - d. Dominance thus contributes to additive variance, which is why the distinction between gene action and variance components matters.
Even when gene action involves strong epistasis, much of the resulting genetic variance in a population can appear as additive variance, because average effects absorb part of the interaction. This helps explain why additive models predict selection response well over the short term even when underlying gene action is complex.
The polygenic model treats genes as many and small, but mapping studies show that a few loci of moderate effect often account for part of the variation, with many small effects making up the rest. The statistical theory of breeding values and variance components applies whatever the distribution of effects, which is why it remains the basis of selection even in the genomic era.
The phenotypic variance is partitioned as sigma squared P = sigma squared G + sigma squared E (+ covariance and interaction between genotype and environment when they exist), and the genotypic variance as sigma squared G = sigma squared A + sigma squared D + sigma squared I, the additive, dominance and interaction variances. Environmental variance includes general (permanent) and special (temporary) components, which matter for repeatability. Narrow-sense heritability h squared = sigma squared A / sigma squared P determines resemblance between parents and offspring and response to selection; broad-sense heritability H squared = sigma squared G / sigma squared P matters for clonal and inbred-line selection, where whole genotypes are transmitted.
Components are estimated from covariances between relatives (chapter 9) using designed matings (half-sib and full-sib families, diallels and North Carolina designs, as in the biometrical techniques subject) or from pedigreed data with mixed models and REML (mixed-models subject). Genotype x environment interaction variance is separated by testing in several environments (multi-environment trials subject). Estimates are specific to the population, the environment and the trait definition, and their standard errors are typically large unless many families are used.
In self-pollinated crops, selection among inbred lines exploits additive and additive x additive variance, so the relevant heritability for line selection includes the latter. In clonal crops (potato, sugarcane, fruit trees), whole genotypes are propagated, so broad-sense heritability is relevant. In outbred animals, only additive variance is transmitted, so narrow-sense heritability matters. Using the wrong type leads to wrong predictions of response.
When genotypes are tested in several environments, genotypic variance estimated from a single trial includes genotype x environment variance, inflating heritability. Separating sigma squared GE requires multi-environment data, and heritability on an entry-mean basis over environments, as in the multi-environment trials subject, is the appropriate measure for variety selection.
The precision of variance components depends mainly on the number of families or parents: a heritability from 20 sires has a much larger standard error than one from 200 sires, however many progeny each has. Designs for parameter estimation therefore emphasise many families of moderate size.
Environmental variance includes micro-environmental variation within a trial, macro-environmental differences between trials and genotype x environment interaction. Good experimental design, as covered in this track's design subjects, reduces the first and so increases heritability on a plot basis, which in turn increases response to selection; this is the practical link between experimental design and genetic gain.
Relatives resemble each other because they share genes. In a random-mating, non-inbred population with no epistasis and no common environment, the covariance between a parent and its offspring is (1/2) sigma squared A; between half-sibs (1/4) sigma squared A; between full-sibs (1/2) sigma squared A + (1/4) sigma squared D; and between identical (monozygotic) twins sigma squared A + sigma squared D. Generally, the covariance is 2 theta sigma squared A + delta sigma squared D, where theta is the coefficient of coancestry (kinship) and delta the probability that both alleles of the two relatives are identical by descent. Full sibs have theta = 1/4 and delta = 1/4; half-sibs theta = 1/8 and delta = 0.
Relatives often share environments as well as genes: full-sibs share the maternal environment and litter, and families may be reared in the same pen or plot. Such common environmental effects inflate full-sib covariances in particular, which is why half-sib and parent-offspring designs are preferred for estimating additive variance. Maternal effects (differences among dams in the environment they provide) add further covariance between offspring of the same dam and can be estimated in mixed models with maternal genetic and permanent environmental terms, as is routine in animal breeding.
Livestock genetic parameters are estimated mostly from half-sib families (many progeny per sire), parent-offspring pairs and full pedigrees analysed by the animal model. Full-sib designs are common in poultry, pigs and fish, where common environment effects (litter, tank, hatch) must be modelled separately. Each design gives information on specific combinations of variance components, and combining them in one pedigree-based analysis improves precision.
The additive relationship between two individuals is twice their coefficient of coancestry: 0.5 for parent and offspring and for full sibs, 0.25 for half sibs and grandparent-grandchild. The numerator relationship matrix A used in BLUP contains these values, and the covariance of breeding values is A sigma squared A.
With inbreeding, covariances between relatives involve additional components, such as the covariance between additive and dominance effects and the variance of inbreeding depression, and the simple formulas of the non-inbred case no longer hold. This matters for self-pollinated crops, where families are partly inbred, and is handled by appropriate genetic models or by restricting analyses to designs where the formulas are known.
The inbreeding coefficient F is the probability that the two alleles at a locus in an individual are identical by descent; it equals the coefficient of coancestry of its parents. Wright's path method computes F for an individual from a pedigree as the sum over all paths through common ancestors of (1/2)^(n1 + n2 + 1)(1 + F_A), where n1 and n2 are the numbers of generations from each parent to the common ancestor A and F_A its own inbreeding. The offspring of half-sibs (one common grandparent) have F = (1/2)^3 = 1/8; the offspring of full-sibs have F = 1/4.
Under continued self-fertilisation, heterozygosity halves each generation: F_t = 1 - (1/2)^t, so after five generations F = 0.969 and only about 3% of the original heterozygosity remains. Full-sib mating approaches homozygosity more slowly, with F_t = (1/4)(1 + 2F_(t-1) + F_(t-2)), and half-sib mating more slowly still. In a finite random-mating population of effective size N_e, inbreeding increases by delta F = 1/(2N_e) per generation; with unequal numbers of breeding males and females, N_e = 4 N_m N_f / (N_m + N_f): 10 males and 40 females give N_e = 32 and delta F = 1/64 = 0.0156 per generation. Effective population size is a central quantity in conservation of breeds and gene banks.
Inbreeding increases homozygosity, redistributing genetic variance: variance among inbred lines rises (to 2F sigma squared A in the absence of dominance) while variance within lines falls. With directional dominance, the mean changes in proportion to F: M_F = M_0 - 2dpqF summed over loci, which is inbreeding depression and the mirror image of heterosis. Inbreeding depression is pronounced for fitness traits in outcrossing species and animals, and the rate of inbreeding is a key constraint in breeding programmes.
Consider an individual X whose parents are full sibs, with grandparents unrelated and not inbred. There are two common ancestors (the two grandparents), each reached by a path of length 1 from each parent, so F_X = (1/2)^3 + (1/2)^3 = 1/4. If X's parents were half sibs sharing only their sire, F_X = 1/8. If the common ancestor were itself inbred with F_A = 0.25, its contribution would be multiplied by 1.25.
Starting from non-inbred parents, repeated full-sib mating gives F = 0.25, 0.375, 0.5, 0.594 and 0.672 in generations 1 to 5, by the recurrence F_t = (1/4)(1 + 2F_(t-1) + F_(t-2)). Selfing reaches F = 0.5 after one generation and 0.969 after five, showing how much faster selfing fixes lines, which is why it is the basis of pure-line and inbred-line development in crops.
Breeding programmes control inbreeding by limiting the contributions of individual parents, by optimum-contribution selection that maximises genetic gain subject to a constraint on the rate of inbreeding, and by mating plans that avoid close relatives. Gene banks and conservation programmes aim to maintain a large effective population size by equalising family contributions.
Within fully inbred lines, all genetic variance disappears and remaining variation is environmental, which is why inbred lines and their F1 hybrids are uniform. This uniformity is valuable for variety registration and for use as checks and testers in experiments.
Heritability is estimated from resemblance between relatives. The regression of offspring on one parent estimates (1/2) h squared, so h squared = 2b; the regression on the mid-parent value estimates h squared directly; in dairy cattle the daughter-dam regression doubled is the classical estimate. From half-sib families, the intraclass correlation t estimates (1/4) h squared, so h squared = 4t: t = 0.08 gives h squared = 0.32. Full-sib correlations estimate at most (1/2) h squared plus dominance and common-environment contributions, so they give upper limits. Realised heritability is the ratio of response to selection differential in selection experiments. Mixed-model (REML) estimation with the full pedigree (animal model) uses all relationships at once and is now standard. Heritability estimates have large standard errors with small samples and should be interpreted as properties of a particular population and environment.
When a trait is measured repeatedly on the same individual (lactations, fleece weights, yearly yields of trees), the repeatability r is the intraclass correlation between repeated records: r = (sigma squared G + sigma squared Eg) / sigma squared P, where sigma squared Eg is the permanent environmental variance. Repeatability sets an upper limit to broad-sense heritability and predicts the value of additional records: the accuracy of predicting an individual's future performance from n records rises with n, and the gain is largest when repeatability is low.
The genetic correlation between two traits is r_A = Cov_A(x, y) / (sigma_Ax sigma_Ay), the correlation between breeding values, estimated from the covariance between one trait in one relative and the other trait in another (for example the cross-covariance between parent and offspring) or from the analysis of covariance among half-sib families. It arises from pleiotropy and linkage disequilibrium and determines correlated response to selection (biometrical techniques subject). Standard errors of genetic correlations are even larger than those of heritabilities, and multivariate REML analyses are preferred.
Twenty sires each have 15 progeny. The ANOVA gives MS between sires = 30 and MS within sires = 20. The sire variance component is (30 - 20)/15 = 0.667, the intraclass correlation t = 0.667/(0.667 + 20) = 0.032 and h squared = 4t = 0.13. With so few sires the standard error is large, and the estimate should be combined with published values or with larger data before use in planning.
Offspring-parent regression is unaffected by selection of parents on the trait (selection of parents changes the variance of x but not the regression), whereas sib analyses are biased if families are selected. Regression estimates are inefficient but robust; REML animal-model estimates use all information and handle selection when the pedigree and data on which selection was based are included. Reports should state the method, the data structure and the standard error.
Heritability changes with allele frequencies, with the environment (more uniform management lowers environmental variance and raises heritability), with the trait definition and with the population. A value reported for one breed, region or set of varieties should be applied elsewhere only with caution.
Genetic correlations near plus or minus one indicate that two traits are largely controlled by the same genes, so selection on one carries the other; near zero, they can be improved independently. Antagonistic correlations, such as between milk yield and fertility in dairy cattle, explain unintended declines in some traits under single-trait selection and motivate multi-trait selection indices.
Students need statistical genetics for examinations dominated by numerical problems (chi-square tests of ratios, recombination fractions, allele frequencies, inbreeding coefficients, heritability from regressions) and for theses that estimate genetic parameters or map genes. Understanding what heritability does and does not mean prevents common misinterpretations.
A five-year quantitative geneticist estimates heritabilities and genetic correlations with mixed models, monitors inbreeding and effective population size in breeding herds and gene banks, checks genotype data for Hardy-Weinberg departures and linkage, and advises breeders on expected responses. They translate theory into the parameters that selection indices and BLUP evaluations need.
A senior geneticist sets policies on mating and inbreeding limits, conservation of breeds and germplasm, and the use of genomic information, relying on effective population size, linkage disequilibrium and genetic-parameter estimates. They judge whether published parameter estimates are reliable enough to base programmes on.
A PhD with long experience develops theory and methods for estimation of genetic parameters, models of selection and inbreeding, and links to genomic data, and advises national programmes on long-term genetic management.
Derivations of the F2 maximum-likelihood estimators of linkage, of the full-sib inbreeding recurrence and of the general covariance between relatives with epistasis are outlined only. Molecular marker mapping, QTL analysis and genomic relationships are in the statistical genomics subject. Selection theory is given for single loci with constant fitness; frequency-dependent and multi-locus selection are not covered.
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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