Partitioning Phenotypic Variance

A quantitative trait shows continuous variation because it is influenced by many loci (polygenic) and by environmental factors, so individuals cannot be sorted into discrete phenotypic classes. To study such traits genetically, we cannot track single genotypes; instead we study the statistical distribution of phenotypes in a population and ask what fraction of its spread — its variance — is due to genetic versus environmental causes. The foundational decomposition is:

V_P = V_G + V_E where V_P is the total phenotypic variance measured in a population, V_G is the variance attributable to genotypic differences among individuals, and V_E is the variance attributable to environmental differences. A more complete model adds a genotype-by-environment interaction term, V_GxE, capturing cases where genotypes respond differently to different environments (for example, a genotype that yields well in irrigated soil but poorly in drought). For most of this article we treat V_GxE as folded into V_E or assume it is negligible, as is conventional in an introductory treatment — real agricultural and medical genetics must often model it explicitly. The genotypic variance V_G is not indivisible. Building on the additive model, where each allele contributes an average effect to the phenotype, geneticists further partition V_G into three components:

V_G = V_A + V_D + V_I

• V_A (additive variance): variance due to the average, independent effects of alleles, summed across loci — the portion of genetic variance that behaves "linearly" and is faithfully transmitted from parent to offspring, because a parent passes on single alleles, not whole genotypes.
• V_D (dominance variance): variance arising from interactions between the two alleles at the same locus (dominance deviations) — a heterozygote's phenotype may deviate from the simple average of the two homozygotes, and this deviation is broken apart at meiosis and not reliably inherited as a unit.
• V_I (epistatic/interaction variance): variance arising from interactions between alleles at different loci — combinations of alleles across loci that produce phenotypes not predictable from each locus's additive and dominance effects alone. Like dominance deviations, allele combinations across loci are shuffled by recombination and independent assortment, so V_I is also poorly transmitted.

This distinction matters enormously in practice: V_A determines how much offspring resemble parents and how fast a population responds to artificial or natural selection, while V_D and V_I, though they inflate the phenotypic variance we observe, largely evaporate at the population level from one generation to the next because Mendelian segregation breaks up genotypic combinations even as it faithfully passes on individual alleles.

Step 1 — Start from the additive genetic model. For a single locus with alleles A₁ and A₂, define genotypic values on a scale where the midpoint between the two homozygotes is zero: A₁A₁ = +a, A₂A₂ = −a, and the heterozygote A₁A₂ = d. If d = 0, the locus is purely additive; d ≠ 0 reflects dominance. Each allele carries an average effect on the population — the average deviation of offspring produced when that allele is transmitted, averaged over all the genotypes it might combine with in the current gene pool. This average-effect concept is what lets us assign a "breeding value" to each individual: twice the sum of average effects of the alleles it carries (for a diploid, the breeding value is the expected mean value of its offspring, doubled to reflect the two gametes an individual could theoretically contribute).

Step 2 — Decompose an individual's genotypic value. Any individual's genotypic value G at a locus can be written as: G = breeding value (A) + dominance deviation (D) + (across multiple loci) epistatic deviation (I) The breeding value A is the sum, across loci, of the average effects of the alleles carried. The dominance deviation D is the extra departure caused by how the two alleles at a locus combine (heterozygote advantage/disadvantage relative to the additive prediction). The epistatic deviation I is the extra departure caused by how alleles at different loci combine with each other, beyond what each locus's additive and dominance effects predict alone.

Step 3 — Variance of a sum is the sum of variances (when components are uncorrelated). Statistically, if A, D, and I are defined as uncorrelated (which is the case under the standard assumptions of random mating, linkage equilibrium, and no epistasis-driven covariance), then: Var(G) = Var(A) + Var(D) + Var(I) → V_G = V_A + V_D + V_I

Step 4 — Add environmental variance. The phenotype P = G + E, where E is the environmental deviation. Assuming genotype and environment are uncorrelated and do not interact (no V_GxE, no Cov(G,E)): V_P = V_G + V_E = V_A + V_D + V_I + V_E

Step 5 — Estimate the components from covariances between relatives, not individuals. You cannot look at one person and say "40% of your height is additive genetic." Variance components are population-level statistical quantities estimated by comparing phenotypic resemblance among relatives of known genetic relatedness:

Because parents transmit single alleles, not whole genotypes, to their offspring, the parent-offspring covariance captures only additive variance — dominance and epistatic combinations are broken up every generation by segregation and independent assortment. This is the deep mechanistic reason V_A, not V_G as a whole, predicts resemblance between generations and response to selection (formalized in the breeder's equation, covered under Heritability: Broad and Narrow Sense).

Step 6 — Half-sib and full-sib breeding designs isolate components experimentally. In a half-sib design, one sire is mated to many unrelated dams; offspring from the same sire share on average only ¼ of their additive genetic variance (they share one common parent), and share negligible dominance variance (their dams differ, so heterozygous combinations differ). The variance among sire families estimates roughly ¼V_A, cleanly excluding V_D. In a full-sib design, offspring share both parents; the variance among full-sib families estimates ¼V_A + ¼V_D (approximately), so comparing full-sib and half-sib family variances lets researchers solve for V_D by subtraction. This logic — deliberately breeding relatives of different, precisely known kinship — is the experimental backbone of variance component estimation.

Variance partitioning is a statistical framework, so its "classic experiments" are breeding designs that empirically confirmed the components behave as the theory predicts — chiefly that V_A, and not total V_G, governs resemblance between relatives and predictable response to selection.

Sewall Wright and the theoretical foundation (1920s). Wright developed the method of path analysis and the concept of average effects and average excess of alleles, laying the mathematical groundwork later formalized by Ronald Fisher. Fisher's landmark 1918 paper, "The Correlation between Relatives on the Supposition of Mendelian Inheritance," is the founding document of this entire framework: Fisher showed mathematically that if a continuous trait is controlled by many Mendelian loci, the correlation between relatives (parent-offspring, siblings) could be predicted precisely from additive and dominance variance components, resolving the long-standing conflict between biometricians (who studied continuous variation statistically, ignoring Mendelism) and Mendelians (who studied discrete traits). Fisher's derivation is, in essence, the mathematical proof that discrete Mendelian genes, summed over many loci, reproduce the continuous, roughly normal distributions biometricians observed — and it is Fisher who first partitioned genetic variance into additive and dominance parts explicitly.

Long-term artificial selection experiments provided the empirical test. In the Illinois Long-Term Selection Experiment for oil and protein content in maize, begun in 1896 by Cyrus Woll and continued for over a century, researchers selected each generation for the highest and lowest oil content, without knowing the underlying genes. Oil content responded to selection for more than 100 generations, moving from roughly 4–5% to over 20% in the high-oil line, and down to near 0% in the low-oil line — a result predictable only if a substantial fraction of the original variance was additive genetic variance, since response to selection depends on V_A (formally, response R = h²S, the breeder's equation, where h² is built from V_A/V_P). If all genetic variance had been dominance or epistatic, sustained directional response of this magnitude and persistence would not be expected, because those components are shuffled apart each generation rather than accumulated.

Half-sib/full-sib breeding designs in livestock (mid-20th century onward), pioneered in the animal breeding programs analyzed by geneticists such as Jay Lush, directly estimated V_A and V_D by comparing variance components among controlled degrees of relatedness. A sire mated to many dams (half-sib design) produces offspring sharing only ¼ of additive variance and negligible dominance variance; a sire mated to one dam repeatedly (full-sib design) produces offspring sharing ¼V_A + ¼V_D. Comparing the variance among half-sib families to the variance among full-sib families in dairy cattle milk yield and swine litter size allowed breeders to solve simultaneously for V_A and V_D using the analysis of variance (ANOVA) framework — a direct empirical decomposition, not just a theoretical one. These estimates matched Fisher's predicted algebra for relative covariances closely enough to validate the entire model, and they underwrote decades of successful selective breeding in agriculture (milk yield in Holstein cattle roughly quadrupled across the 20th century under sustained selection, consistent with substantial V_A).

Twin studies in human quantitative genetics, beginning systematically with the work of researchers following Francis Galton's early twin-comparison idea and formalized statistically through the 20th century, compare monozygotic (MZ) twins (who share ~100% of genotype) with dizygotic (DZ) twins (who share on average 50% of segregating alleles, like ordinary siblings). If a trait's phenotypic correlation in MZ twins is roughly double that in DZ twins, this is consistent with a largely additive genetic architecture (since DZ twins share half the additive variance MZ twins do); if the MZ correlation is more than double the DZ correlation, this signals a contribution from dominance or epistatic variance (because DZ twins share only a quarter of dominance variance, one-half of what MZ twins share, so dominance shrinks faster than additive variance as relatedness decreases). This falls out directly from the expected-covariance table above and has been used across hundreds of twin studies to estimate broad- and narrow-sense heritability for traits from height to blood pressure to behavioral measures.

Example 1 — A dairy cattle breeding design with asymmetric family sizes. A breeder measures milk yield (kg/lactation) in a herd. She uses one sire (Sire X) mated to 20 unrelated dams (half-sib design: 20 half-sib families, each with several daughters), and separately keeps records on a second sire (Sire Y) mated repeatedly to the same single dam across several lactations, producing full-sib daughters. She computes, from an ANOVA of daughter yields:

• Variance among half-sib families (Sire X's daughters, grouped by dam): σ²_HS = 180 kg²
• Variance among full-sib families (Sire Y's daughters, all full sibs): σ²_FS = 340 kg²
• Total phenotypic variance in the herd: V_P = 1200 kg²

Theory gives: σ²_HS ≈ ¼V_A σ²_FS ≈ ¼V_A + ¼V_D Step 1: Solve for V_A. ¼V_A = 180 → V_A = 720 kg² Step 2: Solve for V_D using the full-sib estimate. ¼V_A + ¼V_D = 340 180 + ¼V_D = 340 ¼V_D = 160 → V_D = 640 kg² Step 3: Compute V_E by subtraction (assuming V_I ≈ 0, a common simplifying assumption in livestock genetics since epistatic variance is hard to estimate and usually small relative to V_A and V_D for milk yield). V_G = V_A + V_D = 720 + 640 = 1360 kg² But V_G cannot exceed V_P — this immediately signals a problem: 1360 > 1200. In practice, this discrepancy arises because half-sib and full-sib variance estimates carry substantial sampling error with modest family numbers (only 20 half-sib families, few full-sib records), and because the simplifying assumption V_I = 0 and idealized coefficients (¼, ¼) assume no inbreeding, random mating, and no common maternal environmental effects shared by full sibs (calves from the same dam often share pre-natal and rearing conditions, inflating σ²_FS beyond ¼V_A + ¼V_D). A realistic reanalysis would add a maternal common-environment term, σ²_FS ≈ ¼V_A + ¼V_D + V_Ec(maternal), reducing the inferred V_D once V_Ec is estimated from other data (e.g., comparing full sibs raised together versus cross-fostered). This worked example illustrates a genuinely important, non-trivial point: raw variance-component arithmetic must be checked against the constraint V_G ≤ V_P, and violations are diagnostic of confounded environmental covariance or estimation noise — a very common real-world complication in breeding value estimation.

Example 2 — Predicting response to selection with different V_A/V_P splits. Suppose two traits in a wheat breeding program have identical total phenotypic variance, V_P = 100 (grain protein content, arbitrary units²), but different genetic architectures:

• Trait 1 (grain protein %): V_A = 60, V_D = 10, V_I = 5, V_E = 25
• Trait 2 (grain yield per plant): V_A = 20, V_D = 30, V_I = 10, V_E = 40

Narrow-sense heritability h² = V_A / V_P: Trait 1: h² = 60/100 = 0.60 Trait 2: h² = 20/100 = 0.20 A breeder applies the same selection intensity to both traits, selecting the top 10% of plants, giving a selection differential S = 8 units for both traits (same phenotypic scale, same intensity). Using the breeder's equation, R = h²S (foreshadowing the formal treatment in Heritability: Broad and Narrow Sense): Trait 1: R = 0.60 × 8 = 4.8 units per generation Trait 2: R = 0.20 × 8 = 1.6 units per generation Even though both traits show identical total phenotypic variance and are selected with equal intensity, Trait 1 responds three times faster because a much larger share of its variance is additive. Trait 2's high dominance and epistatic variance inflate its phenotypic spread — making it look, superficially, like a promising target for selection — but most of that variance is "genetic noise" from the breeder's standpoint: it contributes to how different individuals look, but not to how reliably that difference is passed to offspring. This is precisely why breeders routinely estimate h² before committing resources to a selection program, rather than judging breeding potential from phenotypic variance alone.

Ronald Aylmer Fisher, working largely alone in the 1910s while a young statistician with an interest in eugenics and evolutionary theory, resolved one of the fiercest disputes in early 20th-century biology: biometricians (led by Karl Pearson and W. F. R. Weldon), who studied continuous traits statistically and doubted Mendelism could explain smoothly varying characters, versus Mendelians (led by William Bateson), who studied discrete traits and dismissed biometrical methods as unable to reveal underlying genetic mechanisms. Fisher, trained in mathematics at Cambridge and steeped in both statistics and genetics, showed in his 1918 paper that the two camps were not describing different phenomena at all — a trait controlled by many independent Mendelian loci, each with small effect, naturally produces a continuous, approximately normally distributed phenotype in a population, and the statistical correlations biometricians measured between relatives could be derived exactly from Mendelian principles. In doing so he introduced the additive/dominance variance partition, invented the term "variance" itself in this context, and effectively founded the field of quantitative genetics as a rigorous mathematical discipline, later extending this work in his 1930 book The Genetical Theory of Natural Selection, which also formalized the fundamental theorem of natural selection linking additive genetic variance in fitness to the rate of evolutionary change.

Variance partitioning underlies nearly every quantitative breeding and clinical genetics program in use today. Agricultural breeding programs for crops and livestock routinely estimate V_A via half-sib/full-sib and pedigree-based designs (now often replaced or supplemented by genomic relationship matrices from SNP data) to calculate "estimated breeding values" (EBVs) used to select bulls, boars, and elite crop lines — the entire genomic selection revolution in dairy cattle breeding, which has driven rapid gains in milk yield since the 2000s, is a direct descendant of this framework, now estimating V_A directly from genome-wide marker data rather than pedigree alone.

Human medical genetics uses the same logic to interpret twin and family studies for disease risk and quantitative traits (blood pressure, BMI, cholesterol), estimating heritability to judge how much genome-wide association study (GWAS) discovery is likely to yield, and to explain the "missing heritability" problem: GWAS-identified common variants typically explain only a fraction of the V_A implied by twin studies, a gap partly attributable to V_D, V_I, rare variants, and gene-environment interaction not well captured by additive single-marker models.

Conservation biology uses estimates of V_A in wild populations (e.g., via "animal models" fitted to pedigrees of tracked individuals, such as long-term studies of Soay sheep or great tits) to predict whether populations retain the additive genetic variance needed to adapt to rapidly changing environments (climate shift, novel pathogens) — populations with depleted V_A due to small size and inbreeding are at elevated risk of failing to track environmental change. Plant breeding for climate resilience likewise depends on identifying traits with sufficient V_A/V_P (heritability) in target environments before investing in multi-year selection programs, since traits dominated by V_D, V_I, or V_E respond poorly to simple recurrent selection and instead require strategies like hybrid breeding (which specifically exploits V_D, i.e., heterosis) or marker-assisted approaches that can track specific epistatic combinations.

Ask the tutor: How would adding a genotype-by-environment interaction term (V_GxE) change the breeder's equation and the interpretation of heritability across environments?