Estimating Heritability from Resemblance and Selection

Narrow-sense heritability partitions total phenotypic variance, $V_P$, into components. In the simplest model, $V_P = V_G + V_E$, where $V_G$ is genetic variance and $V_E$ is environmental variance. Genetic variance itself is further partitioned, $V_G = V_A + V_D + V_I$, where $V_A$ is additive genetic variance, $V_D$ is dominance variance, and $V_I$ is interaction (epistatic) variance. Narrow-sense heritability is defined as

This is distinct from broad-sense heritability, $H^2 = V_G/V_P$, which includes all genetic variance, dominance and epistasis included. The distinction matters because only the additive component is reliably transmitted from parent to offspring in a predictable, linear way. A dominant allele's beneficial interaction with its partner allele, or an epistatic interaction between two loci, can produce an outstanding phenotype in one individual, but that particular combination is not guaranteed to reassemble in the next generation, because meiosis and random mating break up genotype combinations while additive effects of alleles pass through largely intact. This is why breeders and evolutionary biologists care specifically about $h^2$, not $H^2$: it is $V_A$ that determines how efficiently selection can shift a population's mean. The two methods in this article estimate $h^2$ from complementary angles. Parent-offspring regression is an observational method: it measures a naturally occurring resemblance, without any deliberate intervention, and infers $h^2$ from the regression slope. The breeder's equation is a predictive and retrospective tool: applied prospectively, it forecasts the response to an imposed selection regime, and applied retrospectively, after a selection experiment concludes, it yields a "realized heritability" by working the equation backward. Both methods rest on the same additive genetic model, and under ideal conditions they should agree, a fact that itself constitutes a test of the underlying quantitative genetic theory, connecting directly to the broader concept of Heritability: Broad and Narrow Sense.

Step 1: Why additive variance drives parent-offspring resemblance. Consider a trait controlled by many loci, each contributing a small additive effect, plus environmental noise. A parent transmits a random half of its alleles to each offspring. Because only additive effects combine in a simple, linear, and predictable way across generations, dominance deviations and epistatic interactions are effectively "reshuffled" and do not transmit efficiently. Under random mating and no genotype-environment correlation, the covariance between one parent's phenotype and one offspring's phenotype is:

The factor of one-half arises because a parent shares, on average, half its alleles identical by descent with its offspring. The regression coefficient of offspring phenotype on a single parent's phenotype is then:

so that $h^2 = 2 b_{OP}$. If instead offspring phenotype is regressed on the mid-parent value, the average of both parents' phenotypes, $\tfrac{1}{2}(P_{mother}+P_{father})$, the averaging of two independent parental values halves the variance of the mid-parent value relative to a single parent (assuming both sexes have equal phenotypic variance), and the covariance term simplifies so that the slope of offspring on mid-parent directly equals $h^2$:

This mid-parent regression, first used systematically for continuous traits, is the cleaner and more common design because it removes the noise contributed by only sampling one parent and it automatically corrects for sex differences in mean and variance.

Step 2: Practically fitting the regression. A quantitative geneticist measures a large sample of parent pairs and their offspring (or offspring means, in species that can be reared in replicate). Each family contributes one point: mid-parent value on the x-axis, mean offspring value on the y-axis. Ordinary least-squares regression yields a slope, and that slope, under the assumptions in Step 4, is the estimate of $h^2$. The standard error of the slope, from ordinary regression theory, gives a confidence interval on the heritability estimate, and $n$ (number of families) must typically be in the hundreds for a reasonably tight interval, because trait variance is large relative to the genetic signal.

Step 3: The breeder's equation, mechanistically. Suppose a population has phenotypic mean $\bar{P}$ and a breeder or natural selective pressure chooses only individuals above some threshold to reproduce. The mean phenotype of the selected breeders is $\bar{P}_S$. The selection differential is:

This is the amount by which the selected parents differ, phenotypically, from the population average, entirely an observable, phenotypic quantity, requiring no genetic assumptions to measure. The response to selection, $R$, is the difference between the mean phenotype of the offspring of the selected parents, $\bar{P}'$, and the mean phenotype of the original (unselected) parental generation, $\bar{P}$:

The breeder's equation links the two:

Mechanistically this follows directly from the same additive model as parent-offspring regression: since offspring phenotype regresses on parental phenotype with slope $h^2$ (mid-parent convention), the offspring generation's mean shifts by exactly $h^2$ times however far the selected parents' mean was displaced from the population mean. If heritability is 1 (all variance additive, no environmental noise), the offspring mean matches the selected parents' mean exactly, full response. If heritability is 0 (no additive genetic variance), selecting extreme individuals produces no shift at all in the next generation, because the extreme phenotypes were purely environmental accidents that are not transmitted.

Step 4: The assumptions that make both methods valid. Both methods depend on a shared set of conditions: 1. Mating is random with respect to genotype (no assortative mating, or its effects on covariance are explicitly modeled). 2. Parents and offspring do not share special environmental effects beyond what is captured by $V_E$, i.e., no genotype-by-shared-environment correlation inflating the parent-offspring covariance (a classic confound: better parents may also provide better environments to offspring, mimicking genetic transmission). 3. The trait is measured on the same scale, under comparable environmental conditions, across generations. 4. Selection differentials are modest, because the breeder's equation is a linearization; very strong or repeated selection can change allele frequencies enough that $V_A$ itself shifts, causing the realized response to depart from a naive linear extrapolation. 5. No genotype-environment interaction that changes the additive variance across the environments compared.

Step 5: Realized heritability, working the equation backward. After an actual selection experiment, both $S$ and $R$ are directly measurable phenotypic quantities (no genetic model needed to measure them). Rearranging:

Typically an experiment runs selection over several generations, cumulative selection differential plotted against cumulative response, and the slope of that line (rather than a single-generation ratio) is taken as the more robust realized heritability estimate, since it averages out generation-to-generation sampling noise.

Falconer's foundational synthesis and mouse selection experiments. Douglas Falconer, working at the Institute of Animal Genetics in Edinburgh in the 1950s and 1960s, conducted long-running selection experiments on body weight in laboratory mice that became the textbook demonstration of the breeder's equation. Falconer selected mice for large body size and, in parallel lines, for small body size, generation after generation, while maintaining unselected control lines. He measured the selection differential $S$ each generation, simply the difference between the mean weight of the mice chosen to breed and the mean weight of the whole population they were drawn from, and measured the realized response $R$, the shift in offspring mean weight relative to the unselected controls (controls are essential, because they correct for any environmental drift, such as improved husbandry, across generations, that would otherwise be mistaken for a response to selection). Over many generations, cumulative selection differential plotted against cumulative response gave a straight line whose slope, the realized heritability, was approximately:

for body weight in mice, depending on the specific line and study, consistent with independent estimates obtained from parent-offspring regression and from sib analysis on the same mouse populations. This concordance across independently derived estimates, resemblance-based and selection-based, was the key empirical validation: the additive genetic model was not just a convenient fiction, it made a falsifiable, quantitative prediction that different experimental approaches, run on the same population, delivered the same numeric answer.

Illinois Long-Term Selection Experiment on corn oil content. Beginning in 1896 at the University of Illinois, agronomists selected maize for high and low oil content in the kernel, a program that has continued for over a century, one of the longest-running selection experiments in existence. Starting from a population with roughly 4.7% oil content, the high-oil line was pushed above 20% oil content after around 100 generations of selection, while the low-oil line was driven down to near 0.1%, essentially the floor of biological plausibility. The response did not proceed at a perfectly constant rate predicted by a fixed $h^2$; it decelerated somewhat over many generations as additive genetic variance was depleted by selection (alleles pushed toward fixation), and in some phases actually accelerated again, likely due to new mutation input and occasional recombination generating fresh additive variance. This long experiment is historically important both as a spectacular confirmation that selection acting on standing quantitative variation can produce enormous, sustained phenotypic change, and as a caution: the breeder's equation is a short-term linear approximation, and $h^2$ itself is not a permanent constant of the population, since selection erodes the very variance on which it is based.

Testing for confounding: cross-fostering and adoption designs. A serious risk in parent-offspring regression is that resemblance may arise from shared environment rather than shared genes, since biological parents typically also raise their offspring. Behavioral geneticists and animal breeders have tested this directly using cross-fostering designs, in which offspring are raised by non-biological parents (foster parents), while true genetic parentage is tracked separately. If offspring still resemble their biological parents (not their foster parents) after cross-fostering, the resemblance is attributed to genetics rather than shared rearing environment. Such designs, along with the classical twin study framework (monozygotic versus dizygotic twin comparisons), have repeatedly confirmed that a substantial portion of parent-offspring resemblance for many quantitative traits survives the removal of shared environment, supporting the additive genetic interpretation, though the magnitude of $h^2$ estimated is always specific to the population and environmental range sampled.

Example 1: Parent-offspring regression in a plant-breeding population. A breeder measures seed weight (mg) in a population of bean plants. She records mid-parent seed weight and mean offspring seed weight for 200 families:

The regression slope of offspring on mid-parent is:

Since mid-parent regression slope estimates $h^2$ directly:

Now suppose the breeder wants to predict the result of choosing the heaviest-seeded 10% of plants as parents for the next generation. Measurements show the population mean seed weight is 150 mg, with the selected top 10% averaging 168 mg. The selection differential is:

Applying the breeder's equation:

The predicted offspring mean is:

Notice the pleasing internal consistency: the covariance value used to compute the regression slope (12.6 mg²) numerically matches the predicted response (12.6 mg) in this constructed example, illustrating that both formulas are drawing on the same underlying additive genetic covariance structure, though real data sets will not show this exact numeric coincidence, since $S$ is generally chosen by the breeder while covariance is a fixed population property.

Example 2: Realized heritability with an asymmetric, multi-generation selection experiment. A poultry geneticist selects for increased 8-week body weight in chickens over four generations, imposing unequal selection intensity each generation because only limited hatchery capacity was available in some years. She records:

Cumulative selection differential: $210+95+260+140 = 705$ g. Cumulative response: $68+34+79+51 = 232$ g. Realized heritability from the cumulative ratio:

Compare this to a naive average of the per-generation ratios, which would be:

The two approaches agree closely here (0.329 vs 0.337), but the cumulative-regression method is generally preferred because it weights generations by the actual amount of selection applied, rather than treating each generation's noisy small-sample ratio as equally informative; a generation with weak selection (generation 2, $S=95$) contributes a noisier ratio and should carry less weight, which the cumulative method achieves automatically. If a fifth generation showed an unusually large $R$ relative to a modest $S$, that might indicate an environmental trend (better feed formulation, warmer housing) rather than a true jump in heritability, precisely the kind of artifact an unselected control line is designed to catch.

Douglas Scott Falconer (1913–2004) trained originally in zoology at Oxford and turned to mouse genetics after war service, joining the Institute of Animal Genetics in Edinburgh, where he built one of the most influential quantitative genetics programs of the twentieth century. Falconer's long-running selection lines for mouse body weight, maintained with meticulous unselected controls, provided some of the cleanest experimental confirmations that the additive genetic model could make accurate, falsifiable, quantitative predictions about evolutionary change under selection. His 1960 textbook, Introduction to Quantitative Genetics, systematized the algebra of variance components, parent-offspring regression, and the breeder's equation into the standard curriculum still taught today, and the notation and worked derivations used in this article descend directly from that synthesis, itself building on foundational work by Sewall Wright and Ronald Fisher on the partitioning of genetic variance.

Plant and animal breeders use the breeder's equation daily to design selection programs: dairy cattle breeding schemes estimate $h^2$ for milk yield, fat content, and fertility from large pedigreed populations, then calculate expected genetic gain per generation to decide which sires to use for artificial insemination, directly trading off selection intensity against generation interval and heritability in the "breeder's equation" extended form, $R = h^2 i \sigma_P / L$, where $i$ is selection intensity and $L$ is generation length. Crop breeders apply the same logic to yield, disease resistance, and drought tolerance, and the century-long Illinois maize oil experiment remains a live reference dataset for studying how additive variance is depleted and replenished over deep time. In evolutionary biology, field biologists have applied parent-offspring regression to wild pedigreed populations, most famously Darwin's finches on the Galápagos (studied by Peter and Rosemary Grant) and red deer on the island of Rum, to estimate $h^2$ for beak size, body size, and antler traits, and to test whether observed evolutionary responses to natural selection events (such as droughts altering finch beak size) match the predictions of the breeder's equation, a direct test of natural selection as a quantitative, predictive process rather than a purely qualitative narrative. In conservation genetics, heritability estimates inform whether a threatened population retains enough additive genetic variance to adapt to environmental change (climate shifts, novel pathogens), guiding decisions about genetic rescue and captive breeding management. In human genetics and medicine, narrow-sense heritability estimated from family and twin data (and increasingly from genome-wide SNP data, a modern molecular echo of the classical resemblance methods) shapes how researchers interpret risk for complex diseases such as type 2 diabetes and hypertension, motivates the search for the underlying loci through Quantitative Trait Loci and QTL Mapping, and cautions clinicians against overinterpreting population-level heritability as individual genetic determinism, a caution that follows directly from understanding what parent-offspring regression and the breeder's equation actually measure, variance among individuals in a specific population and environment, not fixed, universal causal fractions, a distinction that also connects to how Genetic Variance Components are partitioned in the first place.

Ask the tutor: How does genome-wide SNP data let researchers estimate narrow-sense heritability without needing known pedigrees at all?