What it does
The breeder’s equation predicts how much a population changes in one generation of selection:
R = h² × S
R is the response (how far the offspring mean moves), h² the narrow-sense heritability (the share of the variation passed on through additive genes), and S the selection differential (how far the selected parents’ mean is above the population mean).
If you know what share of the population you keep rather than S, the tool works S out for you: S = i × σP, where σP is the phenotypic standard deviation and i the selection intensity. For a trait with a normal distribution and the best proportion p kept, i = φ(z) ÷ p, where z is the point that cuts off the top p and φ is the height of the normal curve there. Keeping the best 10 % gives i = 1.755; keeping 1 % gives 2.665.
The tool also gives the response per year (R ÷ the generation interval in years), which is how breeding programmes with different cycle lengths are compared, and, if you give the mean, the new mean and the gain as a percentage.
The second card estimates heritability from your own data:
- Offspring on parent regression: the slope b of offspring values on mid-parent values is h²; on one parent, h² = 2b. You get the standard error, a 95 % interval and a p-value.
- Variance components: plants measured in groups. For lines or clones the between-group variance is genetic, giving broad-sense H² (on a single-plant and a line-mean basis). Half-sib families give h² = 4σ²f ÷ σ²P; full-sib families give 2σ²f ÷ σ²P, an upper limit.
How to use it
- Enter h², and either the proportion selected and σP, or S directly.
- Pick Both parents if you choose plants before they flower and cross only among them; One parent only if you choose seed parents after open pollination (the response is halved, because the pollen is unselected).
- Enter the generation interval and, if you like, the population mean.
- For heritability, pick a method and paste or open your data. Use this h² above copies the estimate into the first card.
Worked example
Press Use the example: h² = 0.4, the best 10 % selected from a made-up population with mean 100 and σP = 8, both parents selected, 2 years per generation. The page shows i = 1.7550, z = 1.282, S = 14.040 and R = 5.616 per generation, which is 2.808 per year; the offspring mean is 105.616, a gain of 5.62 %.
The example parent-offspring data (20 made-up mid-parent and offspring plant heights) give a slope of 0.4461 ± 0.1665, so h² = 0.446, with a 95 % interval of 0.096 to 0.796 and p = 0.015. Switch the method to lines or clones and the six made-up inbred lines give H² = 0.844 on a single-plant basis and 0.954 on a line-mean basis.
FAQ
Why is my h² above 1? Sampling error with few families, or groups that are not the relatives you chose. Heritability cannot be above 1; collect more data.
Does the response stay the same every generation? Only roughly, over a few generations. Selection uses up variation and changes gene frequencies, and h² is specific to one population and environment.
Is my data uploaded? No. Everything runs in your browser and works offline.
Where the numbers come from
- Breeder's equation, selection intensity for truncation selection, parent-offspring regression and variance-component heritability as in Falconer and Mackay, Introduction to Quantitative Genetics (4th ed.), chapters 10 and 11; n0 for unequal group sizes from the one-way analysis of variance (Snedecor and Cochran).
- Checked against scipy (selection intensity norm.pdf(norm.ppf(1 − p))/p for p from 0.1 % to 90 %; linregress for the regression slope, its standard error and p-value) and statsmodels anova_lm for the unbalanced variance-component example.
- The example population, parent-offspring pairs and line heights are made up.
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Last updated 2026-11-09.