# Interpretation of narrow-sense heritability over one (using R/S = h^2)

Here is my data:

Mean height score of the total parental population: 5.2

Mean height score of selected parents (those chosen for breeding due to their higher height): 6.4

Mean height score of the offspring (resulting from the selected parents): 7.8

Selection differential (S) for height: S = Mean height score of selected parents - Mean height score of the total parental population S = 6.4 - 5.2 = 1.2

Response to selection (R): R = 7.8 - 5.2 = 2.6

h^2 = R / S

h^2 = 2.6 / 1.2 = 2.167

I confirmed that the data are not wrong. What is the interpretation of this h^2 value?

I am looking for explanations for why h^2 would be above one.

• Simpler explanations: 1) noisy phenotyping. 2) (my bet) environmental effects in the selected population, unless you somehow can exclude this via a common garden with parents and offspring raised side by side at the same time under the same conditions. Overall this smells like an experiment where something went wrong. "Confirmed the data are not wrong" not sure what this means, you mean you went back and recalculated? Apr 1 at 18:11

It looks like you're using a formula that assumes linear, additive variance. Your data suggest a system that does not follow these assumptions, so the equations are not valid for the situation you describe.

See also https://en.m.wikipedia.org/wiki/All_models_are_wrong - much of the difficulty in applying simple mathematical formulae to real world data is in deciding/recognizing when your model is sufficiently wrong that it's no longer useful.

• What could explain that? Dominance? I thought of that, but I assumed that would decrease h^2, not increase it Feb 29 at 4:31
• @BigMistake I would start by assuming the trait is simply nonlinear. No reason that things have to be linear in the real world just because it's sometimes convenient to model. Feb 29 at 4:38
• They don't, they make the equation you are using to calculate it invalid. Feb 29 at 5:36
• Because you're using an invalid equation to calculate it because the assumptions that make the equation hold are not met - that's why it's too high. R is a bigger response than is possible under the model that makes that equation hold. Feb 29 at 13:07
• R is large. That's it. In another situation with violated assumptions it could be small instead. The direction isn't important what is important is that it's not meaningful. If you tried to calculate how much money you have in the bank by multiplying what you deposit by what you spend you get the wrong answer because that equation is wrong. It's not important to explain why the result is too big or too small, it's just not meaningful to use that equation in that situation. Feb 29 at 18:55

From this paper discussing the pitfalls of using the breeder's equation:

The breeder's equation provides a useful framework for conceptualizing the process of adaptive evolution by natural selection: selection causes phenotypic changes in a population, and genetic variation transmits these changes to future generations. This is not wrong, but given the assumptions we have discussed, it may generally be very inappropriate to apply this framework as a predictive tool in nature. There is consequently a problem with the enthusiastic way in which evolutionary biologists, ourselves included, have transferred this model from being a tool to conceptualise trait evolution and analyse artificial selection experiments (a context in which the inherent assumptions can be more readily approximated) to a predictive model in the field. Estimates of selection based on selection gradients are undoubtedly a conceptual step forward, but they do not escape reliance on the assumption that all traits and environmental factors that jointly influence studied traits and fitness have been identified, meaningfully measured and adequately modelled.

So if this is indeed an artificial selection experiment and you have actually designed the experiment to control for all possible sources of e.g. environmental variation, then it may be appropriate to use the breeder's equation model. However, we don't know anything about the source of your data, so it's difficult to evaluate whether your data are from an adequately controlled e.g. common garden style experiment.

Moreover:

No model can be expected to provide accurate predictions if its assumptions are seriously violated. The simple approaches we discuss here for predicting evolutionary change actually make many simplifying assumptions (e.g. constant population demography, discrete generations, constant environmental conditions; Merilä et al., 2001a), and we are certainly not the first to highlight the potential for predictions to fail when when these models are applied to natural systems (Hadfield, 2008; Price et al., 1988; van Tienderen & de Jong, 1994) or to call for caution in this regard.

They are clearly focusing on natural populations, which I don't think you are. But on the off chance that your "selected" parental population is not selected explicitly on your height measurement, then that is probably the reason why your estimates are wonky.

If you are in fact doing an artificial selection experiment, your estimates are probably wonky because some of those bolded factors were not controlled for.

For example, if these are bean plants, the parental and offspring generations may have been grown in different years with different climatic conditions. Even in a laboratory environment with well controlled conditions, unless the parents and offspring are grown at literally the exact same time side by side (which presents obvious logistical challenges) with positional rotation and constant care, it can be hard to fully reject the hypothesis of some unknown environmental factor impacting your data.