# Are there two different senses of the word 'heritability' in use?

The heritability value was taken as the regression coefficient between final height and midparental height

Is this the same sense of the word 'heritability' that is in use in https://en.wikipedia.org/wiki/Heritability and in answers on this site, namely Var G/Var P? Or are there two different senses in use simultaneously?

Edit: see https://www.jstor.org/stable/2531017 ... heritability h^2 is defined as follows:

Note the superscript (1/2) in the equation, indicating a square root. So the slope of the regression does not seem to be a ratio of variances.

FWIW, Wikipedia cites the S.E.P., which cites Behavioral Genetics by Plomin.

• Commented Oct 26, 2023 at 17:09
• Does this answer your question? Why does the slope of parent-offspring regression equals the heritability in the narrow sense? Commented Oct 26, 2023 at 17:12
• Actually this question answers your question. Basically if you do a bunch of math, the slope reduces to the (narrow-sense) heritability. Var(G)/Var(P) is usually broad-sense heritability, but it's hard to do infinitesimal math with the broad sense heritability. Commented Oct 26, 2023 at 17:14
• @MaximilianPress I'm not sure that it does answer my q. It cites web.archive.org/web/20210420073545/http://userwww.sfsu.edu/efc/…, which states (without proof or reference) that "The slope of regression of offspring on parents will tell you about resemblance between relatives, h^2". The maths in the Q/A don't add anything to that assertion. Commented Oct 26, 2023 at 21:58

The page you cite https://en.wikipedia.org/wiki/Heritability expresses the general concept in the first sentence:

Heritability is a statistic used in the fields of breeding and genetics that estimates the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population.

This is a conceptual definition, not a mathematical one. Variance is only one statistical measure of variation; see also https://stats.stackexchange.com/questions/88348/is-variation-the-same-as-variance

I would consider a statement like "Heritability = H2 = Var G/Var P" to be not a different sense but rather an operational definition that says "in this context, we'll use this mathematical representation of the concept".

There are multiple ways to estimate or compute heritability, some of them addressed in that same article under the heading "Estimating heritability", including methods based on correlations and regression. For a simple, univariate relationship, the Pearson correlation r and the coefficient of determination are related, in that the coefficient of determination is r2 and the same as R2 and the same as the "variance explained".

• So... when you see in a paper a statement like "___ has a heritability of 0.7", how on earth are you meant to make sense of that? What about meta-analyses which look at the heritability of different traits? Commented Oct 26, 2023 at 21:59
• @Mohan In a paper producing numbers, they should be clear about what the number means; in a sense, it's a lot like how you could say something "weighs 60", but this isn't really something you can make sense of, they need to provide the units, too, so you know they're talking about 60 kg or 60 g or 60 lbs. In this case, there are no units, but you still need to know what the quantity is to make sense of it, and especially if some people are giving you a square-root version of what other people are giving you (e.g., r rather than r^2). Commented Oct 26, 2023 at 22:06
• Thank you. Having looked into this further, I think what's confusing in this case is that not just the term 'heritability' but also the specific expression h^2 is being used to denote two different quantities, one of which is the square of the other. [See my edit to my Q.] Sometimes a single paper seems to use both senses -- see e.g. the Plomin textbook I cited. I almost wonder whether someone looked at eq(5) in the paper I quoted and missed the square root symbol, and then that error has propogated through the literature? Commented Oct 26, 2023 at 22:20
• @Mohan I don't know enough about the history of the term to know why h^2 is used; on the statistics side, R^2 is useful because it's interpretable as the fraction of variance explained, and also because it doesn't make sense for sign to enter into it; a squared quantity will always be positive, when you take a square root it's ambiguous whether you started with positive or negative numbers. Commented Oct 26, 2023 at 22:28
• Another place it comes up a lot for some overlapping reasons is in variances versus standard deviations; standard deviations are useful because they are on the measurement scale. Variances are useful because that's the scale on which variation sums. Commented Oct 26, 2023 at 22:30