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From Wikipedia:

Heritability estimates are often misinterpreted if it is not understood that they refer to the proportion of variation between individuals on a trait that is due to genetic factors. It does not indicate the degree of genetic influence on the development of a trait of an individual. For example, it is incorrect to say that since the heritability of personality traits is about .6, that means that 60% of your personality is inherited from your parents and 40% comes from the environment.

So what is the plain english interpretation of heritability?

Is it supposed to be the genetics version of $R^2$? If so, please explain it to me anyway because I forgot my statistics.

I found this so from what I understand, one might say:

If height is estimated to be 80% heritable, and if we get 10 people, get their heights and then compute the variance, we say that around 80% of that variance is likely due to genetic factors while the remaining 20% is due to other things.

Such interpretation of heritability seems to work for non-categorical traits. What about categorical or binary traits?

Hair and eye colour is categorical (I think?). So what, we assign a number for each colour and then compute the variance?

ADHD is binary (I think?). Do we assign 0 for not having and 1 for having then compute variance?

I seem to recall regressing binary or categorical variables needs some kind of adjustments and hence interpretations of estimates (such as intercepts or slopes) may be different.

Also, the book linked above says

heritability represents the degree to which the variance in a trait is attributable to genetics in the population on average

Combining that with the Wiki paragraph above, I don't think someone saying

Eye colour is around 98% heritable. I have brown eyes. Therefore, around 98% of the reason why I have brown eyes is genetics.

is far off from saying

If we have a million people who near-identically flip a million near-identical coins a hundred times, and we compute the average of the coin flips that turn out to be heads to be 58.6, then the probability that you will flip heads when you near-identically flip this near-identical coin is around 58.6%.

Is it? (Of course 'near', 'around', etc are not always used similarly...lol)

So yeah, technically the 60/40 interpretation in the Wiki example is technically wrong but practically it's around 60% $\pm$ some standard deviation?

BCLC
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    Possible duplicate of Why is a heritability coefficient not an index of "how genetic" something is? – Remi.b Feb 08 '16 at 01:23
  • @Remi.b $R^2$ is there? – BCLC Feb 08 '16 at 01:30
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    No, it is true that the other post does not explicitly talk about the relationship between heritability and it's measure in parent-offspring regressions... I haven't decided yet how I would deal with that. I might retract my vote close soon but I must read your edit first :) – Remi.b Feb 08 '16 at 01:34
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    I would advice that you restrict your question to the regression interpretation of heritability and ask separately about the heritability of a non-quantitative trait. – Remi.b Feb 08 '16 at 02:31
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    Heritability (as shown a long time ago by Galton) is the slope of the parent-offspring regression. However, I don't understand it well enough for the moment to formulate an answer. +1 btw – Remi.b Feb 08 '16 at 02:31
  • @Remi.b Are you sure it is slope? Since it is a ratio of variances, it sounds closer to an $R^2$, though some manipulations might give you slope – BCLC Feb 08 '16 at 16:51
  • Yes I am pretty sure. I would expect both the slope and the $R^2$ to get closer to 1 as heritability increases. I would vote to close as too broad until you reduced you post t only a single question (currently you talk about the interpretation of heritability as a regression and about the case of non-quantitative traits). Note that you might want to talk about which regression (which variables) you are referring to. – Remi.b Feb 08 '16 at 17:55
  • @Remi.b approaching 1 through values from the left? Anyway, go ahead and close for now, I guess. I'll split up and edit later on. Thanks ^-^ – BCLC Feb 08 '16 at 19:36
  • What do you mean "approaching 1 through values from the left"? If you agree that the question should be closed, you should probably just delete it (or vote to close as duplicate). – Remi.b Feb 08 '16 at 19:46
  • @Remi.b I'll edit it later. I'm asking if you mean for both slope and R2 that we have 0.8, 0.9, 0.99, 0.999, etc rather than 1.1, 1.05, 1.01, 1.001, 1.00001, etc or 0.9, 1.1, 0.99, 1.0001, 0.999, etc – BCLC Feb 08 '16 at 19:48
  • Yes from the left. You should not have a slope steeper than 1 and it makes no sense to have a correlation coefficient greater than 1. You seem to describe the dynamic of a system (especially when seeing to hypothesize cyclic behaviour) though and I don't get it. – Remi.b Feb 08 '16 at 19:51
  • @Remi.b R2 and correlation coefficient I get. But why can't we have a slope steeper than 1? Is it a bio thing? I seem to recall slopes steeper than 1 in areas outside bio – BCLC Feb 08 '16 at 20:05
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    Yes it is a bio thing! For a parent-offspring regression, only sampling error could cause slope to be higher than 1. In the whole population the slope is bounded between 0 and 1. We will have to delete all these comments once you've edited your post. – Remi.b Feb 08 '16 at 20:31
  • @Remi.b Thanks! ^-^ What am I supposed to do again? Not really into Bio SE hehe – BCLC Mar 04 '18 at 06:23
  • The post Why does the slope of parent-offspring regression equals the heritability in the narrow sense? can help. Otherwise, I would recommend you having a look at the section on quantitative genetics of Gillespie's book. – Remi.b Mar 04 '18 at 21:04

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