100% h2 means all trait variation is of genetic origin. Among other things, it does not consider the trait mean value in any way. Anyway, h2 is more quantitative genetics issue. The probability sounds bit more like qualitative genetics question. Sometimes the difference is not very strict.

Forgive my nativity why wouldn’t probability be quantitative

>The way I read that 100% heritable corresponds phenotype dependent 100% on genes

This is not correct. It means 100% of the phenotypic variation depends on genes, which is quite different. As a classic example, the heritability of "leggedness" in humans is very low, close to 0. This is because when people don't have two legs, it is usually for environmental reasons (accidents, amputation etc.). However, I'm sure you would agree that having two legs is specified in the human genome.

Heritability is not a conditional probability as you describe it (I know of no such statistic, although I suppose you could empirically calculate one). It is more like a r^2 value in statistics if you are familiar with that.

Another note - h^2 is not necessarily fixed, and only applies in a particular environmental context. Change the environment, and you could in theory change h^2.

The key is that heritability is statistical, especially with a big population.

Imagine a trait like height, which is continuous and polygenic. Very tall people and very short people (absent any endocrine or developmental pathology) will mostly have alleles for tall and short, respectively, while average height people will have a mix. If your parents are very tall and very short, you'll get alleles from both sides and likely wind up average. Conversely, if your parents are both average, there's a slim chance you could inherit mostly tall or short alleles, but chances are you'll be average. If both parents are at a height extreme, though, variability is lower.

So to estimate heritability, you regress your height against the average of your parents' heights. With enough people, you get a cloud of points that looks like a football at an angle - sloped, mostly points in the middle, and the middle points are further away from the regression axis than either end.

The instances you're interested in are those in the middle of the graph, far off the axis. And they do exist, but, statistically are balanced out by the ones on the other side of the axis. I'm sure there's more math to be done, but that's where my expertise ends.

>It means 100% of the phenotypic variation depends on genes, which is quite different.

More precisely, it usually refers to share of the variation within the specific population being studied. For example, when measuring the heritability of height in a wealthy country, you will get a very high heritability estimate, perhaps 0.8-0.9. When measuring the heritability of height in a global population, you'll get a lower heritability estimate, because a significant fraction of your sample will have had their growth somewhat limited by environmental factors like undernutrition or disease. Conversely, if you're studying a population of clones, the heritability will be zero, because there's no genetic variation and all variation must be due to environment.

None of these estimates is more correct than the other, because heritability can only be defined for specific populations with specific distributions of genetic and environmental factors. There is no "ideal heritability."

>So to estimate heritability, you regress your height against the average of your parents' heights.

No, you can't estimate heritability that way, because this can't distinguish between genetic and environmental transmission of traits.

Traditionally, heritability is estimated with twin studies, using Falconer's formula. You compare the correlation between pairs of monozygotic twins to the correlation between pairs of same-sex dizygotic twins. You can exploit the fact that MZ twins are twice as genetically similar as DZ twins but MZ and DZ twins are raised in equally similar environments to determine heritability.

So if the MZ correlation is 0.7 and the DZ correlation is 0.4, this implies that 60% (2 * (0.7 - 0.4)) of the variation in the trait can be attributed to genetics, 30% (1.0 - 0.7) to non-shared environment (environmental factors that differ between twins) and the remaining 10% to shared environment (environmental factors that are the same for both twins).

There are some additional adjustments you can do for things like gene-environment correlation, but that's the simplified version.

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>It means 100% of the phenotypic variation depends on genes, which is quite different.

I think this is what I was trying to say, but let me restate it to make sure I am on the right track. 100 heritability implies variation in trait amongst a population is wholly accounted for by variation in genetics of population and there is no correlation between variation in environment causing additional variation in traits.

Probability of a qualitative (categorical) trait, no? Or what was the probability you asked, did I miss something? Now if you think of quantitative genetic trait... such as height... one could have probability of being above a given height or such. Disease traits are sometimes polygenic in one way (risk) while the trait can be categorical (sick or not).

>100 heritability implies variation in trait amongst a population is wholly accounted for by variation in genetics of population

Correct.

>and there is no correlation between variation in environment causing additional variation in traits.

I would phrase it as "environmental variation does not cause additional variation in the trait".

Thanks!

Oh I see. You mean Bernoulli trial. Interesting, would call a something discrete qualitative too or just Bernoulli?

When I asking about the conditional probability of trait given parents have trait, I was really thinking of something like height (continuous), but figured find out if such statistic exists in the simpler situation where we consider a yes or no trait.

>No, you can't estimate heritability that way, because this can't distinguish between genetic and environmental transmission of traits.

What do you mean by environmental transmission of traits?

Parent-offspring regression is definitely one way to estimate heritability. It has flaws and biases, but no more than estimates derived from twin-studies, which tend to overestimate narrow-sense heritability.

That would be maybe more like the expected height given phenotype or genotype of parents, or posterior probability of height in certain range? The oldest quantitative prediction is that progeny is the average of parents. Clearly hides a lot of variation under that simple mean. Maybe check things like estimated breeding value. Might help. Good luck.

Thanks

Are you under the impression that heritability of height is defined as the correlation between children's heights and the average of their parents' heights? Obviously you can determine that by calculating said correlation, but that's not what heritability means.

Heritability refers specifically to share of variation in a trait attributable to genetic variation. I suppose it's possible that there's some field other than genetics in which the term is used to refer to the degree to which children are similar to their parents, but the original question specifically referred to the definition used in genetics, and you definitely can't calculate that by comparing children to their parents. If you could, twin studies would never have been invented. That's the exact problem they were invented to solve.

Somewhat surprised that you're familiar with the concept of heritability, but not yet come across regression of offspring on mid-parents as a method to estimate it. For example:

Parent-offspring regression to estimate the heritability of an HIV-1 trait in a realistic setup

Comparing parent–offspring regression with frequentist and Bayesian animal models to estimate heritability in wild populations: a simulation study for Gaussian and binary traits