I found something regarding heritability in Frank (1997) that I think deals with the artificially high covariance term. He starts with the other definition of heritability:

B_g’z = Cov(g’,z)/Var(z),

where g’ is the breeding value transmitted to offspring, and z is the phenotype of the parent. He showed that this could be further partitioned into

B_g’z = B_g’g * B_gz = (Cov(g’g) / Var(g)) * (Var(g) / Var(z)).

Thus, heritability has two components, the fidelity of transmission of breeding value B_g’g, which will be ~1 for clonal cells (like I said), and B_gz, which will be ~0 for clonal cells (like you said), since there is little genetic variance.

Ref:

Frank (1997) The Price Equation, Fisher’s Fundamental Theorem, Kin Selection, and Causal Analysis

Thanks very much for the blog post and your answer. I’ve been doing some research and I’ve managed to derive something that looks like GP^{-1}, using the ‘effective’ heritability. As you said, I’ve had to use least squares to get something useful out! I’m a bit rushed at the moment so don’t have time to properly respond to the ideas in your blog post, other than to say they sound interesting! I will have a proper read next week when I’m back off holiday and get back to you.

Thanks again,

Michael

]]>I didn’t actually answer your specific question in my blog post. Basically, yes, in general matrices can be added together. If you want it to be meaningful they need to be orthogonal, however. This is why you need to do some form of least squares partitioning of the genetic effects. When you add them, you simply add the matching elements.

]]>I thought that a detailed response to your question was justified, and I upgraded it to a full blog post:

]]>Thanks very much for your response. I’m currently trying to get my head around all these different definitions and what you said made a lot of sense. I’m wondering, do you know how the ‘effective’ heritability fits in with the framework of quantitative genetics? If I have multiple phenotypic traits of interest, the response to selection is

delta z = GP^-1 s

where G is the genetic variance-covariance matrix, P is a phenotpic variance-covariance matrix and s is a vector of trait-fitness covariances. I suppose the GP^-1 part here corresponds to heritability of the ‘Fisherian’ kind as you defined it? Is it legitimate to substitute G for a matrix of parent-offspring covariances to arrive at something of the ‘effective’ kind as I defined it?

The reason I ask is that I’m currently working on developing a multicellularity model, and as you said the VA isn’t well defined in my model for within-organism selection.

Best,

Michael

]]>Yes, It is an interesting problem. Consider, in a world with no somatic mutations there is a very large phenotypic variance in the cells of a mammal (for example), ranging from liver cells, to nerve cells, to skin cells. However, since they are all genetically identical there is no additive genetic variance. Thus, the heritability is h^2 = VA/VP = 0/VP = 0. So using heritability in the sense defined by Fisher the heritability is zero. With somatic mutations that will presumably add a little VA, so it will be non-zero but small.

The problem of course, is that Fisher’s definition of VA doesn’t really apply to non-sexual organisms, which is what a somatic cell is. What you are using is more consistent with a phenotypic definition of heritability, which is the extent to which offspring resemble their parents. In that case, yes, offspring cells do tend to resemble their parents, although that resemblance has almost nothing to do with genetics.

I sometimes call these more general definitions of heritablity “effective heritability”. Fisher defined heritability, and as Ewens would say, if he defined it, his definition is the correct one. That definition is the covariance between the average effect and the average excess. Given that Fisher’s definition is useless in this situation we are stuck with effective heritability, and since you and I were thinking along different lines at the time we each used different definitions without being explicit about what those definitions were. My definition was the genetic variance divided by the phenotypic variance, yours was the parent offspring covariance divided by the phenotypic variance. Mine was more useful for the point I was trying to make, but honestly, I will admit that yours is more correct.

Thanks for bringing up this important point, and in the process clarifying my thinking.

]]>Please could I just clarify something you say in this piece, as it relates to something I’m working on at the moment. You say:

‘From the perspective of individuality, what this does is that it lowers the heritability at the cellular level to nearly zero.’

This confused me, since the heritability at the cell level via mitosis is nearly one, not nearly zero, isn’t it? If we take h^2 = Cov(zi,zi’)\Var(zi), where zi is parent cell phenotype, and zi’ is offspring cell phenotype (we have regressed parent phenotype against offspring phenotype and taken the gradient of the regression line to be the heritability). Assuming high fidelity, we have Cov(zi,zi’) approx = Cov(zi,zi) = Var(zi). Putting this back in we get h^2 = Var(zi)/Var(zi) = 1, and thus h = 1.

Thanks,

Michael

]]>Thanks. I am a huge fan of Wright’s work, as should be obvious. I only had the privilege of meeting him on a few occasions.

]]>“Where m is less than 1/2N there is a tendency toward chance fixation of one or the other allelomorph. Greater migration prevents such fixation. How little interchange would appear necessary to hold a large population together may be seen from the consideration that m=1/2N

means an interchange of only one individual every other generation, regardless of the size of the subgroup. However, this estimate must be much qualified by the consideration that the effective N of the formula is in general much smaller than the actual size of the population or even than the breeding stock, …”

pp. 126-127 in 1931 “Evolution in Mendelian Populations” Genetics 16:97-159. ]]>