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I am trying to understand the heredity of intelligence between generations in general, and how regression to the mean works in detail in particular. The answer I'm looking for should preferably answer some of the following questions.

When looking at the genetic basis for intelligence, it seems to me that the additive contribution of the genes should not be subject to regression to the mean, and that regression to the mean should rather be due to recessive genes and emergenesis. Is this correct? If so, what do we know about the extents to which these two mechanisms contribute to intelligence?

Towards which mean will a specific child be expected to regress? I have seen various suggestions here, ranging from the mean of the racial groups of the parents, to that of the child's grandparents.

Is it even possible to write down an (approximate) equation for the expected intelligence of a child, in terms of for instance the intelligence of parents, siblings and gransparents?

If someone coould answer these questions, that would be awesome!

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    $\begingroup$ I don't think you need this question to be specific to intelligence - you are asking really about regression to the mean and additive genetic variation as a general problem $\endgroup$ – rg255 Jun 17 '16 at 11:24
  • $\begingroup$ As @rg255 said, this question has not reason to be specific about intelligence. I think the question is on-topic and not too broad but a very good source of information is the chapter "quantitative genetics" in A short guide to population genetic by Gillespie. $\endgroup$ – Remi.b Jun 17 '16 at 11:32
  • $\begingroup$ Thank you for your comments, and the reading suggestion. I agree that parts of this question could be answered in a more general framework, but for instance our knowledge of additivity vs emergenesis could be trait specific, no? $\endgroup$ – Daniel Jun 17 '16 at 11:42
  • $\begingroup$ Your comment make me think that you might want to start by reading the definition of heritability (see this post). I never heard of the term "emergenesis" before. Reading from wikipedia, it seems to be the equivalent of what we call epistasis in biology. I will try to formulate an answer a bit later today. $\endgroup$ – Remi.b Jun 17 '16 at 14:05
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I think the question mainly highlight either of two misunderstandings:

  1. misunderstanding about the definition of heritability
  2. misunderstanding why the slope of the regression line is equal to heritability (in the narrow sense).

Definition of heritability

Please have a look at the post Why is a heritability coefficient not an index of “how genetic” something is?.

Notation

Please note that, here I use $h^2 = \frac{V_A}{V_P}$ to designate the heritability in the narrow sense, where $V_A$ is the additive genetic variance and $V_P$ is the (total) phenotypic variance. If anything of what I just said is unclear, please refer to the aforementioned post.

Without any loss of generalization, we will set our scale so that the average phenotype in any studied case is equal to zero. From this starting value, we note X (with a subscript) the contribution of a given haplotype (or allele) to deviate the phenotype of an individual. We will note $E$ (with a subscript) the contribution of a given environmental state to deviate the phenotype of an individual.

An individual phenotype

Let's consider an individual O (stands for offspring, it will makes sense later). The phenotype of this individual is given by the addition of the contribution of each haplotype and of the environment. The expected phenotype of O (called $P_O$) is therefore

$$P_O=X_m + X_p + e_O$$

, where $X_m$ is the contribution of the maternal haplotype, $X_p$ is the contribution of the paternal haplotype and $e_O$ is the contribution of the environment.

Just a note

It might be worth skipping this note and coming back to it once you understood the below calculations (but I did not want to move this note to the end).

In the above calculation, one must assume absence of epistasis and dominance effect. I am not exactly sure how this might eventually cancel out later or whether results would then differ. I would welcome a follow-up in the comments to clarify this point but it might be worth its own post. As the question refers to the heritability of intelligence (assuming this term is defined), I would like to highlight that it is unlikely that we have much understanding about the exact pattern of epistasis underlying this trait as measure of epistatic interaction are often difficult to achieve.

Phenotype of O's parent

Let's note $P_P$, the phenotype of one parent of O'. The below notation applies to the mother but calculations would be the same for the father.

$$P_P = X_m + X_m' + e_P$$

, where $X_m'$ is the haplotype of the mother that has NOT been transmitted to the offspring and $e_P$ is the contribution of the environment experienced by the parent (mother).

Covariance between $P_O$ and $P_P$

The covariance between $P_O$ and $P_P$ (noted $cov(P_O, P_P)$ is just the sum of the covariances of all contributions, that is

$$cov(P_O,P_P) = cov(X_m,X_m) + cov(X_m,X_m') + cov(X_m,e_m) + cov(X_P,X_m) +\ cov(X_P,X_m') + cov(X_P,e_m) + cov(e_O,X_m) + cov(e_O,X_m') + cov(e_O,e_m)$$

I let you take a few minutes to think about the values of each of these covariances.... They are almost all equal to zero (are independent). There are only two covariances that (under a simple model) can potentially be different from 0; $cov(X_m,X_m)$ and $cov(e_O,e_m)$. $cov(e_O,e_m)$ is teh correlation between the environment experienced by the mother and the one experienced by the father. For simplicity, we will make the assumption that these two environments are independent and therefore $cov(e_O,e_m)=0$. The above, long and hairy equation therefore reduces to

$$cov(P_O,P_P) = cov(X_m,X_m) = var(X_m) = \frac{V_A}{2}$$

Correlation between $P_O$ and $P_P$

By definition the correlation (which is the slope of the regression line) is just the covariance divided by the total phenotypic variance. Therefore the correlation between $P_O$ and $P_P$ (noted $cor(P_O,P_P)$) is

$$cor(P_O,P_P) = \frac{V_A}{2} \frac{1}{V_P} = \frac{h^2}{2}$$

Generalization to any correlation

Following the above logic, one can show that in general the correlation between any two individual phenotypes $P_X$ and $P_Y$ is

$$cor(P_X,P_Y) = r h^2$$

, where $r$ is the coefficient of relationship. For parent-offspring $r=\frac{1}{2}$.

Particular case of mid-parent vs offspring relationship

This is just a study case to show that when consdiering a made-up phenotype such as a hypoethetical mid-parent phenotype, then the math are just a little bit more tricky. If you consider the mid-parent phenotype $P_{mid}$, then you must consider that the total phenotypic variance is half the actual phenotypic variance. The correlation therefore becomes

$$cor(P_{mid},P_O) = \frac{cov(P_{mid},P_O)}{\sqrt{\frac{V_P^2}{2}}} = \frac{h^2}{\sqrt{2}}$$

Conclusion

All of the above mean that if you plot the phenotypes of a series of individual pairs, where each pair is related by $r$, then the slope of the regression line is $r h^2$. For example, if you plot the phenotype of the mothers with their offspring, then the slope of the regression is equal to $\frac{h^2}{2}$. If $h^2=0.4$ (which is a typical value of heritability), then the mother-offspring regression will have a slope of $0.2$.

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  • $\begingroup$ This is a great answer. You could improve it by explaining some of the equations in more detail and give a conclusion in the end. $\endgroup$ – AlexDeLarge Jun 17 '16 at 16:44
  • $\begingroup$ I can probably make a conclusion from the last paragraph in ** Generalization to any correlation**. Can you tell me exactly what equation would deserve more details you think? Details about the logic behind the equation, the algebra or the underlying mathematical statistics such as the definition of a covariance? Thanks $\endgroup$ – Remi.b Jun 17 '16 at 16:48
  • $\begingroup$ Maybe conclusion was not the right term. I was pointing in the direction that it would actually make sense to not stop with a correlation equation but give a sentence or two what this verbally means with regard to the question asked. With asking for more detailed explanations I was thinking of the terms 'covariance' and 'correlation' that make sense to statisticians/population geneticists but maybe difficult to grasp for a more general readership. I also corrected a typo in the cor(P_o,P_p) section. Please re-check as I am not an expert in the field. :) $\endgroup$ – AlexDeLarge Jun 17 '16 at 17:04
  • $\begingroup$ Thanks a lot for the elaborate answer! As Alex observed, things got a bit abstract though, so before I accept could I perhaps ask you to help me work out an example? Say, if h^2=0.6, and we know that both parents are 2 standard deviations above the mean, if I understand things correctly that would make the expected value of the child 0.6*2=1.2 standard deviations above the mean, correct? What about if we have the additional information that all grandparents are 1 standard deviation above the mean? How would the expected value change? $\endgroup$ – Daniel Jun 18 '16 at 0:15

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