... measured.¹

Assuming that we've randomized siblings across environments. If we haven't, siblings may resemble one another because of similarities in the environment they experienced, too.

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... measured.²

You'll see the reason for the quotes around genotype in this paragraph and the last a little later. It's a little more complex than what I've suggested.

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... familiar.³

Remember our mother-offspring combinations with Zoarces viviparus?

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... produce.⁴

To check your understanding of all of this, you might want to try to produce the appropriate table.

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....⁵

$\sigma^2_W$, $\sigma^2_D$, and $\sigma^2_S$ are often referred to as the observational components of variance, because they are estimated from observations we make on phenotypic variation. $V_a$, $V_d$, and $V_e$ are often referred to as the causal components of variance, because they represent the gentic and environmental influences on trait expression.

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... half-sibs.⁶

To see why consider this is so, consider the following: The mean genotypic value of half-sib families with an $A_1A_1$ mother is $px_{11} + qx_{12}$; with an $A_1A_2$ mother, $px_{11}/2 + qx_{12}/2 + px_{12}/2 + qx_{22}/2$; with an $A_2A_2$ mother, $px_{12} + qx_{22}$. The equation for the variance of these means is identical to the equation for the covariance among half-sibs.

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... groups.⁷

With $x_{ij} = a_i + \epsilon_{ij}$, where $a_i$ is the mean group effect and $\epsilon_{ij}$ is random effect on individual $j$ in group $i$ (with mean 0), $Cov(x_{ij},x_{ik}) = E(a_i + \epsilon_{ij} - \mu)(a_i + \epsilon_{ik} - \mu) =... ...u^2) + a_i(\epsilon_{ij} + \epsilon_{ik}) + \epsilon_{ij}\epsilon{ik} = Var(A)$.

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... you.⁸

What did you expect from real data? This example is extracted from Falconer and Mackay, pp. 169-170. See the book for details.

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... environment.⁹

Notice that this doesn't affect our analysis of half-sib families, i.e., the progeny of different sires, since each father was bred with several females

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....¹⁰

See Falconer for details.

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