# Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability?

I'm working on a problem where I'm given that the phenotypic variance of the parent, $p$, for trait $A$ is $175.2$ and the phenotypic variance of the offspring, $o$, for trait $A$ is $146.1$. However, I'm also given that the covariance between parent and offspring of a different trait, $B$, is $14.6$.

Can I still calculate the narrow-sense heritability of $A$ using $$\begin{eqnarray} \frac{h^2(A)}{2}&=&\frac{\text{Cov}(p,o)}{\text{Var}(p)}=\frac{14.6}{175.2}\approx0.0833 \\ h^2(A)&=&2\times0.0833\approx0.167 \end{eqnarray}$$ even though they're different traits?

I'm also told that any resemblance between parents and offspring is completely due to genetics, ie. without environmental effects. I've read that if this is the case, then $h^2$ is $0.50$, but I'm a little confused with it.

Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability?

Can I still calculate the narrow-sense heritability of AA [..] even though they're different traits?

No, you can't. A measure of heritability makes sense only for

• a given trait
• a given population
• at a given time
• in a given environment

[I'm also told that if any resemblance between parents and offspring is completely due to genetics, ie. without environmental effects, then $h^2$ is 0.5, but I'm a little confused with it.]

The above is not a exact quote from you but the original sentence made no sense. I rephrase what I suppose you tried to say.

The formulation is inaccurate and could therefore be misleading. It is wrong to say that any covariance between parent and offspring is only due to genetics. In reality, environments between parent and offspring often covary. There might also have covariance due to epigenetic reasons and potentially many other reasons.

You cannot make any claim about what value should $h^2$ takes for the special case where covariance between parent and offspring is solely due to genetic variance (or any other special case considering only genetic variance). You still need to know the relative importance of the genetic variance to the total phenotypic variance. If any of this is unclear, then you should keep on with the recommended reading below.