>
>I think I've resolved this question with a colleague. We likened the
>heritability of a given trait, for example, jump height, to the
>relationship between that trait and some other explanatory variable,
>such as leg length. The R^2 for leg length explaining jump height
>might be 0.36. Now, 0.36 is a gross underestimate of the effect of
>leg length on jump height. We should use root(0.36), i.e. 0.60,
>because, if you experimentally change leg length by one standard
>deviation with growth hormone during development, or if you move from
>one individual to another who differs by one standard deviation in
>leg length, you will find that jump height changes or differs on
>average by 0.60 of a standard deviation for jump height. Trust me,
>it's true. 0.60 is large on Cohen's scale, whereas 0.36 is moderate.
>
>Now let's bring in heritability. For the sake of simplicity, let's
>assume leg length is entirely inherited and is the only inherited
>factor explaining jump height. Therefore we would find that the H^2
>for inheritance explaining jump height is 0.36. So we should
>interpret H, but not H^2, when we talk about magnitude of
>heritability, and we should do it in the following way: there is
>some variable, the values of which are determined by heredity, such
>that a change in one standard deviation of the variable results in a
>change of 0.6 standard deviations in jump height. Of course, there
>are lots of variables contributing to jump height, but you can
>combine them into one composite virtual variable for the sake of
>understanding what H means.
>
I have been looking at Cleveland's "Visualising Data" and he introduces
an "residual-fitted"/"r-f" spread plot that I'd like to see next time I
read an article on hereditability. This is two graphs side by side on
the same scale. One is a quantile plot of (fitted values - mean fitted
values). The other is a quantile plot of residuals. Comparing these two
plots gives you a very good idea of how predictive the fit is. It looks
a bit like this:
---- Fitted ---- |-- Residuals --
* | *
* | *
* | *
* | *
* | *
* | *
* | *
* |*
--------------- ----------------
Now if we had Y = X + E where Y is N(0, 1) and E is N(0, 1)
we would be plotting fitted Y = X and the residual = E, so
we would plot two identical graphs - as above, except that they
wouldn't be straight lines, supporting an R^2 measure, which
says 1/2, which does fit the symmetry of the situation. If you say there
is really 1/sqrt(2) of relationship here when you know X but not E, then
it looks a bit odd when you realise that there is another 1/sqrt(2) of
relationship left if you know E but not X.
--
A. G. McDowell
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