Rich, thanks for those comments. I have a few remarks in reply.
>If you have a criterion (reaction time, etc.) where you average dozens
>or hundreds of observations to make a point to be analyzed, the
>"effect size" is magnified by averaging. That is, if you can change
>an average by .01, that fraction is a lot bigger fraction of the
>between-Subject variance (of averages) than it is of the
>between-trial variance.
Effect size in experiments or cross-sectional studies is the change
or difference in the mean of something expressed as a fraction or
multiple of the between-subject standard deviation. As such, it is
not biased by sample size. Some people erroneously use the standard
deviation of the change score or of the residuals in experiments, or,
as Rich appears to be suggesting, the standard error of a mean. Is
that what you mean, Rich? I don't quite follow you. Between-trial
variance is not an issue here.
Actually, the effect size is biased a little high for small samples.
For example, the difference in IQ between a group of 10 boys and 10
girls, expressed as a fraction of the standard deviation of the boys
or the girls (or the appropriate average thereof), is on average
about 5% higher than the population value, I think, according to one
formula I found. As far as I know, this bias arises from the fact
that the standard deviation is itself biased for small samples. Yes,
it's true! The variance is unbiased, so the SD is biased.
The within-subject standard deviation (usually the SD of the
residuals) is also actually the appropriate denominator for studies
of athletic performance, but that is a specialized application that
few people know about. In the usual population studies, the
appropriate SD is the between-subject within-group SD, or an
appropriate average thereof when there are several groups.
> > So my question is this: should we take the square root of
>> heritability to get an idea of the contribution of inheritance to a
>> particular trait?
>
>I think "to get the idea" you have to be engaged comparing several
>traits. What are you comparing them on? What is it supposed to
>relate to? - If you want something that is *interval* (better by how
>much) rather that *ordinal* (which is better), that is not trivial.
>
>The underlying question is: What is your basis of defining linearity?
>The genetic contribution, in at least one sense, is linear in the
>squared term. But there is probably another sense where the square
>root fits better.
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.
Will
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