Re: st: RE: analysis of mixture experiments

[Austin Nichols <austinnichols@gmail.com>]

Dan Kahan <dmkahan@gmail.com>:

No special techniques are required; just include the proportions as
regressors (the abbreviation IV for RHS vars or covariates is frowned
upon in some circles, since it ordinarily stands for Instrumental
Variables in those circles).  But the interpretation may be a bit odd,
or you may have to come up with some clever marginal effect
calculations, as it no longer makes sense to speak of the effect of X1
on E(y|X) holding all other X constant--an impossibility for a set of
X vars that have a fixed sum (e.g. sum to one).  Now there are
infinite ways to model the effect of a one percentage point increase
in X1, for example increase X1 by .01 and decrease X2 and X3 by .005,
or increase X1 by .01 and decrease X2 by .01, and so on.  You could
write a little routine that models the effect of each covariate by
incrementing that one and decrementing each other one in proportion to
its current level, at which point marginal effects differ by
observation and you can employ the mean marginal effect or any other
measure discussed in -help margins-.

On Thu, Sep 23, 2010 at 7:27 AM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
> You are correct. I am so used to seeing similar questions about response variables that I missed your very clear statement than the problem is on the other side.
>
> There is a literature on _compositional data analysis_ that may help. Google that term for some references, including much material on the internet.
>
> John Aitchison suggested various transformations for bundles of compositional variables. A while back I wrote Mata code for some, which I don't seem to have made public. Examples follow my signature and may serve at a minimum to show that they are straightforward to compute.
>
> John A. Cornell has books on mixtures. Go to the Wiley website and search for "Cornell mixtures".
>
> The main problem with most of the multivariate transformation methods I have seen is what to do with observed zeros for any of the components. Much of the compositional data analysis literature deals with geological examples in which it is plausible that an observed zero falls just below some detection limit and that it should be fudged upwards. Most of the examples I have looked at in my own fields of interest are not quite so simple and zeros often appeal to be real (exact, essential, structural, fixed).
>
> Nick
> n.j.cox@durham.ac.uk<snip>

> Dan Kahan
>
> thanks. I know dirifit; I am very fond of it. But here the proportions are my
> IVs, not the DV, which is a continuous variable (one to which I would
> ordinarily fit an OLS linear regression, except that that seems
> intuitively wrong to me where my IVs are proportions).

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