RE: st: RE: analysis of mixture experiments

[Nick Cox <n.j.cox@durham.ac.uk>]

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 

// compositional data analysis 

mata : 

mata drop cda_*() 

// NJC 1 Sept 2008 
// rows scaled to sum to 1 
real matrix function cda_closure(real matrix X) { 
	return(X :/ rowsum(X)) 
	
} 

// NJC 1 Sept 2008 
// ln(all but last column / last column) 
real matrix function cda_alr(real matrix X) { 
	real scalar c, cm1  
	c = cols(X); cm1 = c - 1 
	return(ln(X[, (1 .. cm1)]) :- ln(X[, c])) 
}

// NJC 1 Sept 2008 
// ln(all / row geometric means) 
real matrix function cda_clr(real matrix X) { 
	return(ln(X) :- mean(ln(X'))') 
} 

// NJC 1 Sept 2008 
// centring 
real matrix cda_centre(real matrix X) { 
	real rowvector centre, invcentre
	centre = cda_closure(exp(mean(ln(X)))) 
	invcentre = cda_closure((1 :/ centre))  
	return(cda_closure(X :* invcentre)) 
} 

// NJC 3 Sept 2008 
// column geometric means 
real matrix cda_colgmean(real matrix X) { 
	return(exp(mean(ln(X)))) 
} 

// NJC 3 Sept 2008 
// row geometric means 
real matrix cda_rowgmean(real matrix X) { 
	return(exp(mean(ln(X'))')) 
} 

// NJC 2 Sept 2008 
// multiplicative replacement for rounded zeros 
real matrix cda_mrzero(real matrix X, real rowvector delta, | real scalar total) { 
	real matrix iszero  
	if (total == .) total = 1 
	iszero = X :== 0 
	return((iszero :* delta) + ((!iszero) :* X :* (1 :- rowsum(iszero :* delta) :/ total)))
}

// NJC 10 Oct 2008 
// isometric log-ratio transformation 
real matrix function cda_ilr(real matrix X) { 
	real scalar c, j  
	real matrix Y, lnX 
	c = cols(X)
	Y = X[, (1 .. c - 1)]; lnX = ln(X)
	for (j = 1; j < c; j++) { 
		Y[, j] = rowsum(lnX[, (1 .. j)]) - j * lnX[, j + 1] 
		Y[, j] = (1 / sqrt(j * (j + 1))) * Y[, j]
	} 
	return(Y) 
}  

end	

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).

On Wed, Sep 22, 2010 at 3:20 PM, Nick Cox <n.j.cox@durham.ac.uk> wrote:
>
> Look at -dirifit- from SSC.
>

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