rogers@ARSUN.UTAH.EDU (Alan R. Rogers) (05/30/90)
The confusion about phylogenetic autocorrelation was all mine. Jim Cheverud
called yesterday and explained what I had misunderstood. I was right in
arguing that rP and rS could not be determined from path analysis, but wrong
(as Sam Scheiner suggested) in thinking that they were in fact obtained that
way. Their method finds the maximum likelihood estimator of the
"phylogenetic autocorrelation", p, in the model
y = p W y + e
where y is a vector of observed character values, W a known matrix
describing phylogenetic relatedness, and e a vector of error terms. Having
found p, they estimate the phylogenetic component of character by p W y, and
the specific component by (I - p W) y. The correlations, rP and rS are
calculated from these estimates, not from the path analysis. The method
seems to work pretty well, as the plot just below shows. The horizontal
axis is the true value of the correlation between X1 and X2 in 1000
simulated data sets, each with 30 species. The vertical axis is the
estimate obtained using the method of Cheverud, Dow, and Leutenegger.
I am still skeptical that theirs is the best approach, but I can't see much
to complain about in the results below.
1 ......................................................................
. . * ** *********** .
0 . * ******************** .
o 0 . *** ****** ***************** ** .
u . . * **************************** .
t 5 . * * * *************************** * .
$ . . ** ** ** ****** ****************** * .
a 0 . ** *** *** ***************** ***** * .
c . * * *********************** * * *** * .
. . ********** *************** ** .
s . ************************ ** * .
- . * ************** * ** * .
1 . ************** .
. ......................................................................
0 -1.0 -0.5 0.0 0.5 1.0
out$rho
Alan Rogers
INTERNET: rogers@arsun.utah.edu
USMAIL : Dept. of Anthropology, Univ. of Utah, S.L.C., UT 84112
PHONE : (801) 581-5529