[R-sig-phylo] Multiple regressions with continuous and categorical data
tgarland at ucr.edu
tgarland at ucr.edu
Mon Apr 7 06:17:54 CEST 2008
6 April 2008
Hi All,
I agree with most of what Simon has written, and would just like to make a few follow-up points.
1. A lot of the issues about implementing phylogenetically independent contrasts with categorical independent variables and so forth were discussed fairly thoroughly many years ago, before people were much using GLS and related approaches. See Garland et al. (1992, 1993), Martins (1992). See also Grafen (1989), a paper that was ahead of its time and suffered from doing so much that most people could not grasp it (self included) at the time. We now have a somewhat newer generation of users/hackers who have gone forward with PGLS types of things without fully appreciating the development of independent contrasts that came before. Reading the old papers is worthwhile.
2.
> > >With contrasts you also have
> > > something that is perhaps a bit more "tangible" to look at,
> > > i.e., contrasts for the different variables that can be
> > > plotted as scattergrams (2D or 3D).
>
> >Hmm. I've always had a hard time thinking in terms of contrasts.
> >Perhaps it is my simple mind. Anyway, I don't think I agree here. You
> >can plot the variables, and should do so (isn't that a general mantra
> >of applied stats, to always do bivariate plots of your data?). Once
> >you've transformed your variables, you can easily plot them. You
> >should also look at plots of the data before phylogenetic
> >transformation. The more you know your data the better.
>I agree here. Maybe Ted can understand contrast plots better than raw
>data plots, but I certainly can't.
Looking at your raw data is certainly the most intuitive, but if you are analyzing it with anything other than a star phylogeny (and with contemporaneous tips!) then it is also misleading. In other words, looking at plots of raw tip data (or ensuing residuals) only makes sense if you are doing a conventional, nonphylogenetic analysis (and not even conventional weighted regression!). Once you perform a phylogenetic analysis, then the plots of raw tip data are misleading, sometimes very much so. In a plot of raw tip data, everything is, in effect, being compared with everything else in an entirely unprincipled (phylogenetically speaking) way, and this is appropriate only for an analysis that assumes a star phylogeny.
Once you start doing a phylogenetic analysis, plots of contrast can be quite informative. At least that is my experience based on analyzing literally dozens of real data sets from a variety of perspectives (e.g., correlation, multiple regression, ANCOVA, PCA). For example, a bivariate plot of contrasts in some trait versus contrasts in body size (often using tip data that had been log transformed) often reveals one or more points that seem far from the general trend. What are those points? With contrasts, you can see what is being contrasted with what, and it often turns out to be contrasts involving tip data points as opposed to internal nodes. One nice example involves hindlimb lengths of Carnivora versus body mass. A single very deviant point turns out to be polar bear versus grizzly bear, thus indicating a very high rate of evolution in one or the other branch since they diverged (see Garland and Janis, 1993). Alternatively, the branch lengths could be in error (shorter than they really are) and/or one or other tip data points could be in error. Whatever may be the case, you have a good start on figuring this out by being able to identify individual contrasts that are something compared with something else and then standardized in relation to branch lengths (square roots of sums of corrected branch lengths -- see Felsenstein, 1985; Garland et al., 1992). See also Garland and Adolph (1994), Garland and Ives (2000), papers by Purvis and colleagues, and by Nunn and colleagues. The point is that with contrasts you can see the fitted line (through the origin) and the points (contrasts) to which it was fitted. With respect to measurement error in the tip data, see Ives et al. (2997, Systematic Biology).
With PGSL and phylogenetic regression transform models, you don't have comparable data points to examine, or at least that is not what people typically do. Rather, a bunch of matrices are slammed together (ouch - no pun intended!) and estimates of slopes and so forth are examined pretty much without reference to a plot of any type. Even residuals can be quite tricky (see Lavin et al., 2008 in press; see also Grafen, 1989). Tony Ives could jump in here!
All for now and cheers,
Ted
---- Original message ----
Date: Mon, 07 Apr 2008 12:38:59 +1000
From: Simon Blomberg <s.blomberg1 at uq.edu.au>
Subject: Re: [R-sig-phylo] Multiple regressions with
continuous and categorical data
To: Marguerite Butler <mbutler at hawaii.edu>
Cc: tgarland at ucr.edu, r-sig-phylo at r-project.org,
arives at wiscmail.wisc.edu
>On Sat, 2008-04-05 at 23:19 -1000, Marguerite Butler
wrote:
>> Hi All,
>>
>> On Apr 5, 2008, at 6:17 AM, tgarland at ucr.edu wrote:
>>
>> > 5 April 2008
>> >
>> > Hi Luke,
>> >
>> > I was hoping that Tony Ives may comment here, but
I'll
>> > give it a try.
>> >
>> > In general, you can view Felsenstein's (1985)
>> > phylogenetically independent contrasts algorithm as
"simply"
>> > a way to implement PGLS (by which I mean models that
do not
>> > include transformation of branch lengths) without
having to
>> > invert matrices and so forth.
>>
>> I have always found it easier to think of Independent
Contrasts as a
>> means for statistically removing the covariance among
species. To a
>> greater or lesser extent, species share parts of their
evolutionary
>> history. How this translates into our statistical
analysis is that we
>> assume a Brownian motion model of evolution, which
predicts that
>> pairs of species should differ, on average, in
proportion to how long
>> they have been evolving independently. So species which
share 95% of
>> their history and only diverged in the last 5% should
be much closer
>> in phenotype than species which share only 20% of their
history. (The
>> linear dependence on time is a direct prediction of
Brownian motion.)
>>
>> OK, so the problem is that this "correlation" among the
species
>> (i.e., data points) simply because of their
phylogenetic relationship
>> violates statistical assumptions. In statistical
parlance, they have
>> "correlated errors" -- (the "errors" here is the
phylogeny!). It's
>> funny terminology, but that's what it is. All
statistical methods
>> assume that there are no "correlated errors", or that
the data are
>> independent.
>
>No they don't. Time-series models, geostatistical models,
and
>mixed-effects models all model correlated errors.
>
>> It is a simple thing to correct for the phylogenetic
>> covariance resulting from a BM model of evolution, you
just divide it
>> out of the data. This is exactly the same thing that
you do with
>> other statistical procedures where you have unequal
variances. For
>> example, if you want to do a t-test between two
populations that have
>> unequal variances, you can normalize the data by
dividing each
>> observation by the sample standard deviation.
>
>This is only an approximation, probably not the best one.
Nobody knows
>how to get the exact solution to this problem (It is
known as the
>Fisher-Behrens problem in the statistical literature).
Weighted
>regression for unequal variances and GLS are closely
related, since a
>variance is just the covariance of a variable with
itself. Your are
>right to point this out.
>
>>
>> It's only slightly more complicated here -- you have a
different
>> expected covariance for each pair of species
corresponding to their
>> shared history. That is why Felsenstein has all those
equations which
>> relate to various branch lengths. I find it much
simpler to do the IC
>> calculations in matrix form. You just create a species
similarity
>> matrix based on the phylogeny (i.e., 1- distance
matrix), take the
>> root of this expected variance matrix (SAS, R, matlab,
etc. all have
>> Cholesky decomposition and other routines to do this),
and then pre-
>> multiply your species data vector. You can calculate
all of the
>> contrasts at once. This is explained in more detail in
Butler,
>> Schoener, Losos (2000).
>
>This is exactly GLS.
>
>>
>> Phylogenetic regression methods assuming Brownian
motion do exactly
>> the same thing.
>>
>> > The numbers from an analysis
>> > with independent contrasts will match those from a
PGLS
>> > analysis if and only if you use the same tree for
>> > calculating contrasts for all of the variables. That
also
>> > points out a difference between contrasts and GLS.
With the
>> > former, you can actually use a different set of
branch
>> > lengths (but always the same topology) for each
variable in
>> > the analysis (Y and/or the Xs). With the PGLS
>> > implementation, you only enter a single matrix to
represent
>> > the expected variances and covariances of the
residuals,
>> > derived (obviously) from a single tree.
>>
>> Applying different sets of branch lengths to each
variable in the
>> analysis sounds like a very confusing proposition to
me. It may be
>> statistically valid (however, the only reason I can
think of to do
>> this is that you're trying to improve the fit of the
Brownian motion
>> model to your variables?), but it will be a nightmare
to interpret.
>> Adjusting the branch lengths is tantamount to altering
the BM model.
>> Are we saying that the different characters are
evolving according to
>> different BM models? I would hope that we have a very
good reason for
>> suspecting this, and that the form of the branch length
scaling
>> corresponds to how we envision the warping of the BM
model?
>
>We have very good reasons for suspecting this. Brownian
motion
>(actually, the Weiner process) is almost certainly a poor
model for the
>evolution of any trait you care to name. It assumes a
linear
>relationship between variance and time, there are no
bounds to
>evolution, and although the process is continuous it is
nowhere
>differentiable, making the calculation of instantaneous
rates of
>evolution impossible.
>
>> For
>> example, if branch lengths are scaled such that those
closer to the
>> root are shorter than the original, and those nearer to
the tips are
>> longer, this might make sense if evolutionary changes
tend to magnify
>> later evolutionary changes like perhaps in the
accumulation of junk
>> DNA where the evolution of repeats could lead to larger
duplication
>> or deletion events -- in other words, in this example
the
>> evolutionary "steps" in the phenotype get bigger and
bigger. But
>> still, this is a rough short-cut because the scaling is
on the
>> phylogeny (i.e., time) and not the character (e.g., why
not do a
>> transformation on the data that reflects the
non-linearity in the
>> evolutionary changes you hypothesize?).
>
>You face the same problem in interpretation either way.
Transforming the
>branch lengths (model of evolution) and not transforming
the character
>data may mean that interpretation of the character
results may be a bit
>easier. (Usually we are less interested in the model of
evolution than
>in the predictors of the mean response.) Also, they don't
do the same
>thing. The GLS model is:
>
>y = X %*% beta + epsilon, epsilon ~ sigma^2 * Lambda,
>
>So transforming X (the character data) and/or Lambda (the
var-covar
>matrix) are different procedures with different results
and
>interpretations. One affects the mean response, the other
affects the
>covariance.
>
>> So then the interpretation
>> of the regression or correlation at the end of all this
>> "transforming" gets increasingly difficult. What
exactly have we
>> corrected for? (Also, there is a danger of correcting
out the very
>> effect that we're trying to study).
>
>As you can see by inspection of the model, the structure
of Lambda will
>affect the results ("correct out") when beta is small and
sigma^2 is
>large. This is precisely what we want, as we would
otherwise be more
>inclined to make a Type I error without including
knowledge of Lambda.
>In extremely sloppy language, the observed effect would
be due to the
>effect of phylogeny.
>
>> I wouldn't know what to tell you
>> if the transformations are haphazard.
>>
>> Yet another issue is that the phylogeny is estimated
from independent
>> data, with great effort. Why ignore the information
content in the
>> phylogeny and branch lengths?
>
>The branch lengths are typically estimated with MUCH less
precision than
>the topology. I don't know any systematists who trust
their branch
>length estimates.
>
>>
>> I don't quite understand why you (Ted) accentuate the
same topology
>> when you imply that the branch lengths are more free to
change.
>
>Because regression results are much more sensitive to
variation in
>topology than to variation in branch lengths. If you
change the
>topology, you are essentially changing the rank-order of
the
>covariances. This is a much more vicious transformation
than adjusting
>the branch lengths. Changing the branch lengths just
changes the model
>of evolution, which we know to be wrong a priori anyway.
>
>
>> All
>> BM cares about is time since the species become
independent. If you
>> are making some branches longer and some shorter, you
are really
>> monkeying around with the BM predictions.
>
>Yes!
>
>> I suppose that if you
>> constrain the topology to be fixed, you'll never have a
change in
>> ordering of independent evolution times, but it's not
so clear to me
>> why this is so much worse than allowing branch lengths
to take any
>> value (my point is not that we can just allow phylogeny
to be
>> anything, and thus essentially ignore phylogeny -- my
point is rather
>> that allowing branch lengths to take any value is a
pretty big liberty).
>
>Not any value. Values from a more realistic model of
evolution are to be
>preferred!
>
>>
>> > In principle, you
>> > could enter a matrix that was a sort of conglomerated
>> > version of several different matrices, each
representing a
>> > different set of branch lengths for (potentially)
each
>> > different variable in the model. Coming up with how
to
>> > create such a matrix would be nontrivial in cases
with lots
>> > of variables. In principle, though, it could be done.
This
>> > gets at the lack of an existing formal proof for the
>> > complete identify of independent contrasts and PGLS
(see
>> > Garland and Ives, 2000; Rohlf, 2001, 2006).
>> >
>> Anyone can verify that IC and PGLS are the same thing.
You can
>> translate the matrix equations to equations to look
like
>> Felsenstein's, you just write out the terms of the
matrix as branch
>> lengths and multiply them out. They will be the same.
>
>Then please do this! The problem is that Felsenstein
presented an
>algorithm, not a set of equations relating the response
and predictor
>variables. I'm sure you could do what you say, but as it
involves the
>choleski decomposition of the vcv matrix, which is itself
a complicated
>linear transformation of the branch lengths, I don't
think it is
>trivial. But if it is, I would like to see the proof!
>
>>
>> > So, as presently implemented by any existing
software,
>> > contrasts and "phylogenetic regressions" (as in
Grafen, 1989
>> > or Lavin et al., 2008) have slightly different
advantages.
>> > With contrasts, you can use different trees for
different
>> > variables. I have found this to be particularly
appealing
>> > when I have one or more independent variables that is
a
>> > nuisance variable and clearly not phylogenetically
related
>> > on first principles. For example, Perry and Garland
(2002)
>> > analyzed home ranges of lizards. For number of
sightings,
>> > study duration, and calculation method, we computed
>> > contrasts on a star phylogeny (one created by
collapsing the
>> > original tree down to a single giant polytomy). These
>> > contrasts were then used in a big multiple regression
along
>> > with contrasts for body size and other biological
>> > independent variables.
>>
>> I guess you're talking about in terms of software
implementation?
>> Because I *think* all this would involve is not doing
the matrix
>> transformation above on your variables that are not
related to the
>> phenotype of the species, but doing it on your
phenotypic data, and
>> then running the analyses. You don't really have to
compute contrasts.
>
>Yes, that makes sense. I'm not really sure what that will
mean for the
>analysis. I suspect that it blows away the orthogonality
of your
>explanatory variables. But in these sorts of studies,
which are usually
>observational, multicollinearity is bound to be an issue
anyway. So it
>will make model selection (even) more difficult.
>>
>> Yes, I agree on the point that there is no phylogenetic
dependence if
>> the variables are not related to phenotypes:).
>>
>> > With contrasts you also have
>> > something that is perhaps a bit more "tangible" to
look at,
>> > i.e., contrasts for the different variables that can
be
>> > plotted as scattergrams (2D or 3D).
>>
>> Hmm. I've always had a hard time thinking in terms of
contrasts.
>> Perhaps it is my simple mind. Anyway, I don't think I
agree here. You
>> can plot the variables, and should do so (isn't that a
general mantra
>> of applied stats, to always do bivariate plots of your
data?). Once
>> you've transformed your variables, you can easily plot
them. You
>> should also look at plots of the data before
phylogenetic
>> transformation. The more you know your data the better.
>
>I agree here. Maybe Ted can understand contrast plots
better than raw
>data plots, but I certainly can't.
>> >
>> > With phylogenetic regressions, on the other hand, it
>> > is easier to "automate" analyses and implement branch
length
>> > transforms to optimize model fit (see also Garland an
Ives,
>> > 2000; Garland et al., 2005).
>> >
>>
>> Automation is really easy if you learn some R:).
>
>Agreed!
>
>>
>> Take care,
>> Marguerite
>>
>> > All for now and cheers,
>> > Ted
>> >
>> > cc: arives at wiscmail.wisc.edu
>> >
>> > ---- Original message ----
>> >
>> > Date: Wed, 2 Apr 2008 09:03:15 -0700
>> > From: Luke Harmon <lukeh at uidaho.edu>
>> > Subject: Re: [R-sig-phylo] Multiple regressions with
>> > continuous and categorical data
>> > To: <tgarland at ucr.edu>
>> >
>> >> And, a question - since it seems like you've been
putting
>> > a lot of
>> >> thought into this stuff! My understanding is that
PGLS
>> > considers all
>> >> the predictor variables fixed, and so only accounts
for
>> > phylogenetic
>> >> signal in (the residuals of) the dependent variable.
Is
>> > this
>> >> interpretation correct? What about phylogenetic
signal in
>> > the
>> >> predictors, esp. the discrete ones? Is there any
sensible
>> > way to deal
>> >> with that?
>> >>
>> >> Thanks for including the paper with that last email,
I
>> > just downloaded
>> >> it. It seems very interesting. And your
participation in
>> > the mailing
>> >> list is providing an amazing resource for all of us.
>> > Archived
>> >> messages from the r list have a very long "shelf
life,"
>> > which I think
>> >> will be the case with these as well - this will
benefit
>> > students for
>> >> quite some time.
>> >>
>> >> Luke
>> >>
>> >>
>> >> On Apr 2, 2008, at 8:38 AM, <tgarland at ucr.edu>
wrote:
>> >>
>> >>> 2 April 2008
>> >>>
>> >>> Dear Alejandro,
>> >>>
>> >>> Yes, you can do all of that with PGLS. We have
recently
>> > released a
>> >>> Matlab program, Regressionv2.m, that performs
multiple
>> > regressions
>> >>> in several versions:
>> >>>
>> >>> 1. conventional, nonphylogenetic (i.e., assuming a
star
>> >>> phylogeny). These results will match what you might
get
>> > from SPSS,
>> >>> SAS, etc.
>> >>>
>> >>> 2. PGLS (i.e., using a user-inputted phylogeny).
These
>> > results
>> >>> will match what you can get from performing
>> > Felsenstein's (1985)
>> >>> phylogenetically independent contrasts, so long as
you
>> > use the same
>> >>> topology and branch lengths to compute contrasts
for
>> > each variable.
>> >>> However, when you have independent variables that
are
>> > categorical
>> >>> and have several categories it can be somewhat
>> > laborious to
>> >>> implement contrasts. The Regresisonv2.m program
will
>> > automatically
>> >>> code your categorical independent variables into
the
>> > appropriate
>> >>> number of 0-1 dummy variables. You can also
>> > automatically test for
>> >>> interactions between the categorical and/or
>> > continuous-valued
>> >>> independent variables.
>> >>>
>> >>> 3. Regression with the residual error term modeled
as
>> > an Ornstein-
>> >>> Uhlenbeck process ("RegOU"), the
>> > accelerating-decelerating model of
>> >>> Blomberg et al. (2003)("RegACDC"), Grafen's (1989)
rho
>> > or Pagel's
>> >>> (1997) lambda. As discussed in Lavin et al. (2008
--
>> > see below),
>> >>> sensible terminology for all of the above models is
>> > complicated. We
>> >>> try to avoid mixing estimation procedures (e.g.,
OLS
>> > and GLS) with
>> >>> transform models (e.g., OU).
>> >>>
>> >>> The program has a number of other useful features,
>> > including:
>> >>>
>> >>> a. provides likelihoods, AIC, and AICc to allow for
>> > model
>> >>> comparisons.
>> >>>
>> >>> b. automatically handles missing values
>> >>>
>> >>> c. likelihood-based phylogenetic comparative
methods
>> > can have poor
>> >>> statistical properties when sample sizes are small
>> > (e.g. Freckleton
>> >>> et al. 2002, Blomberg et al. 2003). Therefore,
>> > Regressionv2.m
>> >>> incorporates bootstrap methods (Efron and
Tibshirani
>> > 1993) to obtain
>> >>> confidence intervals for all parameters, including
the
>> > transform
>> >>> parameter in #3 above.
>> >>>
>> >>> The program is available on request and accompanies
>> > this paper,
>> >>> which I will attach separately (if anybody else
wants
>> > it, just click
>> >>> on the link).
>> >>>
>> >>> Lavin, S. R., W. H. Karasov, A. R. Ives, K. M.
>> > Middleton, and T.
>> >>> Garland, Jr. 2008. Morphometrics of the avian small
>> > intestine,
>> >>> compared with non-flying mammals: A phylogenetic
>> > perspective.
>> >>> Physiological and Biochemical Zoology. In press.
>> >>>
>> >
[http://biology.ucr.edu/people/faculty/Garland/Lavin_et_al_2008.pdf]
>> >>>
>> >>> Obviously, we would like to see all of this
available
>> > in R. Much
>> >>> of this could be done in APE, but not all of it. It
>> > would be nice
>> >>> to compare results from APE with those from
>> > Regressionv2.m.
>> >>>
>> >>> Sincerely,
>> >>> Ted Garland
>> >>>
>> >>> Theodore Garland, Jr., Ph.D.
>> >>> Professor
>> >>> Department of Biology
>> >>> University of California
>> >>> Riverside, CA 92521
>> >>> Phone: (951) 827-3524 = Ted's office (with
answering
>> > machine)
>> >>> Phone: (951) 827-5724 = Ted's lab
>> >>> Phone: (951) 827-5903 = Dept. office
>> >>> Home Phone: (951) 328-0820
>> >>> FAX: (951) 827-4286 = Dept. office
>> >>> Email: tgarland at ucr.edu
>> >>> http://biology.ucr.edu/people/faculty/Garland.html
>> >>> List of all publications with PDF files:
>> >>>
>> > http://www.biology.ucr.edu/people/faculty/Garland/
>> > GarlandPublications.html
>> >>>
>> >>> ---- Original message ----
>> >>>
>> >>> Date: Wed, 2 Apr 2008 10:11:09 +0200
>> >>> From: "Alejandro Gonzalez Voyer"
>> >>> <alejandro.gonzalezvoyer at ebc.uu.se>
>> >>> Subject: [R-sig-phylo] Multiple regressions with
>> >>> continuous and categorical data
>> >>> To: <r-sig-phylo at r-project.org>
>> >>>> Dear R-phylo community,
>> >>>>
>> >>>>
>> >>>>
>> >>>> First let me say I think the wiki site and mailing
>> > list
>> >>> are a great and very
>> >>>> helpful idea.
>> >>>>
>> >>>>
>> >>>>
>> >>>> I want to run some comparative analyses in a
multiple
>> >>> regression framework
>> >>>> using a PGLS approach. I have both continuous and
>> >>> categorical data, also if
>> >>>> at all possible, I would like to include some
>> > dichotomous
>> >>> variables. My
>> >>>> response variable is continuous and the
explanatory
>> >>> variables continuous and
>> >>>> categorical. My aim is to run a multiple
regression
>> >>> analysis with a backward
>> >>>> stepwise elimination of non-significant
explanatory
>> >>> variables. My question
>> >>>> is twofold: first should categorical variables be
>> > coded
>> >>> as sets of
>> >>>> dichotomous dummy variables? Second, can I use
PGLS
>> > for
>> >>> this sort of
>> >>>> analyses, is it able to run robust GLS analyses
even
>> > if
>> >>> most of the
>> >>>> variables are categorical or dichotomous? Is there
a
>> > rule
>> >>> of thumb for how
>> >>>> many dichotomous/categorical variables one can
>> > include?
>> >>>>
>> >>>> I get the feeling I am not alone facing this sort
of
>> >>> problem, and from my
>> >>>> understanding of Martins and Hansen (1997) there
>> > should
>> >>> be a way of running
>> >>>> these sort of analyses with PGLS.
>> >>>>
>> >>>>
>> >>>>
>> >>>> Thank you for any tips or suggestions.
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>> Cheers,
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>> Alejandro
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>> Dr. Alejandro Gonzalez Voyer
>> >>>>
>> >>>> Post-doc.
>> >>>>
>> >>>>
>> >>>>
>> >>>> Animal Ecology, Department of Ecology and
Evolution
>> >>>>
>> >>>> Evolutionary Biology Centre (EBC)
>> >>>>
>> >>>> Uppsala University
>> >>>>
>> >>>> Norbyv�gen 18D
>> >>>>
>> >>>> 75236 Uppsala
>> >>>>
>> >>>> Sweden
>> >>>>
>> >>>>
>> >>>>
>> >>>> e-mail: alejandro.gonzalezvoyer[AT]ebc.uu.se
>> >>>>
>> >>>> Tel: ++46-18-471-2930
>> >>>>
>> >>>> Web page:
>> >>>
>> >
http://www.iee.uu.se/zooekol/default.php?type=personalpage
>> >>>>
>> > <http://www.iee.uu.se/zooekol/default.php?
>> > type=personalpage&id=146&lang=en
>> >>>>>
>> >>>> &id=146&lang=en
>> >>>>
>> >>>>
>> >>>>
>> >>>>
>> >>>> [[alternative HTML version deleted]]
>> >>>>
>> >>>> ________________
>> >>>> _______________________________________________
>> >>>> R-sig-phylo mailing list
>R-sig-phylo at r-project.org
>> >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-phylo
>> >>> _______________________________________________
>> >>> R-sig-phylo mailing list
>> >>> R-sig-phylo at r-project.org
>> >>> https://stat.ethz.ch/mailman/listinfo/r-sig-phylo
>> >>
>> > _______________________________________________
>> > R-sig-phylo mailing list
>> > R-sig-phylo at r-project.org
>> > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo
>>
>> ____________________________________________
>> Marguerite A. Butler
>> Department of Zoology
>> University of Hawaii
>> 2538 McCarthy Mall, Edmondson 259
>> Honolulu, HI 96822
>>
>> Phone: 808-956-4713
>> FAX: 808-956-9812
>> Dept: 808-956-8617
>> http://www2.hawaii.edu/~mbutler
>> http://www.hawaii.edu/zoology/
>>
>>
>> [[alternative HTML version deleted]]
>>
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>--
>Simon Blomberg, BSc (Hons), PhD, MAppStat.
>Lecturer and Consultant Statistician
>Faculty of Biological and Chemical Sciences
>The University of Queensland
>St. Lucia Queensland 4072
>Australia
>Room 320 Goddard Building (8)
>T: +61 7 3365 2506
>http://www.uq.edu.au/~uqsblomb
>email: S.Blomberg1_at_uq.edu.au
>
>Policies:
>1. I will NOT analyse your data for you.
>2. Your deadline is your problem.
>
>The combination of some data and an aching desire for
>an answer does not ensure that a reasonable answer can
>be extracted from a given body of data. - John Tukey.
>
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