I agree with Julien about this. If you have very few observations per species in your original data, it might even be useful to just compute a single pooled estimate of the within-species variance, and then use that to obtain estimates of the standard error for each species mean. To do this, first calculate a pooled variance in the normal way (https://en.wikipedia.org/wiki/Pooled_variance), and then divide the square-root of this variance by the square root of the sample size for each species to get an estimate of the standard error for that species. This works better if your data have been log-transformed in which case an assumption of homogeneity of the variance is more likely to hold.

I also agree with Dave, though, that if the Brownian model doesn't fit your data reasonably well, then sigma^2 is pretty difficult to interpret. (I hate to point to one of my own papers, but a couple of years ago some colleagues & I published a study in which we made a set of 'phylogenetic Anscombe' datasets - each of which showed a really different among-species pattern, and yet that all had the same value of sigma^2: https://dx.doi.org/10.1111/2041-210X.13067. The point of this article, like Anscombe 1973, was about graphing - but it may also apply to your problem.)

All the best, Liam

Liam J. Revell
University of Massachusetts Boston
Universidad Católica de la Ssma Concepción
web: http://faculty.umb.edu/liam.revell/, http://www.phytools.org

Academic Director UMass Boston Chile Abroad:
https://www.umb.edu/academics/caps/international/biology_chile

On 9/24/2020 1:46 PM, Julien Clavel wrote:
[EXTERNAL SENDER]

Hi Karla,

As pointed out by Liam you can account for the sampling error with 
fitContnuous. Pagel's lambda corresponds to BM + noise at the tips (e.g. 
sampling error). If after accounting for this nuisance (for a set of 
competitive models) you find that BM is better than the other models, then it 
will makes sense to consider the sigsq as a reasonable measure of rate even if 
your Pagel's lambda estimate was low.

Best wishes,

Julien

De : R-sig-phylo <r-sig-phylo-boun...@r-project.org> de la part de Karla Shikev 
<karlashi...@gmail.com>
Envoyé : jeudi 24 septembre 2020 03:41
À : Liam J. Revell <liam.rev...@umb.edu>
Cc : R Sig Phylo Listserv <r-sig-phylo@r-project.org>; Theodore Garland 
<theodore.garl...@ucr.edu>
Objet : Re: [R-sig-phylo] evolutionary rate for species with low phylogenetic 
signal

Thanks, Liam! That was really helpful. But what would be an alternative to
estimate evolutionary rates in those cases? Simply the sigsq estimate under
a white noise model? (which I assume is simply the variance in the original
data?)

Karla

On Wed, Sep 23, 2020 at 10:23 PM Liam J. Revell <liam.rev...@umb.edu> wrote:

Hi Karla.

   > I just wanted to make sure that the estimate of sigma^2 in a BM model
   > in fitContinuous is a meaningful measure of the rate of evolution of
   > the trait even if its phylogenetic signal is low.

I would say that the answer to this question (if it were posed as one)
is 'no.' Another way to put this is that sigma^2 is probably not a
meaningful measure of the rate of evolution for your character if
Brownian motion is a (very) poor fit to your data.

It's kind of equivalent to saying 'is the slope of a fitted linear
regression a good measure of the rate of change in y as a function of x
if a linear model is a very poor fit to my data?' (That is, you can
still find a value of the slope that maximizes the probability of your
data under a linear model, and this slope accurately give the mean
effect of x on y over the range of your observations for x - but it may
not mean a whole lot about how x & y genuinely co-vary.)

All the best, Liam

Liam J. Revell
University of Massachusetts Boston
Universidad Católica de la Ssma Concepción
web: http://faculty.umb.edu/liam.revell/, 
https://nam10.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.phytools.org%2F&amp;data=02%7C01%7Cliam.revell%40umb.edu%7Cc32f6ae954334ee27bbf08d860a9554a%7Cb97188711ee94425953c1ace1373eb38%7C0%7C0%7C637365627682724545&amp;sdata=mPJpNrUUHdUZ7nGRvCSc0yJSGpec32VK8%2F5ijH3ATTw%3D&amp;reserved=0

Academic Director UMass Boston Chile Abroad:
https://www.umb.edu/academics/caps/international/biology_chile

On 9/23/2020 9:36 PM, Karla Shikev wrote:
**[EXTERNAL SENDER]

Dear All,

Thanks for your fantastic feedback. Just a bit more background: I'm
interested both in the phylogenetic signal and the evolutionary rate. I
just wanted to make sure that the estimate of sigma^2 in a BM model in
fitContinuous is a meaningful measure of the rate of evolution of the
trait even if its phylogenetic signal is low.

thanks again!

Karla

On Wed, Sep 23, 2020 at 8:45 PM Theodore Garland
<theodore.garl...@ucr.edu <mailto:theodore.garl...@ucr.edu>> wrote:

      Carla, I can also suggest looking at some of the papers on this,
      such as:

      Blomberg, S. P., T. Garland, Jr., and A. R. Ives. 2003. Testing for
      phylogenetic signal in comparative data: behavioral traits are more
      labile. Evolution 57:717–745.


      Ives, A. R., P. E. Midford, and T. Garland, Jr. 2007. Within-species
      variation and measurement error in phylogenetic comparative methods.
      Systematic Biology 56:252–270.


      Revell, L., L. Harmon, and D. Collar. 2008. Phylogenetic signal,
      evolutionary process, and rate. Systematic Biology 57:591–601.


      Hardy, O. J., and S. Pavoine. 2012. Assessing phylogenetic signal
      with measurement error: a comparison of Mantel tests, Blomberg et
      al.’s K, and phylogenetic distograms. Evolution 66:2614–2621.


      Cheers,

      Ted Garland




      On Wed, Sep 23, 2020 at 4:33 PM Liam J. Revell <liam.rev...@umb.edu
      <mailto:liam.rev...@umb.edu>> wrote:

          Dear Karla.

          What do you want to use fitContinuous to do?

          fitContinuous fits multiple models, some of which will result in
          low
          values of lambda in a test of phylogenetic signal. In fact, this
is
          really the main point of fitContinuous: to compare alternative
          models
          for trait evolution. If you just want to estimate the
          most-likely rate
          of evolution assuming a Brownian evolutionary process, you could
          just
          compute the mean squared PIC value (this is the REML estimate of
          sigma^2).

          fitContinuous can also take into account sampling error in the
          estimation of species means, which can be a cause for low
measured
          values of phylogenetic signal.

          Hope this is of some help Karla.

          All the best, Liam

          Liam J. Revell
          University of Massachusetts Boston
          Universidad Católica de la Ssma Concepción
          web: http://faculty.umb.edu/liam.revell/,
          
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          Academic Director UMass Boston Chile Abroad:
          https://www.umb.edu/academics/caps/international/biology_chile

          On 9/23/2020 8:21 PM, Karla Shikev wrote:
           > [EXTERNAL SENDER]
           >
           > Dear friends,
           >
           > I guess this is as much a philosophical question as it is
          methodological. I
           > want to estimate the rate of evolution of a trait (e.g. using
the
           > fitContinuous function), but the trait has low phylogenetic
          signal (e.g.
           > lambda close to 0). Does it make sense to use fitContinuous
          on a trait
           > evolving like that?
           >
           > Thanks for your input.
           >
           > Karla
           >
           >          [[alternative HTML version deleted]]
           >
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