I'm a little surpised not to see A.W.F. Edwards book on Likelihood cited 
in connection 
with the possibility of a distinctive "likelihoodist" viewpoint, as I 
think the book was influential, 
including for some biologists.  (The first edition of the book was 
sometimes known as 
"Little Red Likelihood.") 

Since a Bayesian would hold that the data enter a parametric analysis via 
the likelihood, 
it seems that a Bayesian as such would be interested in the likelihood 
surface.  A divergence might 
tend to occur in how nuisance parameters are eliminated (by optimization 
or by integration). 








"Paulo Inácio de Knegt López de Prado" <prado@ib.usp.br> 
Sent by: r-sig-ecology-bounces@r-project.org
09/28/2008 08:55 PM

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r-sig-ecology@r-project.org
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Subject
[R-sig-eco] Are likelihood approaches frequentist?






Dear r-sig-ecology users

Here follow the messages I exchanged with Ben Bolker last week about the
likelihood and frequentist approaches. We both would like to open this 
topic
for discussion in the list.

Best wishes

Paulo

----------------------------------------------------------------------------
Dear Dr. Bolker, 

I am puzzled why some authors treat likelihood approaches as 
frequentist, as it seems you did in page 13 of your book 'Ecological 
Models
and Data'. 
This sounds odd to me because  what brought my attention to likelihood was
Richard Royall's book 'Statistical Evidence'. His framing of a paradigm
based on the likelihood principle, and the clear distinction he makes
between this paradigm and frequentist and Bayesian approaches looks
quite convincing to me.

I agree with him that we use likelihood criteria to identify, among
competing hypotheses, which one attribute the highest probability to  a
given dataset. If I understood correctly, this is what Royal calls the
'evidence value' of a data set to a hypothesis 'vis a vis' other
hypotheses. I also like his idea that the role of statistics in science
is just to gauge this evidence value, no less, no more.

This approach differs from the frequentist because the sampling
space is irrelevant, that is, other datasets that might be observed do not
affect the evidence value of the observed data set. My favourite example 
is
the comparison of binomial and negative binomial experiments on coin
tossing, in the sections 1.11 and 1.12 of his book.

I am not an "orthodox likelihoodist"; on the contrary, I agree with the
pragmatic view you express in your book. I'd just like to understand
the key differences among the available statistical tools, in order to 
make
a good pragmatic use of them. I'd really appreciate if you can help me
with this.

Best wishes

Paulo
-----------------------------------------------------------------------------
>   Very well put.  Royall, and Edwards (author of _Likelihood_, Johns
> Hopkins 1992) are what I would call "pure", or "hard-core",
> or "orthodox", likelihoodists. They are satisfied with a statement
> of relative likelihood, and don't feel the need to attach a p-value
> to the result in order to have a decision rule for hypothesis rejection.
>
>   Far more commonly, however, people impose (? add ?) an additional
> layer of frequentist procedure on top of this basic structure, namely
> using the likelihood ratio test to assess the statistical significance
> of a given observed likelihood ratio and/or to set a cutoff value
> for profile confidence intervals.  Using the LRT puts the inference
> back squarely into the frequentist domain, although the sample space
> we are now dealing with (sample space of likelihoods derived from
> coin-tossing experiments) is quite different from the one
> we started with (sample space of outcomes of coin-tossing experiments).
> As far as I can see, Edwards and Royall are almost alone in their
> adherence to "pure" likelihood -- most of the rest of us pander
> to the desire for p-values (or, less cynically, to the desire
> for a probabilistically sound decision rule).
>
>  I would also add that different scientists have different
> goals (belief, prediction, decision, assessing evidence). I too
> think Royall makes a good case for the primacy of
> assessing strength-of-evidence, and he gives the clearest
> explanation I have seen, but I wouldn't completely
> rule out the other frameworks.
>
>   Hope that makes sense -- thanks for the kind words!
>
>   cheers
>    Ben Bolker 

--
Paulo Inácio de Knegt López de Prado
Depto. de Ecologia - Instituto de Biocięncias - USP

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