[R-sig-eco] Are likelihood approaches frequentist?

[Dave Hewitt]

As Ben pointed out, the key difference between pure likelihood approaches and
frequentist approaches is the addition of a layer of "significance"
assessment based on the idea of repeated experimentation. (The term
"frequentist" has been stretched in a variety of directions now, perhaps due
to lazy writing, so sometimes it is unclear what's included under the
umbrella.)

In his 2001 book "In All Likelihood: Statistical Modelling and Inference
Using Likelihood", Yudi Pawitan refers to pure likelihood inference as
"Fisher's third way", a compromise between frequentist and Bayesian
approaches that began with Fisher himself. Inference based strictly on the
likelihood function is not probabilistic, so would not conform to either of
these two other paradigms.

In a model selection context, which can be applied in most ecological
situations, one often (always?) does not need "significance" assessment and
can turn instead to model selection criteria for probabilistic statements
about the evidence. Of course, once you plot estimates with confidence
intervals you've entered the "gray area".


prado wrote:
> 
> 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.
> 
> -----------------------------------------------------------------------------
>>   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.
> 

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