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I've recently reviewed some old papers by Nancy Reid, Barndorff-Nielsen, Richard Cox and, yes, a little Ronald Fisher on the concept of "conditional inference" in the frequentist paradigm, which appears to mean that inferences are based considering only the "relevant subset" of the sample space, not the entire sample space.

As a key example, it is known that the confidence intervals based on the t-statistic can be improved (Goutis & Casella, 1992) if you also consider the sample's coefficient of variation (referred to as an ancillary statistic).

As someone who regularly uses likelihood-based-inference, I have assumed that when I form an asymptotic $\alpha$%-confidence interval, I am performing (approximate) conditional inference, since the likelihood is conditional on the observed sample.

My question is that, apart from conditional logistic regression, I have not seen much use of the idea of conditioning on ancillary statistics prior to inference. Is this type of inference restricted to exponential families, or is it going by another name nowadays, so that it only appears to be limited.

I found a more recent article (Spanos, 2011) that seems to cast serious doubt about the approach taken by conditional inference (i.e., ancillarity). Instead, it proposes the very sensible, and less mathematically convoluted suggestion that parametric inference in "irregular" cases (where the support of the distribution is determined by the parameters) can be solved by truncating the usual, unconditional sampling distribution.

Fraser (2004) gave a nice defense of conditionality, but I am still left with the feeling that more than just a little luck and ingenuity are required to actually apply conditional inference to complex cases...certainly more complex than invoking the chi-squared approximation on the likelihood ratio statistic for "approximate" conditional inference.

Welsh (2011, p. 163) may have answered my question (3.9.5, 3.9.6).

They point out Basu's well-known result (Basu's theorem) that there can be more than one ancillary statistic, begging the question as to which "relevant subset" is most relevant. Even worse, they show two examples of where, even if you have a unique ancillary statistic, it does not eliminate the presence of other relevant subsets.

They go on to conclude that only Bayesian methods (or methods equivalent to them) can avoid this problem, allowing unproblematic conditional inference.

References:

Richard Hardy
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    (+1) For those interested, Spanos is discussing the "submarine" example from [Morey et al. (2015), "The fallacy of placing confidence in confidence intervals", *Psychonomic Bulletin & Review*, pp 1-21](https://learnbayes.org/papers/confidenceIntervalsFallacy/) - & see [What do confidence intervals say about precision (if anything)?](http://stats.stackexchange.com/q/204530/17230) & [Confidence interval for Uniform($\theta$, $\theta+a$)](http://stats.stackexchange.com/q/66407/17230). (This also shows conditional inference isn't restricted to the exponential family - in fact the conditioning ... – Scortchi - Reinstate Monica Jul 21 '16 at 16:25
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    ... procedure here generalizes to the location-scale family.) Another problem, in addition to those you mention, is that conditioning can restrict the sample space rather more than you'd like, & another is when to condition on an approximate ancillary - how do you balance information loss against increased relevance? These issues arise not only in contrived examples: see [Given the power of computers these days, is there ever a reason to do a chi-squared test rather than Fisher's exact test?](http://stats.stackexchange.com/q/14226/17230). – Scortchi - Reinstate Monica Jul 22 '16 at 08:47
  • Comments are not for extended discussion; this conversation has been [moved to chat](http://chat.stackexchange.com/rooms/42972/discussion-on-question-by-bey-is-frequentist-conditional-inference-still-being-u). – Scortchi - Reinstate Monica Jul 25 '16 at 10:27

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It appears that, indeed, likelihood-based inference is conditional, when such an ancillary statistic exists. I got this from p.197 of Yudi Pawitan's "In All Likelihood":

enter image description here

This means that the shape of the likelihood function $L(\theta)$ is determined by the conditional likelihood. Therefore, by performing likelihood inference on $L(\theta)$, we are effectively performing inference on $L(\theta|a)$, even if we don't know a!

Bottom line: **Likelihood of the data $\propto$ likelihood based on conditional model **

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