Discussion: On Arguments Concerning Statistical Principles

1 [ ] 4 Nov 2014 Statistical Science2014, Vol. 29, No. 2, 252 253 article Institute of Mathematical Statistics, 2014 Discussion: On Arguments ConcerningStatistical PrinciplesD. A. S. Fraser(i) Statistical inference after Neyman inference as an alternative to Neyman Pearson decision theory has a long history in sta-tistical thinking, with strong impetus from Fisher sresearch; see, for example, the overview in Fisher(1956). Some resulting concerns in inference theorythen reached the mathematical statistics communityrather forcefully with Cox (1958); this had focus onthe two measuring-instruments example and on usesof conditioning that were compelling.

2 (ii)Birnbaum and logical analysis in Statistical in-ference. Birnbaum (1962) introduced notation forthe Statistical inference available from an investiga-tion with a model and data. This gave grounds toanalyze how different methods or Principles mightinfluence the Statistical inference . As part of this hediscussed how sufficiency, likelihood and condition-ing could differentially affect Statistical of his discussion centered on the argumentfrom conditioning and sufficiency to likelihood, buta primary consequence was the attention attractedto conditioning and its role in inference . While thisinterest in conditioning was substantial for thoseconcerned with the core of statistics, it has morerecently been neglected or overlooked.

3 Indeed, somerecent texts, for example, Rice (2007), seem not toacknowledge conditioning in inference or even themeasuring-instrument example.(iii)Mayo and Statistical Principles . Mayo shouldbe strongly commended for reminding us that theprinciples and Arguments of Statistical inferencedeserve very serious consideration and, we mightadd, could have very serious consequences (FraserD. A. S. Fraser is Emeritus Professor, Department ofStatistical Sciences, University of Toronto, 100 St., Toronto, Ontario M5S 3G3, Canada is an electronic reprint of the original articlepublished by theInstitute of Mathematical StatisticsinStatistical Science,2014, Vol.)

4 29, No. 2, 252 253. Thisreprint differs from the original in pagination andtypographic detail.(2014)). Her primary focus is on the argument (Birn-baum (1962)) that the Principles sufficiency andconditionality lead to the likelihood principle. Thismay not cover some recent aspects of condition-ing (Fraser, Fraser and Staicu (2010)), but shouldstrongly stimulate renewed interest in conditioning.(iv)Contemporary inference theory. Many sta-tistical models have continuity in how parame-ter change affects observable variables or, morespecifically, how parameter change affects coordi-nate quantile functions, the inverses of the coordi-nate distribution functions.

5 This continuity in itsglobal effect is widely neglected in Statistical infer-ence. If this effect on quantile functions is acceptedand used in the inference procedures, then in widegenerality there is a well-determined conditioning(Fraser, Fraser and Staicu (2010)). And likelihoodanalysis then offers an exponential model approxi-mation that is third-order equivalent to the givenmodel, and this in turn provides third-order infer-ence for any scalar component parameters of in-terest. Thus, the familiar conditioning conflicts areroutinely avoided by acknowledging the importantmodel continuity.(v)What is available?The conditioning just de-scribed leads routinely top-value functionsp( )for any scalar component parameter = ( ) ofthe Statistical model.

6 A wealth of Statistical infer-ence methodology then immediately becomes avail-able from suchp-value functions. For example, atest for a value 0is given by thep-valuep( 0),a confidence interval by the inverse ( /2, 1 /2) =p 1(1 /2, /2) of thep-value function, and a me-dian estimate by the valuep 1( ). But quite gen-erally the neededp-value functions are not availablefrom a likelihood function alone!(vi)What are the implications?If continuity is in-cluded as an ingredient of many model-data combi-nations, then, as we have indicated, likelihood anal-ysis producesp-values and confidence intervals, andthese are not available from the likelihood func-tion alone.

7 This thus demonstrates that with such12D. A. S. FRASER continuity-based conditioning the likelihood princi-ple is not a consequence of sufficiency and condition-ing Principles . But if we omit the continuity then weare directly faced with the issue addressed by in part by the Natural Sciences andEngineering Research Council of Canada and SeniorScholars Funding at York , A.(1962). On the foundations of Statistical infer-ence (with discussion).J. Amer. Statist. , D. R.(1958). Some problems connected with Math. , R. A.(1956). Statistical Methods and Scientific In-ference. Oliver and Boyd, , D. A. S.(2014). Why does statistics have twotheories?

8 InPast, Present and Future of Statistical Sci-ence(X. Lin,D. Banks,C. Genest,G. Molenberghs,D. Wang, eds.) 237 252. CRC Press,Boca Raton, , A. M.,Fraser, D. A. , (2010). Second order ancillary: A differen-tial view from , J.(2007).Mathematical Statistics and Data Analysis,3rd ed. Brooks/Cole, Belmont, CA.