I received a copy of a statistical text recently that included a discussion of severity, and this is my first chance to look through it. It’s Introductory Statistical Inference with the Likelihood Function by Charles Rohde from Johns Hopkins. Here’s the blurb:
This textbook covers the fundamentals of statistical inference and statistical theory including Bayesian and frequentist approaches and methodology possible without excessive emphasis on the underlying mathematics. This book is about some of the basic principles of statistics that are necessary to understand and evaluate methods for analyzing complex data sets. The likelihood function is used for pure likelihood inference throughout the book. There is also coverage of severity and finite population sampling. The material was developed from an introductory statistical theory course taught by the author at the Johns Hopkins University’s Department of Biostatistics. Students and instructors in public health programs will benefit from the likelihood modeling approach that is used throughout the text. This will also appeal to epidemiologists and psychometricians. After a brief introduction, there are chapters on estimation, hypothesis testing, and maximum likelihood modeling. The book concludes with sections on Bayesian computation and inference. An appendix contains unique coverage of the interpretation of probability, and coverage of probability and mathematical concepts.
It’s welcome to see severity in a statistics text; an example from Mayo and Spanos (2006) is given in detail. The author even says some nice things about it: “Severe testing does a nice job of clarifying the issues which occur when a hypothesis is accepted (not rejected) by finding those values of the parameter (here mu) which are plausible (have high severity[i]) given acceptance. Similarly severe testing addresses the issue of a hypothesis which is rejected…. . ” I don’t know Rohde, and the book isn’t error-statistical in spirit at all.[ii] In fact, inferences based on error probabilities are often called “illogical” because they take into account cherry-picking, multiple testing, optional stopping and other biasing selection effects that the likelihoodist considers irrelevant. I wish he had used severity to address some of the classic howlers he delineates regarding N-P statistics. To his credit, they are laid out with unusual clarity. For example a rejection of a point null µ= µ0 based on a result that just reaches the 1.96 cut-off for a one-sided test is claimed to license the inference to a point alternative µ= µ’ that is over 6 standard deviations greater than the null. (pp. 49-50). But it is not licensed. The probability of a larger difference than observed, were the data generated under such an alternative is ~1, so the severity associated with such an inference is ~ 0. SEV(µ <µ’) ~1.
[i]Not to quibble, but I wouldn’t say parameter values are assigned severity, but rather that various hypotheses about mu pass with severity. The hypotheses are generally in the form of discrepancies, e..g,µ >µ’
[ii] He’s a likelihoodist from Johns Hopkins. Royall has had a strong influence there (Goodman comes to mind), and elsewhere, especially among philosophers. Bayesians also come back to likelihood ratio arguments, often. For discussions on likelihoodism and the law of likelihood see:
Breaking the Law of Likelihood, to keep their fit measures in line A, B
Royall, R. (2004), “The Likelihood Paradigm for Statistical Evidence” 119-138; Rejoinder 145-151, in M. Taper, and S. Lele (eds.) The Nature of Scientific Evidence: Statistical, Philosophical and Empirical Considerations. Chicago: University of Chicago Press.