In the field of psychology, the practice ofp value null-hypothesis testing is as widespread as ever. Despite this popularity, or perhaps because of it, most psychologists are not aware of the statistical peculiarities of thep value procedure. In particular,p values are based on data that were never observed, and these hypothetical data are themselves influenced by subjective intentions. Moreover,p values do not quantify statistical evidence. This article reviews thesep value problems and illustrates each problem with concrete examples. The three problems are familiar to statisticians but may be new to psychologists. A practical solution to thesep value problems is to adopt a model selection perspective and use the Bayesian information criterion (BIC) for statistical inference (Raftery, 1995). The BIC provides an approximation to a Bayesian hypothesis test, does not require the specification of priors, and can be easily calculated from SPSS output.

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A practical solution to the pervasive problems of p values.

This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953, 1956, 1965) for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969, 1971) presented Bayesian analogs of small-sample frequentist tests for 2 x 2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972). Adopted usually in a hierarchical form, the logit-normal approach allows greater flexibility and scope for generalization. The 1970s also saw considerable interest in loglinear modeling. The advent of modern computational methods since the mid-1980s has led to a growing literature on fully Bayesian analyses with models for categorical data, with main emphasis on generalized linear models such as logistic regression for binary and multi-category response variables.

http://udsmed.u-strasbg.fr/labiostat/IMG/pdf/AGRESTI_bayesian_inference_for_categorical_data_analysis.pdf

Bayesian inference for categorical data analysis

The estimation of variance components in the one-way model is considered from a subjective Bayesian point of view. The situation in which the classical unbiased estimate of the between variance component is negative is explored in some detail. Exact and approximate posterior distributions are obtained in both the balanced and unbalanced case. Common sense aspects of the problem are emphasized, and some contrasts with other approaches. For example, Bayesianly speaking, a large negative unbiased estimate of the between variance component indicates an uninformative experiment in which the effective likelihood for that variance component is extremely flat, instead of strong evidence that the variance component is nearly zero.

/pdf/inference-about-variance-components-in-the-one-way-model-3womlqg2h2.pdf

Inference about Variance Components in the One-Way Model

In the analysis of the risk associated to rare events that may lead to catastrophic consequences with large uncertainty, it is questionable that the knowledge and information available for the analysis can be reflected properly by probabilities. Approaches other than purely probabilistic have been suggested, for example, using interval probabilities, possibilistic measures, or qualitative methods. In this article, we look into the problem and identify a number of issues that are foundational for its treatment. The foundational issues addressed reflect on the position that "probability is perfect" and take into open consideration the need for an extended framework for risk assessment that reflects the separation that practically exists between analyst and decisionmaker.

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Concerns, challenges, and directions of development for the issue of representing uncertainty in risk assessment.

Some unusual features of the likelihood function of the three-parameter lognormal distribution ln(t – γ)∼N(μ, σ2) are explored. In particular, it is shown that there exist paths along which the likelihood function of any sample t 1, …, tn tends to ∞ as (γ, μ, σ2) approaches (t (1), – ∞, + ∞), where t (1) is the smallest of the ti , and hence that in a meaningful sense this is the maximum-likelihood estimate. Estimation is then considered from a Bayesian point of view, and some natural posterior distributions are explored. A statistical model for a point-source epidemic is presented, and the theory developed is used in estimating the time of onset and other parameters.

The Three-Parameter Lognormal Distribution and Bayesian Analysis of a Point-Source Epidemic

Inferences from multinomial data: Learning about a bag of marbles - Discussion

Author(s): L. J. Savage, George Barnard, Jerome Cornfield, Irwin Bross, George E. P. Box, I. J. Good, D. V. Lindley, C. W. Clunies-Ross, John W. Pratt, Howard Levene, Thomas Goldman, A. P. Dempster, Oscar Kempthorne and Allan Birnbaum Source: Journal of the American Statistical Association, Vol. 57, No. 298 (Jun., 1962), pp. 307-326 Published by: Taylor & Francis, Ltd. on behalf of the American Statistical Association Stable URL: http://www.jstor.org/stable/2281641 Accessed: 25-12-2017 20:48 UTC

On the Foundations of Statistical Inference: Discussion

Lawrence Phillips (London School of Economics and Political Science) As we sat down to lunch in the staff dining room at University College London about 30 years ago, Dennis Lindley introduced me to a distinguished looking gentleman who to my delight turned out to be Professor Pearson. My youthful enthusiasm for Bayesian statistics, kindled at the University of Michigan in the 1960s by Ward Edwards and Jimmy Savage, led us to discuss the difficulty of obtaining a prior distribution. Professor Pearson gently explained that he could see no way of determining meaningful prior probabilities from the scientists with whom he had worked. The three papers presented today all say, 'Yes, Professor Pearson, it can be done'. However, the authors are providing perspectives that are very different from those of 30 years ago. First, the papers go beyond exploiting the properties of vague prior opinion to justify using non-informative priors, as in the principle of stable estimation (Edwards et al., 1963). For these examples, prior opinion is not vague and it matters. Second, the key question is not how to assess prior opinion, as O'Hagan recognizes; the more general issue is how to obtain meaningful and useful representations of expert uncertainty expressed in probabilistic form. That could apply to priors, likelihoods, posteriors, predictions or indeed any uncertainties. Third, as human beings, experts naturally experience uncertainty as a feeling, not as a numerical probability, so a good elicitation process helps the experts to learn what numbers are appropriate representations of their feelings and minimizes the biases summarized by Kadane and Wolfson. Fourth, we have access to computing power that can reduce the complexities of elicitation and make consistency checking an easy task, as is demonstrated to excellent effect by the University of Durham authors, as well as the others. Finally, all the authors make it clear that the process of assessing expert uncertainty deserves careful attention, in my view at least as much care as is devoted to the design of experiments. Most of us can distinguish a well-designed experiment from a poor one. But would we agree on criteria for judging the acceptability of an elicitation process? These papers suggest several criteria, which I shall summarize, supplemented by my own experience. The most important consideration is that proper elicitation is a sociotechnical process, as demonstrated in the early handbook by Stael von Holstein and Matheson (1979). The interaction between assessor and expert(s) requires careful handling of both social and technical issues at each of the following three main stages in the assessment process.

George A. Barnard

Papers

Inferences from multinomial data: Learning about a bag of marbles - Discussion

On the Foundations of Statistical Inference: Discussion

Discussion on the papers on 'Elicitation'

Strong Inconsistency from Uniform Priors: Comment

Discussion of the Papers of Messrs. Hald, Wetherill and Cox