Jacques Poitevineau

Implementing Bayesian predictive procedures: The K-prime and K-square distributions

The implementation of Bayesian predictive procedures under standard normal models is considered. Two distributions are of particular interest, the K-prime and K-square distributions. They also give exact inferences for simple and multiple correlation coefficients. Their cumulative distribution functions can be expressed in terms of infinite series of multiples of incomplete beta function ratios, thus adequate for recursive calculations. Efficient algorithms are provided. To deal with special cases where possible underflows may prevent a recurrence to work properly, a simple solution is proposed which results in a procedure which is intermediate between two classes of algorithm. Some examples of applications are given.

The Significance Test Controversy Revisited

This chapter revisits the significance test controversy in the light of Jeffreys’ views about the role of statistical inference in experimental investigations. These views have been clearly expressed in the third edition of his Theory of Probability. The relevant passage is quoted and commented. The elementary inference about the difference between two means is considered, but the conclusions are applicable to most of the usual situations encountered in experimental data analysis.

Listener evaluations of new and Old Italian violins

Significance Old Italian violins are widely believed to have playing qualities unobtainable in new violins, including the ability to project their sound more effectively in a hall. Because Old Italian instruments are now priced beyond the reach of the vast majority of players, it seems important to test the fundamental assumption of their tonal superiority. A recent study found that, under blind conditions, violin soloists generally prefer new violins and are unable to distinguish between new and old at better than chance levels. This paper extends the results to listeners in a hall. We find that they generally prefer new violins over Stradivaris, consider them better-projecting, and are no better than players at telling new and old apart. Old Italian violins are routinely credited with playing qualities supposedly unobtainable in new instruments. These qualities include the ability to project their sound more effectively in a concert hall—despite seeming relatively quiet under the ear of the player—compared with new violins. Although researchers have long tried to explain the “mystery” of Stradivari’s sound, it is only recently that studies have addressed the fundamental assumption of tonal superiority. Results from two studies show that, under blind conditions, experienced violinists tend to prefer playing new violins over Old Italians. Moreover, they are unable to tell new from old at better than chance levels. This study explores the relative merits of Stradivari and new violins from the perspective of listeners in a hall. Projection and preference are taken as the two broadest criteria by which listeners might meaningfully compare violins. Which violins are heard better, and which are preferred? In two separate experiments, three new violins were compared with three by Stradivari. Projection was tested both with and without orchestral accompaniment. Projection and preference were judged simultaneously by dividing listeners into two groups. Results are unambiguous. The new violins projected better than the Stradivaris whether tested with orchestra or without, the new violins were generally preferred by the listeners, and the listeners could not reliably distinguish new from old. The single best-projecting violin was considered the loudest under the ear by players, and on average, violins that were quieter under the ear were found to project less well.

Preamble—Frequentist and Bayesian Inference

This chapter serves as an overall introduction to statistical inference concepts. The basic notions about the frequentist and Bayesian approaches to inference are presented and the corresponding terminology is introduced.

Reporting Confidence Intervals: A Paradoxical Situation

This chapter reviews the different views and interpretations of interval estimates. It discusses their methodological implications—what is the right use of interval estimates? The usual confidence intervals are compared with the so-called “exact” or “correct” confidence intervals for ANOVA effect sizes. While the former can receive both frequentist and Bayesian justifications and interpretations, the latter have logical and methodological inconsistencies that demonstrate the shortcomings of the uncritical use of the Neyman-Pearson approach. In conclusion, we have to ask: Why isn’t everyone a Bayesian?

Basic Fiducial Bayesian Procedures for Inference About Means

This chapter presents the basic fiducial Bayesian procedures for a contrast between means, which is an issue of particular importance for experimental data analysis. The presentation is essentially non-technical. Focus is on the computational and methodological aspects.

GHOST: An Officially Recommended Practice

This chapter gives a brief account of the misuses and abuses of Null Hypothesis Significance Testing (NHST). It also examines the most often recommended “good statistical practice,” called here Guidelined Hypotheses Official Significance Testing (GHOST). GHOST is a hybrid practice that appears as an amalgam of Fisherian and Neyman–Pearsonian views. It does not ban the use of significance testing, but the choice of the sample size should be justified, and estimates of the size of effects and confidence intervals should also be reported.

Generalizations and Methodological Considerations for ANOVA

This chapter generalizes the basic fiducial Bayesian procedures to the usual unstandardized and standardized ANOVA effect sizes indicators. Methodological aspects are discussed and appropriate alternatives to these indicators are introduced and illustrated.

Reporting Effect Sizes: The New Star System

This chapter demonstrates the shortcomings of the widespread practice that consists of simply reporting effect size [ES] indicators in addition to NSHT (without interval estimates). It also questions the consequences of restricting the use of ES to standardized measures, as commonly done in psychology and related fields.

The Fisher, Neyman–Pearson and Jeffreys Views of Statistical Tests

This chapter briefly reviews the rationale of the three main views of statistical tests. Current practice is based on the Fisher “test of significance” and the Neyman–Pearson “hypothesis test”. Jeffreys’ approach is a Bayesian alternative based on the use of “objective” prior probabilities of hypotheses. The main similarities and dissimilarities of these three approaches will be considered from a methodological point of view: what is the aim of statistical inference, what is the relevance of significance tests in experimental research? The dangers inherent in uncritical application of the Neyman–Pearson approach will also be stressed.