Fermat, Schubert, Einstein, and Behrens-Fisher: The Probable Difference Between Two Means When σ_1^2≠σ_2^2

The history of the Behrens-Fisher problem and some approximate solutions are reviewed. In outlining relevant statistical hypotheses on the probable difference between two means, the importance of the Behrens- Fisher problem from a theoretical perspective is acknowledged, but it is concluded that this problem is irrelevant for applied research in psychology, education, and related disciplines. The focus is better placed on “shift in location” and, more importantly, “shift in location and change in scale” treatment alternatives

Methods for Comparing Two Means with Application in Adaptive Clinical Trials

In the design of a clinical trial, the study of the effect of an intervention for a given medical condition is frequently of interest to researcher. Also, in recent years, the use of sequential and adaptive design methods in clinical research and development based on accrued data has become very popular due to its flexibility and efficiency. In this thesis, we derive the Behrens-Fisher distribution, and use the distributional result to examine the effect of an intervention by comparing population means of intervention group and control group. Sample size prediction methods proporting to solve the Behrens-Fisher problem are examined. A new method for solving the Behrens-Fisher problem is proposed. Various sequential and adaptive designs are reviewed

A Bayesian Solution to the Behrens-Fisher problem

A simple solution to the Behrens–Fisher problem based on Bayes factors is presented, and its relation with the Behrens–Fisher distribution is explored. The construction of the Bayes factor is based on a simple hierarchical model, and has a closed form based on the densities of general Behrens–Fisher distributions. Simple asymptotic approximations of the Bayes factor, which are functions of the Kullback–Leibler divergence between normal distributions, are given, and it is also proved to be consistent. Some examples and comparisons are also presented.Open access funding provided by Universidad de Málaga/CBUA

Fermat, Schubert, Einstein, and Behrens-Fisher: The Probable Difference Between Two Means When σ\u3csub\u3e1\u3c/sub\u3e\u3csup\u3e2\u3c/sup\u3e≠ σ\u3csub\u3e2\u3c/sub\u3e\u3csup\u3e2\u3c/sup\u3e

The history of the Behrens-Fisher problem and some approximate solutions are reviewed. In outlining relevant statistical hypotheses on the probable difference between two means, the importance of the Behrens- Fisher problem from a theoretical perspective is acknowledged, but it is concluded that this problem is irrelevant for applied research in psychology, education, and related disciplines. The focus is better placed on “shift in location” and, more importantly, “shift in location and change in scale” treatment alternatives

Efficient experimental design for the Behrens-Fisher problem with application to bioassay

A common approach in the design of experiment for the problem of comparing two means from a normal distribution is to assume knowledge of the ratio of the population variances. The optimal sampling ratio is proportional to the square root of this quantity. In this paper it is demonstrated that a misspecification of this ratio can cause a substantial loss in power of the corresponding tests. As a robust alternative a maximin approach is used to construct designs, which are efficient, whenever the experimenter is able to specify a specific region for the ratio of the population variances. The advantages of the robust designs for inference in the Behrens-Fisher problem are illustrated by means of a simulation study and an application to the design of experiment for bioassay is presented. --Behrens-Fisher problem,bioassay,design of experiment,local optimal design,robust designs

Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence

The Behrens-Fisher problem concerns the inferences for the difference between the means of two normal populations without making any assumption about the variances. Although the problem has been extensively studied in the literature, researchers cannot agree on its solution at present. In this paper, we propose a new method for dealing with the Behrens-Fisher problem in the Bayesian framework. The Bayesian evidence for testing the equality of two normal means and a credible interval at a specified level for the difference between the means are derived. Simulation studies are carried out to evaluate the performance of the provided Bayesian evidence

Robustness And Power Comparison Of The Mood-Westenberg And Siegel-Tukey Tests

The author examined how, in the context of experimental design, one might become aware of the Behrens-Fisher problem (heteroscedasticity) in order to apply an approximate solution, such as the Yuen\u27s statistic (1974). It was expected that both the Mood-Westenberg dispersion test (1948) and the Siegel-Tukey test (1960) would remain robust with respect to Type I and Type II error properties (and associated power levels) for detecting variance changes when their assumptions of equal means was slightly violated (i.e., the Behrens-Fisher problem). With the use of Monte Carlo Simulations, the author reviewed 34,606 permutations composed of interactions between various sample sizes, alpha levels, distributions/data sets, variance changes and means shifts. While the Mood-Westenberg (1948) and Siegel-Tukey (1960) tests both remained robust under certain conditions with respect to Type I and II error properties, the Siegel-Tukey test (1960) was by far the most robust of the two statistics, able to handle a more diverse set of conditions and would therefore be the statistic of choice in identifying the Behrens-Fisher problem