A fully objective Bayesian approach for the Behrens-Fisher problem using historical studies

For in vivo research experiments with small sample sizes and available historical data, we propose a sequential Bayesian method for the Behrens-Fisher problem. We consider it as a model choice question with two models in competition: one for which the two expectations are equal and one for which they are different. The choice between the two models is performed through a Bayesian analysis, based on a robust choice of combined objective and subjective priors, set on the parameters space and on the models space. Three steps are necessary to evaluate the posterior probability of each model using two historical datasets similar to the one of interest. Starting from the Jeffreys prior, a posterior using a first historical dataset is deduced and allows to calibrate the Normal-Gamma informative priors for the second historical dataset analysis, in addition to a uniform prior on the model space. From this second step, a new posterior on the parameter space and the models space can be used as the objective informative prior for the last Bayesian analysis. Bayesian and frequentist methods have been compared on simulated and real data. In accordance with FDA recommendations, control of type I and type II error rates has been evaluated. The proposed method controls them even if the historical experiments are not completely similar to the one of interest

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

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

Resampling-based inference methods for repeated measures data with missing values

The primary objective of this dissertation was to (i) develop novel resampling approaches for handling repeated measures data with missing values, (ii) compare their empirical power against other existing approaches using a Monte Carlo simulation study, and (iii) pinpoint the limitations of some common approaches, particularly for small sample sizes. This dissertation investigates four different statistical problems. The first is semiparametric inference for comparing means of matched pairs with missing data in both arms. Therein, we propose two novel randomization techniques; a weighted combination test and a multiplication combination test. They are based upon combining separate results of the permutation versions of the paired t-test and Welch test for the completely observed pairs and the incompletely observed components, respectively. As second problem, we consider the same setting but missingness in one arm only. There, we investigate a Wald-type statistic (WTS), an ANOVA-type statistic (ATS), and a modified ANOVA-type statistic (MATS). However, ATS and MATS are not distribution free under the null hypothesis, and WTS suffers from the slow convergence to its limiting 2 distribution. Thus, we develop asymptotic model-based bootstrap versions of these tests. The third problem is on nonparametric rank-based inference for matched pairs with incompleteness in both arms. In this more general setup, the only requirement is that the marginal distributions are not one point distributions. Therein, we propose novel multiplication combination tests that can handle three different testing problems, including the nonparametric Behrens-Fisher problem (Hp 0 : {p = 1/2}). Finally, the fourth problem is nonparametric rank-based inference for incompletely observed factorial designs with repeated measures. Therein, we develop a wild bootstrap approach combined with quadratic form-type test statistics (WTS, ATS, and MATS). These rank-based methods can be applied to both continuous and ordinal or ordered categorical data and (some) allow for singular covariance matrices. In addition to theoretically proving the asymptotic correctness of all the proposed procedures, extensive simulation studies demonstrate their favorable small samples properties in comparison to classical parametric tests. We also motivate and validate our approaches using real-life data examples from a variety of fields

Market linkages, variance spillovers and correlation stability: empirical evidences of financial contagion

We propose a simultaneous equation system with GARCH errors to model the contemporaneous relations among Asian and American stock markets. On the estimated residuals, we evaluate the correlation matrix over rolling windows and introduce a correlation matrix distance, which allows both a graphical analysis and the development of a statistical test of correlation movements. Furthermore, we introduce a methodology that can be used for identifying turmoil periods on a data-driven basis. We employ the previous results in the analysis of the contagion issue between Asian and American stock markets. Our results shows some evidence of contagion and the proposed statistics identifies, on a data-driven basis, turmoil periods consistent with the ones currently assumed in the literature.Financial market contagion, Market linkages, Variance spillovers, Dynamic correlations, Rolling correlations, Transformed correlations

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