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

Misconceptions Leading to Choosing the t Test Over the Wilcoxon Mann-Whitney Test for Shift in Location Parameter

There exist many misconceptions in choosing the t over the Wilcoxon Rank-Sum test when testing for shift. Examples are given in the following three groups: (1) false statement, (2) true premise, but false conclusion, and (3) true statement irrelevant in choosing between the t test and the Wilcoxon Rank Sum test

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

The Impact Of Nested Testing On Experiment-Wise Type I Error Rate

When conducting a statistical test the initial risk that must be considered is a Type I error, also known as a false positive. The Type I error rate is set by nominal alpha, assuming all underlying conditions of the statistic are met. Experiment-wise Type I error inflation occurs when multiple tests are conducted overall for a single experiment. There is a growing trend in the social and behavioral sciences utilizing nested designs. A Monte Carlo study was conducted using a two layer design. Five theoretical distributions and four real datasets taken from Micceri (1989) were used, each with five different samples sizes and conducted with nominal alpha set to 0.05 and 0.01. These were conducted both unconditionally and conditionally. All permutations were executed for 1,000,000 repetitions. It was found that when conducted unconditionally, the experiment-wise Type I error rate increases from alpha = 0.05 to 0.10 and 0.01 increases to 0.02. Conditionally, it is extremely unlikely to ever find results for the factor, as it requires a statistically significant nest as a precursor, which leads to extremely reduced power. Hence, caution should be used when interpreting nested designs

Combining Quantum Mechanical Calculations And A χ^2 Fit In A Potential Energy Function For The CO_2 + O^+ Reaction

In order to compute a highly accurate statistical rate constant for the CO2 + O+ reaction, it is necessary to first calculate the potential energy of the system at many different geometric configurations. Quantum mechanical calculations are very time-consuming, making it difficult to obtain a sufficient number to allow for accurate interpolation. The number of quantum mechanical calculations required can be significantly reduced by using known relations in classical physics to calculate energy for configurations where the oxygen is relatively far from the CO2. A chi-squared fit to quantum mechanical points is obtained for these configurations, and the resulting parameters are used to generate an equation for the potential energy. This equation, combined with an interpolated set of quantum mechanical points to give the potential energy for configurations where the molecules are closer together, allows all configurations to be calculated accurately and efficiently

Statistical Reanalysis of Jewish Priests’ and Non-Priests’ Haplotypes Using Exact Methods

Researchers in an article appearing in Nature used asymptotic (i.e., large sample) chi-square tests in analyzing haplotypes of Y chromosomes using the polymerase chain reaction applied to genomic DNA from male Israeli, North American, and British Jews. The use of classical methods for analyzing extremely sparse contingency tables is frequently done, but with the advent of statistical software capable of conducting exact tests, researchers should certainly cease relying on outdated methods for small sample analyses. A reanalysis was conducted using modern statistical methods. Results and implications for using exact tests are discussed

New Effect Size Rules of Thumb

Recommendations to expand Cohen’s (1988) rules of thumb for interpreting effect sizes are given to include very small, very large, and huge effect sizes. The reasons for the expansion, and implications for designing Monte Carlo studies, are discussed