Statistical inference with paired observations and independent observations in two samples

A frequently asked question in quantitative research is how to compare two samples that include some combination of paired observations and unpaired observations. This scenario is referred to as `partially overlapping samples'. Most frequently the desired comparison is that of central location. Depending on the context, the research question could be a comparison of means, distributions, proportions or variances. Approaches that discard either the paired observations or the independent observations are customary. Existing approaches evoke much criticism. Approaches that make use of all available data are becoming more prominent. Traditional and modern approaches for the analyses for each of these research questions are reviewed. Novel solutions for each of the research questions are developed and explored using simulation. Results show that proposed tests which report a direct measurable difference between two groups provide the best solutions

Preliminary testing: The devil of statistics?

In quantitative research, the selection of the most appropriate statistical test for the comparison of two-independent samples can be problematic. There is a lack of consensus in the Statistics community regarding the appropriate approach; particularly towards assessing assumptions of normality and equal variances. The lack of clarity in the appropriate strategy affects the reproducibility of results. Statistical packages performing different tests under the same name, only adds to this issue.The process of preliminary testing assumptions of a test using the sample data, before performing a test conditional upon the preliminary test, is performed by some researchers; this practice is often criticised in the literature. Preliminary testing is typically performed at the arbitrary 5% significance level. In this paper this process is reviewed, and additional results are given using simulation, examining a procedure with normality and equal variance preliminary tests

Why Welch’s test is Type I error robust

The comparison of two means is one of the most commonly applied statistical procedures in psychology. The independent samples t-test corrected for unequal variances is commonly known as Welch’s test, and is widely considered to be a robust alternative to the independent samples t-test. The properties of Welch’s test that make it Type I error robust are examined. The degrees of freedom used in Welch’s test are a random variable, the distributions of which are examined using simulation. It is shown how the distribution for the degrees of freedom is dependent on the sample sizes and the variances of the samples. The impact of sample variances on the degrees of freedom, the resultant critical value and the test statistic is considered, and hence gives an insight into why Welch’s test is Type I error robust under normality

How to compare the means of two samples that include paired observations and independent observations: A companion to Derrick, Russ, Toher and White (2017)

Standard approaches for comparing the means of two samples, comprising both paired observations and independent observations, involve the discarding of valuable information. An alternative test which uses all of the available data, is the partially overlapping samples t-test. Two variations of the test are available, one assuming equal variances, and one assuming separate variances. Issues with standard procedures, and considerations for choosing appropriate tests in the partially overlapping scenario are discussed. An example with details of how to apply the partially overlapping samples t-test is given along with an R package that implement these new tests

An inverse normal transformation solution for the comparison of two samples that contain both paired observations and independent observations

Inverse normal transformations applied to the partially overlapping samples t-tests by Derrick et.al. (2017) are considered for their Type I error robustness and power. The inverse normal transformation solutions proposed in this paper are shown to maintain Type I error robustness. For increasing degrees of skewness they also offer improved power relative to the parametric partially overlapping samples t-tests. The power when using inverse normal transformation solutions are comparable to rank based non-parametric solutions

Test statistics for the comparison of means for two samples that include both paired and independent observations

©2017 JMASM, Inc. Standard approaches for analyzing the difference in two means, where partially overlapping samples are present, are less than desirable. Here are introduced two test statistics, making reference to the t-distribution. It is shown that these test statistics are Type I error robust, and more powerful than standard tests

Tests for equality of variances between two samples which contain both paired observations and independent observations

Tests for equality of variances between two samples which contain both paired observations and independent observations are explored using simulation. New solutions which make use of all of the available data are put forward. These new approaches are compared against standard approaches that discard either the paired observations or the independent observations. The approaches are assessed under equal variances and unequal variances, for two samples taken from the same distribution. The results show that the newly proposed solutions offer Type I error robust alternatives for the comparison of variances, when both samples are taken from the same distribution

Covalent Post-assembly Modification Triggers Multiple Structural Transformations of a Tetrazine-Edged Fe4L6 Tetrahedron

Covalent post-assembly modification (PAM) reactions are useful synthetic tools for functionalizing and stabilizing self-assembled metal-organic complexes. Recently, PAM reactions have also been explored as stimuli for triggering supramolecular structural transformations. Herein we demonstrate the use of inverse electron-demand Diels-Alder (IEDDA) PAM reactions to induce supramolecular structural transformations starting from a tetrazine-edged FeII4L6 tetrahedral precursor. Following PAM, this tetrahedron rearranged to form three different architectures depending on the addition of other stimuli: an electron-rich aniline or a templating anion. By tracing the stimulus-response relationships within the system, we deciphered a network of transformations that mapped different combinations of stimuli onto specific transformation products. Given the many functions being developed for self-assembled three-dimensional architectures, this newly established ability to control the interconversion between structures using combinations of different stimulus types may serve as the basis for switching the functions expressed within a system.D.A.R. acknowledges the Gates Cambridge Trust. B.S.P. acknowledges the Royal Commission for the Exhibition of 1851 Fellowship and Corpus Christi College, Cambridge. This work was supported by the UK Engineering and Physical Sciences Research Council (EP/M01083X/1)