scholarly journals Statistics of eigenvalue dispersion indices: quantifying the magnitude of phenotypic integration

2021 ◽  
Author(s):  
Junya Watanabe
Keyword(s):  
Sample Size ◽  
Correlation Matrix ◽  
Sampling Bias ◽  
Phenotypic Integration ◽  
Test Statistic ◽  
Moderate Sample Size ◽  
Simulation Based ◽  
Dispersion Indices ◽  
Mean And Variance ◽  
Approximate Expressions

Quantification of the magnitude of covariation plays a major role in the studies of phenotypic integration, for which statistics based on dispersion of eigenvalues of a covariance or correlation matrix—eigenvalue dispersion indices—are commonly used. However, their use has been hindered by a lack of clear understandings on their statistical meaning and sampling properties such as the magnitude of sampling bias and error. This study remedies these issues by investigating properties of these statistics with both analytic and simulation-based approaches. The relative eigenvalue variance of a covariance matrix is known in the statistical literature as a test statistic for sphericity, thus is an appropriate measure of eccentricity of variation. The same of a correlation matrix is exactly equal to the average squared correlation, thus is a clear measure of overall integration. Exact and approximate expressions for the mean and variance of these statistics are analytically derived for the null and arbitrary conditions under multivariate normality, clarifying the effects of sample size N, number of variables p, and parameters on the sampling bias and error. Accuracy of the approximate expressions are evaluated with simulations, confirming that most of them work reasonably well with a moderate sample size (N ≥ 16–64). Importantly, sampling properties of these indices are not adversely affected by high p:N ratio, promising their utility in high-dimensional phenotypic analyses. These statistics can potentially be applied to shape variables and phylogenetically structured data, for which necessary assumptions and modifications are presented.

scholarly journals Parallelized calculation of permutation tests

2020 ◽  
Author(s):  
Markus Ekvall ◽  
Michael Höhle ◽  
Lukas Käll
Keyword(s):  
Dynamic Programming ◽  
Sample Size ◽  
Permutation Test ◽  
Statistical Tests ◽  
Permutation Tests ◽  
Supplementary Information ◽  
Attractive Alternative ◽  
Test Statistic ◽  
Sample Distribution ◽  
Running Time

Abstract Motivation Permutation tests offer a straightforward framework to assess the significance of differences in sample statistics. A significant advantage of permutation tests are the relatively few assumptions about the distribution of the test statistic are needed, as they rely on the assumption of exchangeability of the group labels. They have great value, as they allow a sensitivity analysis to determine the extent to which the assumed broad sample distribution of the test statistic applies. However, in this situation, permutation tests are rarely applied because the running time of naïve implementations is too slow and grows exponentially with the sample size. Nevertheless, continued development in the 1980s introduced dynamic programming algorithms that compute exact permutation tests in polynomial time. Albeit this significant running time reduction, the exact test has not yet become one of the predominant statistical tests for medium sample size. Here, we propose a computational parallelization of one such dynamic programming-based permutation test, the Green algorithm, which makes the permutation test more attractive. Results Parallelization of the Green algorithm was found possible by non-trivial rearrangement of the structure of the algorithm. A speed-up—by orders of magnitude—is achievable by executing the parallelized algorithm on a GPU. We demonstrate that the execution time essentially becomes a non-issue for sample sizes, even as high as hundreds of samples. This improvement makes our method an attractive alternative to, e.g. the widely used asymptotic Mann-Whitney U-test. Availabilityand implementation In Python 3 code from the GitHub repository https://github.com/statisticalbiotechnology/parallelPermutationTest under an Apache 2.0 license. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals The Effects of Manifest Residual Variances, Indicator Communality, and Sample Size on the χ2-Test Statistic of the Metric Invariance Model

2019 ◽  
Author(s):  
Eric Klopp ◽  
Stefan Klößner
Keyword(s):  
Sample Size ◽  
Measurement Invariance ◽  
Signal To Noise Ratio ◽  
Practical Implication ◽  
Residual Variance ◽  
Test Statistic ◽  
Monte Carlo Studies ◽  
Key Factor ◽  
Metric Measurement ◽  
Invariance Model

In this contribution, we investigate the effects of manifest residual variance, indicator communality and sample size on the χ2-test statistic of the metric measurement invariance model, i.e. the model with equality constraints on all loadings. We demonstrate by means of Monte Carlo studies that the χ2-test statistic relates inversely to manifest residual variance, whereas sample size and χ2-test statistic show the well-known pro- portional relation. Moreover, we consider indicator communality as a key factor for the size of the χ2-test statistic. In this context, we introduce the concept of signal-to-noise ratio as a tool for studying the effects of manifest residual error and indicator commu- nality and demonstrate its use with some examples. Finally, we discuss the limitations of this contribution and its practical implication for the analysis of metric measurement invariance models.

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