Inversion of 3×33\times 3 partitioned matrices in investigation of the twoepoch linear model with the nuisance parameters

summary:The estimation procedures in the multiepoch (and specially twoepoch) linear regression models with the nuisance parameters that were described in [2], Chapter 9, frequently need finding the inverse of a $3 \times 3$ partitioned matrix. We use different kinds of such inversion in dependence on simplicity of the result, similarly as in well known Rohde formula for $2\times 2$ partitioned matrix. We will show some of these formulas, also methods how to get the other formulas, and then we applicate the formulas in estimation of the mean value parameters in the twoepoch linear regression model with the nuisance parameters

Robust Principal Component Analysis for Compositional Tables

A data table which is arranged according to two factors can often be considered as a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information would consist of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decomposed into independent and interactive parts, and orthonormal coordinates can be assigned to these parts. However, these coordinates usually require some prior knowledge about the data, and they are not easy to handle for exploring the relationships between the given factors. Here we propose a special choice of coordinates with a direct relation to centered logratio (clr) coefficients, which are particularly useful for an interpretation in terms of the original cells of the tables. With these coordinates, robust principal component analysis (PCA) is performed for dimension reduction, allowing to investigate the relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply robust PCA, which would otherwise suffer from the singularity of clr coefficients.Comment: 20 pages, 2 figure

General approach to coordinate representation of compositional tables

This is the peer reviewed version which has been published in final form at [https://doi.org/10.1111/sjos.12326]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.Compositional tables can be considered a continuous counterpart to the well-known contingency tables. Their cells, which generally contain positive real numbers rather than just counts, carry relative information about relationships between two factors. Hence, compositional tables can be seen as a generalization of (vector) compositional data. Due to their relative character, compositions are commonly expressed in orthonormal coordinates using a sequential binary partition prior to being further processed by standard statistical tools. Unfortunately, the resulting coordinates do not respect the two-dimensional nature of compositional tables. Information about relationship between factors is thus not well captured. The aim of this paper is to present a general system of orthonormal coordinates with respect to the Aitchison geometry, which allows for an analysis of the interactions between factors in a compositional table. This is achieved using logarithms of odds ratios, which are also widely used in the context of contingency tables

The Goldilocks Day for healthy adiposity measures among children and adolescents

BackgroundThe optimal balance of time spent on daily movement behaviors (“The Goldilocks Day”) associated with childhood obesity remains unknown.ObjectiveTo estimate the optimal durations of sleep, sedentary behavior (SB), light physical activity (LPA), and moderate-to-vigorous physical activity (MPVA) associated with excess adiposity in a paediatric population.MethodsAccelerometer-measured 24-h movement behaviors were obtained from 659 Czech children and adolescents (8-18-year-olds). Adiposity indicators were body mass index z-score, fat mass percentage, fat-free mass index, and visceral adipose tissue. Excess adiposity was defined as exceeding the 85th percentile for an adiposity indicator. Compositional regression analyses were used investigate the associations between movement behaviors and adiposity indicators and estimating “The Goldilocks Day.”ResultsThe movement behavior composition was associated with visceral adipose tissue (Fdf1 = 3,df2 = 317 = 3.672, p = 0.013) and fat mass percentage (Fdf1 = 3,df2 = 289 = 2.733, p = 0.044) among children and adolescents. The Goldilocks Day consisted of 8.5 h of sleep, 10.8 h of SB, 3.9 h of LPA, and 0.8 h of MVPA among children and 7.5 h of sleep, 12.4 h of SB, 3.6 h of LPA, and 0.5 h of MVPA among adolescents.ConclusionOptimizing the time spent sleeping, and in sedentary and physical activities appears to be important in the prevention of excess adiposity