Josep Antoni Martín Fernández

Out-of-control signals in three-part compositional T2 control chart

Interpretation of out-of-control signals in multivariate control charts is a persistent problem. Identifying the variables contributing to the signal is of crucial importance for process control. Both procedures turn out to be more complicated when observations are compositions (variables adding to a constant sum). Compositional T2 control chart () is suitable for monitoring compositions because it is based on a transformation of the data, which moves it from restricted to real space. This paper proposes a method for interpreting the out-of-control signals of the control chart for three-part compositions on the basis of an appropriate selection of the transformation. An example of industrial application illustrates the introduced techniques. Copyright

Signal Interpretation in Hotelling's T2 Control Chart for Compositional Data

This work has been partially financed by the Ministerio de Ciencia e Innovaci´on (Ref: MTM2012- 33236) and the Ag`encia de Gesti´o d’Ajuts Universitaris i de Recerca (AGAUR), Generalitat de Catalunya (Ref: 2014SGR551)ABSTRACT Nowadays, the control of the concentrations of elements is of crucial importance in industry. Concentrations are expressed in terms of proportions or percentages, which means that they are Compositional Data (CoDa). CoDa are defined as vectors of positive elements that represent parts of a whole and usually add to a constant sum. The classical T2 Control Chart is not appropriate for CoDa; rather, it is better to use a compositional T2 Control Chart (T2C CC). This article generalizes the interpretation of the out-of-control signals of the individual T2C CC for more than three components. We propose two methods for identifying the ratio of components that mainly contribute to the signal. The first one is suitable for low-dimensional problems and consists in finding the log ratio of the components that maximizes the univariate T2 statistic. The second approach is an optimized method for large-dimensional problems that simplifies the calculation by transforming the coordinates into an sphere. We illustrate the T2C CC signal interpretation with a practical example from the chemical and pharmaceutical industry.

Human development index, children’s health-related quality of life and movement behaviors: a compositional data analysis

PurposeHealth-related quality of life has been related to physical activity, sedentary behavior, and sleep among children from developed nations. These relationships have rarely been assessed in developing nations, nor have behaviors been considered in their true context, as mutually exclusive and exhaustive parts of the movement behavior composition. This study aimed to explore whether children’s health-related quality of life is related to their movement behavior composition and if the relationship differs according to human development index.MethodsChildren aged 9–11 years (n = 5855), from the 12-nation cross-sectional observational International Study of Childhood Obesity, Lifestyle and the Environment 2011–2013, self-reported their health-related quality of life (KIDSCREEN-10). Daily movement behaviors were from 24-h, 7-day accelerometry. Isometric log-ratio mixed-effect linear models were used to calculate estimates for difference in health-related quality of life for the reallocation of time between daily movement behaviors.ResultsChildren from countries of higher human development index reported stronger positive relationships between health-related quality of life and moderate-to-vigorous physical activity, relative to the remaining behaviors (r = 0.75, p = 0.005) than those from lower human development index countries. In the very high human development index strata alone, health-related quality of life was significantly related to the movement behavior composition (p = 0.005), with moderate-to-vigorous physical activity (relative to remaining behaviors) being positively associated with health-related quality of life.ConclusionsThe relationship between children’s health-related quality of life and their movement behaviors is moderated by their country’s human development index. This should be considered when 24-h movement behavior guidelines are developed for children around the world.