The distribution of time that people spend in physical activity of various intensities has important health implications. Physical activity (commonly categorised by the intensity into light, moderate and vigorous physical activity), sedentary behaviour and sleep, should not be analysed separately, because they are parts of a time-use composition with a natural constraint of 24 h/day. To find out how are relative reallocations of time between physical activity of various intensities associated with health, herewith we describe compositional scalar-on-function regression and a newly developed compositional functional isotemporal substitution analysis. Physical activity intensity data can be considered as probability density functions, which better reflects the continuous character of their measurement using accelerometers. These probability density functions are characterised by specific properties, such as scale invariance and relative scale, and they are geometrically represented using Bayes spaces with the Hilbert space structure. This makes possible to process them using standard methods of functional data analysis in the L2 space, via centred logratio (clr) transformation. The scalar-on-function regression with clr transformation of the explanatory probability density functions and compositional functional isotemporal substitution analysis were applied to a dataset from a cross-sectional study on adiposity conducted among school-aged children in the Czech Republic. Theoretical reallocations of time to physical activity of higher intensities were found to be associated with larger and more progressive expected decreases in adiposity. We obtained a detailed insight into the dose-response relationship between physical activity intensity and adiposity, which was enabled by using the compositional functional approach.

• Keywords
• Compositional scalar-on-function regression, isotemporal substitution, physical activity, probability density functions, sedentary behaviour, sleep,
• MeSH
• Adiposity * MeSH
• Bayes Theorem MeSH
• Time Factors MeSH
• Exercise * physiology   MeSH
• Child MeSH
• Humans MeSH
• Obesity * MeSH
• Cross-Sectional Studies MeSH
• Check Tag
• Child MeSH
• Humans MeSH
• Publication type
• Journal Article MeSH

Permutation methods are commonly used to test the significance of regressors of interest in general linear models (GLMs) for functional (image) data sets, in particular for neuroimaging applications as they rely on mild assumptions. Permutation inference for GLMs typically consists of three parts: choosing a relevant test statistic, computing pointwise permutation tests, and applying a multiple testing correction. We propose new multiple testing methods as an alternative to the commonly used maximum value of test statistics across the image. The new methods improve power and robustness against inhomogeneity of the test statistic across its domain. The methods rely on sorting the permuted functional test statistics based on pointwise rank measures; still, they can be implemented even for large data. The performance of the methods is demonstrated through a designed simulation experiment and an example of brain imaging data. We developed the R package GET, which can be used for the computation of the proposed procedures.

• Keywords
• function-on-scalar regression, general linear model, global envelope test, graphical method, multiple testing correction,
• MeSH
• Humans MeSH
• Linear Models MeSH
• Brain * diagnostic imaging   MeSH
• Neuroimaging * MeSH
• Computer Simulation MeSH
• Research Design MeSH
• Check Tag
• Humans MeSH
• Publication type
• Journal Article MeSH
• Research Support, Non-U.S. Gov't MeSH