Interpretation of Compositional Regression with Application to Time Budget Analysis

Abstract

Regression with compositional response or covariates, or even regression between parts of a composition, is frequently employed in social sciences. Among other possible applications, it may help to reveal interesting features in time allocation analysis. As individual activities represent relative contributions to the total amount of time, statistical processing of raw data (frequently represented directly as proportions or percentages) using standard methods may lead to biased results. Specific geometrical features of time budget variables are captured by the logratio methodology of compositional data, whose aim is to build (preferably orthonormal) coordinates to be applied with popular statistical methods. The aim of this paper is to present recent tools of regression analysis within the logratio methodology and apply them to reveal potential relationships among psychometric indicators in a real-world data set. In particular, orthogonal logratio coordinates have been introduced to enhance the interpretability of coefficients in regression models.

Robust Compositional Analysis of Physical Activity and Sedentary Behaviour Data

“… in (4) are unknown regression coefficients and is the random error term in the th model, [ 33 ]; note, however, that these coefficients are not the same as in model (3). In both models above, it is assumed that just the response is measured with errors.…”

“…Then contains all the relative information about with respect to the average contribution of the remaining parts of the subcomposition. Accordingly, our focus is on the interpretation of the estimate from each model, [ 33 ], because it stands for a positive or negative influence of dominating within the subcomposition ( ) on dominance of within the parent composition (as conveyed by ).…”

Compositional data analysis for physical activity, sedentary time and sleep research

“…However, the interpretation of log-ratio regression coefficients is not straight-forward. 23 As daily activity data have a meaningful total, i.e., 24-hours or 1440 minutes, regression coefficients can be interpreted on a meaningful scale.…”

Cox regression survival analysis with compositional covariates: Application to modelling mortality risk from 24-h physical activity patterns

“…Note that only the ilr-coordinates are rotated. For clarity we also re-express equations (19) and (20) in terms of equation 39as…”

“…The log-ratio methodology for compositional data analysis 16 provides a coherent statistical framework which resolves these issues through the use of log-ratio type representations of the compositional variables. Although approaches have been presented for ordinary regression modelling with compositions, 9,[16][17][18][19][20] and survival analysis has been applied in respect of compositional outcomes such as the composition of different strains of bacteria, 12 to our knowledge survival analysis based on the Cox regression model, or indeed survival analysis in general, with compositional explanatory variables remains formally unexplored.…”