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Introduction: Initiatives leading to more exposure variation are often suggested as a remedy supporting occupational health and reducing musculoskeletal disorders. Day-to-day variability within workers is a source of variation that can plausibly impact health. It may differ widely between workers, even if they have the same average exposure across days, but it has rarely been assessed in research. Compositional time-use data such as time spent in physical behaviors (e.g. sitting, standing, moving) need particularly adapted methods for documentation and analysis since they add up to a closed number, e.g. 24 h or 100%. The present study proposes a new metric for quantifying between-days variability in studies of compositional data.

Methods: The proposed ‘Aitchison dispersion index’ is based on the Aitchison distance metric published e.g. in 2000 [1]. For an individual having two or more measurement days, it is determined in three steps:

1. Calculate the mean exposure composition across days for the individual, e.g. percent time sitting, standing and moving
2. Calculate the Aitchison distance between this mean composition and the composition for each day, using the following equation: (attachment) with D: number of exposure coordinates (three in the example); xi and Xi: values of coordinate i (i = 1…D) in the daily and mean exposure compositions, respectively; g(x) and g(X): geometric mean values of the daily and mean exposure compositions, respectively.
3. Calculate the mean value of the daily distances, cf. 2, across all available days.

Results: Assume that sitting, standing and moving has been measured in two workers for two days each. Worker A has the compositions (70,20,10; day 1) and (50,30,20; day 2) with geometric means g(x) = 24.1 and 31.1, and worker B has (85,10,5) and (35,40,25), with g(x) = 16.2 and 32.7. Thus, the mean composition across the two days is the same for A and B, i.e. (60,25,15), with geometric mean g(X) = 28.2, while the variability between days is obviously not. The Aitchison distances to the mean for the two days in worker A are 0.40 and 0.35, and for worker B 1.11 and 0.84 (cf. 2 above). Thus, the average Aitchison distance, i.e. the ‘Aitchison dispersion index’ for the two workers is 0.38 and 0.98 (cf. 3 above), correctly reflecting that days for B are considerably more dispersed than for A.

Discussion: Metrics assessing day-to-day variability in exposure, even across weeks or months, are needed in future studies examining the extent to which this source of variation is predictive of musculoskeletal health and well-being. The proposed ‘Aitchison dispersion index’ is a compositional analogue to the standard deviation for quantifying the amount of day-to-day variability in exposure about its mean. It allows exposures to be multidimensional, and it operates in the geometric ‘simplex’ space required for compositional variables.

Conclusion: This paper proposes a compositional index for quantifying exposure variability between days within workers. Future studies will be needed to confirm its properties and predictive ability in epidemiologic studies of compositional exposures and occupational health.

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