Citation for published version:
Dumuid, D, Stanford, TE, Martin-Fernández, JA, Pedisic, Z, Maher, C, Lewis, L, Hron, K, Katzmarzyk, PT,
Chaput, J-P, Fogelholm, M, Hu, G, Lambert, EV, Maia, J, Sarmiento, OL, Standage, M, Barreira, TV, Broyles,
ST, Tudor-Locke, C, Tremblay, MS & Olds, T 2018, 'Compositional data analysis for physical activity, sedentary
time and sleep research', Statistical Methods in Medical Research, vol. 27, no. 12, pp. 3726-3738.
https://doi.org/10.1177/0962280217710835
DOI:
10.1177/0962280217710835
Publication date:
2018
Document Version
Peer reviewed version
Link to publication
Dorothea Dumuid, Tyman E Stanford, Josep-Antoni Martin-Fernández, Željko Pediši, Carol A Maher, Lucy K
Lewis, Karel Hron, Peter T Katzmarzyk, Jean-Philippe Chaput, Mikael Fogelholm, Gang Hu, Estelle V Lambert,
José Maia, Olga L Sarmiento, Martyn Standage, Tiago V Barreira, Stephanie T Broyles, Catrine Tudor-Locke,
Mark S Tremblay, Timothy Olds, Compositional data analysis for physical activity, sedentary time and sleep
research, Statistical Methods in Medical Research. Copyright © 2017 The Author(s). Reprinted by permission of
SAGE Publications.
University of Bath
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1
1 Introduction: Daily activity behaviours are compositional by nature
1
2
Physical inactivity is considered to be a major risk factor for non-communicable disease
3
and premature death.
1
A global economic analysis has estimated the health-care system
4
cost of physical inactivity in 2013 alone to be $ (INT$) 53.8 billion.
2
To produce such
5
economic estimates, physical inactivity is defined as relatively lower amounts of
6
moderate-to-vigorous-intensity physical activity (MVPA), but underlying analyses fail
7
to account for the fact that when time in MVPA is reduced, a subsequent and equal
8
increase in time must be distributed to the remaining behaviour domains: sleep,
9
sedentary time and light-intensity physical activity (light PA), to represent the finite 24-
10
hours, or 1440 minutes, in any given day. These other non-MVPA behaviours may
11
themselves have positive or negative effects on health and mortality.
3-6
Therefore, the
12
health and economic burden of physical inactivity per se remains unclear. Similar
13
estimates for sedentary time
7
are uncertain for the same reason, i.e., they fail to
14
adequately account for other behaviours. Pedišić
8
argued that studies on health
15
outcomes of sleep duration and light PA can be put under the same scrutiny.
16
17
Time spent in MVPA represents one of the exhaustive and mutually exclusive
18
components of an individual’s 24-h day. The non-MVPA time remaining within an
19
individual’s day can be partitioned into light PA, sedentary time and sleep and all of
20
2
them can be considered as relative contributions to the overall time budget. The defined
1
behaviours (MVPA, light PA, sedentary time and sleep) are therefore compositional
2
data, and have important properties that must be respected.
9
3
4
Consider a vector
with positive components, where
,
5
and is the closure constant. The sample space of the vector can thus be represented
6
by the D-part simplex (S
D
), which is a (D-1)-dimensional subset of the real space (R
D
)
7
due to the constant sum constraint of . Compositional data are scale invariant
9
because
8
the application of a common factor a to the parts
where ensures the
9
relative difference between the parts is maintained, as
. The
10
numerical value of the closure constant (e.g., 24 h, one week, one month) is irrelevant.
11
Daily behaviours could equivalently be measured in hours, minutes or percentages as
12
the data convey only relative information. The property of scale invariance means
13
compositional data are in fact elements of equivalence classes of proportional vectors.
10
14
Accordingly, the simplex is the sample space of representatives of compositional data
15
with a chosen constant sum constraint. Specific properties of compositional data are
16
followed by a natural geometry, known as the Aitchison geometry.
11
The closure
17
constant representation of compositions imposes perfect multi-collinearity among the
18
components, causing the covariance structure of the data to be negatively biased.
9
19
Accordingly, traditional statistical methods for unconstrained variables in real space
20
3
(e.g., t-tests, multiple linear regression) are not predicative with respect to the specific
1
geometry of the simplex sample space, and should not be used for absolute or raw
2
measures of time spent in daily behaviours.
12
3
4
2 The log-ratio approach for compositional data analysis
5
The invalidity of standard multivariate techniques for analyzing untransformed or raw
6
compositional data was recognized in scientific fields decades ago,
13
and in the 1980s
7
Aitchison proposed a new methodology for the analysis of compositional data.
9
The
8
methodology is based on the premise that any composition (e.g., an individual’s daily
9
time budget) can be expressed in terms of ratios of its parts (e.g., duration of sleep,
10
sedentary time, light PA and MVPA). The expression of compositional data as log-ratio
11
coordinates transfers them from the constrained simplex space to the unconstrained real
12
space, where traditional multivariate statistics may be applied.
14
The presence of zeros
13
in a compositional dataset prohibits applying log-ratio coordinates. Several methods
14
have been proposed to deal with zeros;
15
however they are beyond the scope of this
15
paper.
16
17
A number of log-ratio coordinate systems for compositional data have been described.
11
18
One such coordinate system, the additive log-ratio (alr), has coordinates, , defined by
19
. (1)
20
4
However, the alr coordinates are asymmetric, because the components x
1
,x
2
,…, x
D-1
are
1
divided by the component x
D
. Moreover, they are not isometric, i.e., distances and
2
angles in the Aitchison geometry are violated by using the alr coordinates, limiting their
3
use in statistical applications. This means that the system of alr coordinates in the
4
Aitchison geometry is oblique, and traditional statistical methods which assume
5
orthogonal coordinates therefore cannot be directly applied. Another coordinate system
6
is the centred log-ratio (clr) coordinate system, , defined as
7
8
whereis the geometric mean of all the D components of the vector The clr are
9
symmetric and isometric; they produce a singular covariance matrix because
10
. The clr are, strictly speaking, not coordinates but coefficients with respect to a
11
generating system. The covariance matrix of clr coefficients is singular, so the clr
12
coefficients cannot be fully utilized as independent variables in multiple regression
13
analysis.
14
15
The singularity problem of the clr can be overcome by the use of an isometric log-ratio
16
(ilr) coordinate system. Isometric log ratio coordinates form an isometric mapping of
17
the composition from the simplex sample space to the real space.
16
To construct the ilr
18
coordinates, an orthonormal basis coherent with the Aitchison geometry is created in the
19
(D-1)-dimensional hyperplane of the clr coordinates. Many possible orthonormal
20
