## Appendix A

The associations between physical activities at work and leisure time were analyzed by multivariate linear regression models on isometric log-ratio (ilr) coordinates of the 4-part OPA and 5-part LTPA subcompositions of low socioeconomic workers.

Following [

30,

41], for a

D − part composition

$\mathbf{x}=\left({x}_{1},\dots ,{x}_{D}\right)$ we can obtain a real vector

$\mathit{z}=\left({z}_{1},{z}_{2},\dots ,{z}_{D-1}\right)$ of

$D-1$ ilr-coordinates, where

This is a particular choice of ilr-coordinates, recently called pivot coordinates [

58], by which all relative information about the first part of the composition (

${x}_{1}$) is included in the first ilr-coordinate (

${z}_{1}$). The remaining ilr-coordinates (

${z}_{2},{z}_{3},\dots ,{z}_{D}$) contain no information about the first part of the composition. This way,

${z}_{1}$ represents the relative importance or dominance of the part

${x}_{1}$ with respect to an (geometric) average of the remaining parts in the composition. Note that an infinite number of ilr coordinate systems can be defined; however, they are just geometric rotations from each other. This is a useful property in statistical analysis as it enables to use an arbitrary choice of ilr coordinates to obtain the required output.

Accordingly, with the objective of investigating the associations between parts of the OPA and LTPA subcompositions, the parts were sequentially rearranged (permuted) within the respective compositions to place each one of them at the first position before ilr transformation using Equation (A1).

For example, for the regression model 1a below, the ilr-coordinates of the 4-part OPA subcomposition were computed as

Giving rise to an OPA ilr-coordinate vector $ilr\left(\mathit{Z}\right)=\left(\begin{array}{c}{z}_{1}^{\ast}\\ {z}_{2}^{\ast}\\ {z}_{3}^{\ast}\end{array}\right)$.

The ilr-coordinates of the 5-part LTPA composition for model 1a were computed as

Giving rise to a LTPA ilr-coordinate vector $ilr\left(\mathit{Y}\right)=\left(\begin{array}{c}{y}_{1}^{\ast}\\ {y}_{2}^{\ast}\\ {y}_{3}^{\ast}\\ {y}_{4}^{\ast}\end{array}\right)$.

The generic compositional linear regression model was then defined as

Like in ordinary regression, ${\beta}_{0}$ represents the model intercept; whereas ${\beta}_{1}$, ${\beta}_{2}$ and ${\beta}_{3}$ are the regression coefficients associated to each OPA ilr-coordinate in $ilr\left(\mathit{Z}\right)$.

Based on Equation (A2), a total of 16 compositional linear regression models were constructed and fitted as described above, sequentially rearranging the first part of either the OPA or LTPA subcompositions (note that a model with dominance of LTPA time in bed, TIB, as response variable was not considered). Thus, permuting the OPA parts we defined models 1 to 4:

By then re-arranging LTPA parts for each model we obtained versions a, b, c and d:

- 1.
Models 1a, 2a, 3a and 4a used $ilr\left({y}_{1}^{\ast}\right)=\sqrt{\frac{4}{5}}\mathrm{ln}\left(\frac{S{B}_{leis}{}_{i}}{\sqrt[4]{wal{k}_{leis}{}_{i}\ast stan{d}_{leis}{}_{i}\ast HiP{A}_{leis}{}_{i}\ast TI{B}_{i}}}\right)$.

- 2.
Models 1b, 2b, 3b and 4b used $ilr\left({y}_{1}^{\ast}\right)=\sqrt{\frac{4}{5}}\mathrm{ln}\left(\frac{wal{k}_{leis}{}_{i}}{\sqrt[4]{stan{d}_{leis}{}_{i}\ast HiP{A}_{leis}{}_{i}\ast S{B}_{leis}{}_{i}\ast TI{B}_{i}}}\right)$.

- 3.
Models 1c, 2c, 3c and 4c used $ilr\left({y}_{1}^{\ast}\right)=\sqrt{\frac{4}{5}}\mathrm{ln}\left(\frac{stan{d}_{leis}{}_{i}}{\sqrt[4]{HiP{A}_{leis}{}_{i}\ast S{B}_{leis}{}_{i}\ast wal{k}_{leis}{}_{i}\ast TI{B}_{i}}}\right)$.

- 4.
Models 1d, 2d, 3d and 4d used $ilr\left({y}_{1}^{\ast}\right)=\sqrt{\frac{4}{5}}\mathrm{ln}\left(\frac{HiP{A}_{leis}{}_{i}}{\sqrt[4]{S{B}_{leis}{}_{i}\ast wal{k}_{leis}{}_{i}\ast stan{d}_{leis}{}_{i}\ast TI{B}_{i}}}\right)$.

For each model, the most relevant statistical significance test for our purposes was then performed on the ${\beta}_{1}$ coefficient of the regression models, which accounts for the association between the first ilr-coordinates of the OPA and LTPA subcompositions after accounting for all the other covariates. Importantly, although we focus on ${\beta}_{1}$, all the other ilr-coordinates must be included in the model to account for the intrinsic inter-dependences between parts of the respective compositions.

Below are examples of output tables for model 1a and model 2b among men and women, respectively. Note that the

${\beta}_{1}$ coefficient (highlighted in output tables) corresponds to the first ilr-coordinate and to the result shown in

Table 4 and

Table 5. The results of the remaining models (i.e., model 1c, 1d, 2a, 2b, 2c, 2d, etc.) are shown in

Table 4 and

Table 5.

**Table A1.**
Output table for model 1a (among men, N = 400).

**Table A1.**
Output table for model 1a (among men, N = 400).

Variable | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{1}}^{\mathbf{\ast}}\mathbf{\right)}$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{2}}^{\mathbf{\ast}}\mathbf{\right)}$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{3}}^{\mathbf{\ast}}\mathbf{\right)}$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{4}}^{\mathbf{\ast}}\mathbf{\right)}$ |
---|

$\widehat{{\mathit{\beta}}_{\mathbf{1}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{2}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{3}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{4}}}$ | p-Value |
---|

Intercept | 1.207 | <0.001 | −0.105 | 0.650 | 1.138 | <0.001 | −3.339 | <0.001 |

$\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{z}}_{\mathbf{1}}^{\mathbf{\ast}}\mathbf{\right)}$ | **−0.063** | **0.332** | 0.235 | <0.001 | 0.079 | 0.292 | 0.053 | 0.677 |

$ilr\left({z}_{2}^{\ast}\right)$ | 0.014 | 0.736 | −0.107 | 0.002 | −0.049 | 0.292 | −0.002 | 0.766 |

$ilr\left({z}_{3}^{\ast}\right)$ | −0.033 | 0.271 | 0.052 | 0.049 | 0.032 | 0.365 | −0.021 | 0.719 |

Average work hours | 0.065 | 0.433 | 0.079 | 0.272 | 0.055 | 0.565 | −0.243 | 0.133 |

Age | 0.002 | 0.324 | −0.003 | 0.106 | −0.001 | 0.688 | 0.005 | 0.233 |

BMI | 0.022 | <0.001 | 0.011 | 0.016 | 0.009 | 0.159 | −0.031 | 0.002 |

Shift work | 0.002 | 0.978 | 0.013 | 0.785 | −0.065 | 0.312 | −0.105 | 0.334 |

One pain-site | 0.099 | 0.337 | −0.178 | 0.048 | −0.116 | 0.324 | 0.023 | 0.907 |

Two pain-sites | 0.050 | 0.589 | −0.214 | 0.008 | −0.128 | 0.225 | 0.096 | 0.590 |

Three pain-sites | −0.004 | 0.964 | −0.239 | 0.005 | −0.253 | 0.019 | 0.264 | 0.146 |

**Table A2.**
Output for model 2b (among women, N = 400).

**Table A2.**
Output for model 2b (among women, N = 400).

Variable | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{1}}^{\mathbf{\ast}}\mathbf{\right)}$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{2}}^{\mathbf{\ast}}\mathbf{\right)}$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{3}}^{\mathbf{\ast}}\right)$ | Estimates for $\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{y}}_{\mathbf{4}}^{\mathbf{\ast}}\mathbf{\right)}$ |
---|

Estimates for $\widehat{{\mathit{\beta}}_{\mathbf{1}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{2}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{3}}}$ | p-Value | $\widehat{{\mathit{\beta}}_{\mathbf{4}}}$ | p-Value |
---|

Intercept | −1.014 | <0.001 | 0.535 | 0.064 | 0.496 | 0.204 | −1.826 | 0.002 |

$\mathit{i}\mathit{l}\mathit{r}\mathbf{\left(}{\mathit{z}}_{\mathbf{1}}^{\mathbf{\ast}}\mathbf{\right)}$ | **−0.080** | **0.095** | 0.136 | 0.017 | −0.032 | 0.677 | −0.022 | 0.847 |

$ilr\left({z}_{2}^{\ast}\right)$ | 0.101 | 0.041 | −0.097 | 0.096 | 0.161 | 0.041 | −0.210 | 0.071 |

$ilr\left({z}_{3}^{\ast}\right)$ | 0.022 | 0.267 | 0.029 | 0.224 | 0.098 | 0.002 | −0.180 | 0.001 |

Average work hours | 0.026 | 0.686 | 0.090 | 0.245 | −0.112 | 0.284 | 0.149 | 0.332 |

Age | 0.001 | 0.793 | −0.002 | 0.482 | 0.005 | 0.163 | 0.001 | 0.919 |

BMI | 0.013 | <0.001 | 0.007 | 0.077 | 0.011 | 0.034 | −0.014 | 0.079 |

Shift work | −0.016 | 0.725 | −0.023 | 0.677 | 0.006 | 0.939 | −0.078 | 0.479 |

One pain-site | 0.106 | 0.233 | −0.053 | 0.613 | 0.481 | 0.735 | −0.003 | 0.990 |

Two pain-sites | 0.124 | 0.141 | −0.051 | 0.609 | 0.017 | 0.899 | −0.013 | 0.949 |

Three pain-sites | 0.087 | 0.301 | −0.094 | 0.347 | −0.019 | 0.883 | 0.036 | 0.954 |

## Appendix B

As outlined in the manuscript, compositional isotemporal substitution models were used to determine expected changes in workers’ LTPA subcomposition when reallocating fixed time durations to one part of the OPA subcomposition from the remaining parts. A detailed description of the method based on ilr linear regression can be found in Dumuid et al. [

43,

59].

In short, using a reference D-part composition $\mathbf{x}=\left({x}_{1},\dots ,{x}_{D}\right)$ as starting point, we can consider the reallocation of a fixed duration of time, $\Delta t$, from one part of the composition to another. In our case, the reference composition was equal to the compositional geometric mean of the workers’ OPA subcomposition. It was closed to sum up to 1 and hence expressed in proportions to facilitate manipulations. We can then consider a relative increase in ${x}_{1}$ by a factor $1+r$ with $-1<r<\frac{1-{x}_{1}}{{x}_{1}}$.

To maintain a total sum of 1 when multiplying the first compositional part by the constant (

$1+r$), the remaining parts were reduced using a factor (

$1-s$), calculated as

Modifying the parameters $r$ and $s$ we can use compositional linear regression models to estimate the expected LTPA subcomposition from a change in workers’ daily OPA subcomposition.

#### Example: Expected Change in Leisure Time Sedentary Behavior when Increasing Occupational Walking

The compositional geometric mean of daily activities during work for men (closed to 480 min) was (walking = 87 min; standing = 212 min; sitting = 177 min; HiPA = 4 min).

After closing to 1, this corresponded to the set of proportions [0.181; 0.443; 0.368; 0.008].

As proportion walking during work was equal to 0.181, the complementary part of the OPA subcomposition was equal to 1 − 0.181. The time reallocated to the first part of the compositions must be expressed as proportion. Thus, to relatively increase occupational walking by 15 min, we compute

Using Formula (A3) we obtain

Multiplying the first part of the OPA subcomposition (i.e., walking) by $r=0.172$ and the remaining OPA parts by $s=0.038$, we obtained a new OPA subcomposition. This corresponded to OPA time proportions [0.215; 0.431; 0.339; 0.014]. By closing it to 480 min, we obtained [walking = 102 min; standing = 204 min; sitting = 170 min; HiPA = 4 min].

Then, ilr-coordinates were calculated from this new OPA subcomposition including a 15-min relocation to occupational walking.

To investigate the effect on LTPA of this move from the reference to the new OPA subcomposition, we first estimated the LTPA subcomposition based on the reference OPA subcomposition using the following ilr multivariate regression model Equation (A3):

where the LTPA ilr-coordinate vector

$ilr\left(\mathit{Y}\right)=\left(\begin{array}{c}{y}_{1}^{\ast}\\ {y}_{2}^{\ast}\\ {y}_{3}^{\ast}\\ {y}_{4}^{\ast}\end{array}\right)$ consisted of

The following beta coefficients matrix (

**B**) was obtained from model Equation (A3):

Using

**B** in combination with the ilr-coordinates of the reference OPA subcomposition, we obtained expected ilr-coordinates of the LTPA subcomposition (

$\widehat{ilr\left(\mathit{Y}\right)}$ as:

Resulting in the following expected LTPA ilr-coordinates:

This vector was transformed back by ilr-inverse transformation to express it as a composition $[0.25;0.05;0.14;0.003;0.560]$. By closing it to 960 min as originally, we obtained the expected LTPA subcomposition [SB 237 min; walking 50 min; standing 133 min; HiPA 3 min; TIB 538 min].

The same procedure was performed, but now using

**B** in combination with the ilr-coordinates of the new OPA subcomposition after 15-min walking relocation, to obtain the corresponding expected LTPA composition for this case:

Using the parameters

$r=0.172$ and

$s=0.038$, the following expected LTPA ilr-coordinates and subcomposition (in proportions) were obtained:

and

$[0.25;0.05;0.14;0.003;0.555]$, respectively. After closing to 960 min, the expected LTPA subcomposition was [SB 238 min; walking 52 min; standing 135 min; HiPA 3 min; TIB 533 min].

By taking the change in

$\widehat{ilr\left(\mathit{Y}\right)}$ predicted by the reference and new OPA subcomposition, it was possible to estimate the expected change in leisure time activities. The same procedure was performed for each new OPA subcomposition where time had been reallocated. We expressed the change matrices as minutes changed in leisure time activities by taking the inverse ilr transformation of the model predictions (see

Table 5 and

Table 7 for results).