We analyzed the association between total energy expenditure and physical activity, assessed via accelerometry as mean counts per minute per day (CPM/d), using several approaches. We began by using multivariate regression to investigate the relative effects of anthropometric variables (fat-free mass, fat mass, height, age, and sex) and behavioral or lifestyle variables (accelerometry CPM/d, employment in manual labor, and location) on total energy expenditure, using linear regression in R []. By far the strongest anthropometric correlate of total energy expenditure was fat-free mass; fat mass and height were marginally negatively correlated with total energy expenditure, and age and sex had no effect ( Table 1 , model 1). To examine the effect of physical activity on total energy expenditure while controlling for anthropometric effects, we calculated adjusted total energy expenditure, total energy expenditure, for each subject by adding residuals from the total energy expenditure ∼ fat-free mass + fat mass + height + age + sex + study site regression to mean total energy expenditure (model 2 in Table 1 ; see Supplemental Experimental Procedures ). Total energy expenditurewas used for subsequent analyses in the main text. We similarly calculated an adjusted resting metabolic rate, resting metabolic rate, by adding residuals from the resting metabolic rate ∼ fat-free mass + fat mass + height + age + sex + study site regression to mean resting metabolic rate, and we calculated an adjusted activity energy expenditure, activity energy expenditure, as (0.9 × total energy expenditure− resting metabolic rate). We also tested a range of other models correcting for anthropometric and other effects on total energy expenditure and resting metabolic rate, as well as raw (unadjusted) values of total energy expenditure, resting metabolic rate, and activity energy expenditure; results were nearly identical to those reported in the main text (see Supplemental Experimental Procedures and Figures S1 and S2 ).

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32 Cleveland W.S. Robust locally weighted regression and smoothing scatterplots.

31 R Core Team

R: A Language and Environment for Statistical Computing, v3.1.0.

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33 Lerman P.M. Fitting segmented regression models by grid search.

34 Hinkley D.V. Inference in two-phase regression.

35 Efron B.

Tibshirani R. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy.

36 Kim H.J.

Fay M.P.

Feuer E.J.

Midthune D.N. Permutation tests for joinpoint regression with applications to cancer rates.

To examine the shape of the relationship between physical activity and total energy expenditure and compare Additive and Constrained total energy expenditure models, we fit three different regression models to the scatterplot of total energy expenditureagainst CPM/d. First, we fit a robust locally weighed regression (lowess) curve [] using the “lowess” function in R [], with f = 2/3, iter = 5. This nonparametric model allows studying non-linear relationships between continuous variables (e.g., physical activity and total energy expenditure) without assumptions about the shape of the underlying function. Second, to test the fit of a linear, Additive total energy expenditure model, we estimated the linear correlation, via Pearson’s correlation coefficient, between total energy expenditureand physical activity ( Table 1 , models 5 and 6). We used a modified version of this approach for the CPM/d threshold analysis ( Figure 1 B): we evaluated the effect of CPM/d and manual labor on total energy expenditurevia linear regression for all subjects with CPM/d values above a threshold CPM/d = i and iterated this analysis over the range of CPM/d thresholds i = (1, 2, 3…500). The resulting set of β, SE, and model adjusted rvalues were examined with respect to CPM/d threshold ( Figure 1 B). Lastly, we used change-point regression to estimate the association between physical activity and total energy expenditure, controlling for manual labor employment. This model is similar to the Constrained total energy expenditure model, which predicts a plateau in the physical activity:total energy expenditure relationship at higher activity levels ( Figure 1 ) and allows the estimation of a change point, from increasing linear/additive to flat/plateau. The change point was estimated using a computer-intensive grid search approach [], which has been shown to more flexible than the standard method based on maximum-likelihood estimation []. Bootstrap simulations were applied to calculate the SE of the change-point estimator []. We applied an F-like test, based on an approximate permutation test, using a computer-intensive algorithm as described in the literature to formally test whether the Constrained total energy expenditure model (piecewise regression model) was preferred over the Additive total energy expenditure model (traditional linear regression) [].