Fig 1.
Empirical mean, standard deviation, and skewness of the BMI distribution for BRFSS survey data.
BMI mean, SD, and skewness have steadily increased over the course of the obesity epidemic, with growth tempered in recent years. (a)-(c): dots show data points, lines show show regression fits; (d): probability distributions for BMI in 1991 (red triangles show binned data, red solid curve shows smoothed histogram) and 2011 (blue circles show binned data, blue dashed curve shows smoothed histogram).
Fig 2.
Drift and diffusion in the short-term BMI dynamics of individuals in a human population.
The figure shows the average annual change in the BMI of individuals (blue dots), and the standard deviation of the annual change in the BMI of individuals (red triangles), as a function of BMI, for data from our new large NU data set (left panel; 121,574 measurements for 2011) and from the publicly available NHANES survey data set (right panel; 5,624 measurements for 2011–2012). The plots are obtained by binning empirical BMI differences. The blue curves (dots) show that low-BMI individuals on average increase their weight year-over-year, while high-BMI individuals decrease their weight on average, and the dependence on BMI is approximately linear. The red curves (triangles) show that the standard deviation of annual BMI changes, which results from natural short-term fluctuations in an individual’s BMI that may be due to variations in diet or physical activity, increases approximately linearly as a function of BMI. These results establish that BMI dynamics feature a drift towards a set point, and a diffusion that is proportional to the BMI. The black curves are the curves of best fit for all data years to our mathematical models for the drift term (Eq (2), including social effects) and for the diffusion amplitude (Eq (10)), as discussed in the Methods and mathematical models section. Fig A of S1 Appendix repeats this analysis for the NU and NHANES BMI data split up by age range and by gender, confirming the drift-diffusion dynamics identified here. Fig D of S1 Appendix repeats this analysis for the entire data set over all years, confirming the nearly linear relations observed here.
Fig 3.
Results from fitting the 2011 NU empirical BMI distribution (black dots) to our predicted distributions (no social effects; red solid) and peq(x) (with social effects; red dashed), and to a standard log-normal (blue dash-dotted) distribution.
From top to bottom, the first panel illustrates how the BMI distribution results from a balance between drift and diffusion, and is right-skewed. The second panel shows the same BMI distributions in log scale to make tails more visible, and the third panel shows differences between the log-normal distribution as null-model and the other distributions. The second and third panels show that the (red solid) and peq(x) (red dashed) distributions are more successful in fitting the empirical data than the commonly used log-normal distribution, both near the center of the distribution and in the high-BMI tail. This is confirmed in the bottom panel that shows the root mean-square error (RMSE) resulting from fitting NU data to BMI distributions in the range 1997–2014.
Table 1.
Akaike Information Criterion test for model distributions fitted to 2011 empirical BMI distribution data in Fig 3 and Fig B of S1 Appendix.
Relative likelihood ratio exp[(AICmin−AIC)/2] of non-social , social peq(x), and log-normal flog(x) models for 2011 NU, NHANES and BRFSS empirical BMI distributions.