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Model selection
To perform the statistical analyses, we noted that the dependent variable is characterized by nonnegative values only, has a positively skewed distribution (though the mean of 6.4 is not much larger than the median of 6.0), and is not likely to have truly independent values over time. This casts doubt on the appropriateness of ordinary least squares regressions. We therefore tested our model against the broader class of Generalized Estimating Equations (GEE), ADDIN EN.CITE Hardin200376[1]76766Hardin, J.W.Hilbe, J.M.Generalized estimating equations2003Boca RatonCRC Press[ HYPERLINK \l "_ENREF_1" \o "Hardin, 2003 #76" 1] which do not require the assumptions of ordinary least squares regression and can explicitly incorporate the lagged-dependent nature of the data (auto-regressivity). GEE models are an extension of generalized linear models that use a link function g() to relate the expected value of the dependent variable y to a linear function of the explanatory variables and specifies a variance distribution F() for y, such that g{E(y)}=x where y ~ F(y). To assess the fit of different link functions g and distributions F using standard approaches, we used comparisons of the Akaike Information Criterion (a test of which model fits the data with minimal complexity); the modified Park/GLM family tests; Pregibon s link test (which assesses the linearity of the mean function x' ); and a modified Hosmer-Lemeshow test for systematic misspecification (which divides the data into deciles based on x' and conducts an F-test to determine whether the mean residuals across groups differ from zero) (1). We found no significant advantages comparing the model with identity link and Gaussian distribution (the default model presented in the main text and specified below) to the inverse Gaussian, negative binomial, Poisson or gamma variance distributions F with a log or identity g link and either compound symmetry for the correlation structure, autoregressivity or an unstructured correlation matrix. We nevertheless re-ran the regressions using the log link and Gaussian, Poisson and Gamma variance functions to assess the robustness of the main model findings; none of the results significantly changed.
Granger causality test
We found that diabetes prevalence was a function of lagged sugar availability, Diabetest = Sugart-1, and not the other way around (i.e., we found no association in the following regression: Sugart = Diabetest-1).
In standard usage, a variable X is said to Granger cause another variable Y if Y can be better predicted from the past of X and Y together than the past of Y independently. ADDIN EN.CITE Maddala1983113[2]1131136Maddala, GSLimited-Dependent and Qualitative Variables in Econometrics1983CambridgeCambridge University Press[ HYPERLINK \l "_ENREF_2" \o "Maddala, 1983 #113" 2] Note that Granger causality is not proof of true causality, but actually refers to the predictive ability of one variable on another. Granger causality tests precedence, i.e., whether Y precedes X, X precedes Y or they are contemporaneous (3).
The Granger test is specified as:
Yt = 1Yt-1 + 1Xt-1 + t
As shown in the table below, we found that higher sugar availability Granger-causes higher diabetes prevalence but that diabetes prevalence fails to Granger-cause sugar availability. The same pattern was not observed with obesity and diabetes, suggesting that while sugar passes the Granger test for temporal causality, obesity fails (possibly, as discussed in prior literature, due to the fact that obesity is a correlate or marker of diabetes rather than a long-term early driver) ADDIN EN.CITE Bremer201258[3]585817Bremer, A.A.Mietus-Snyder, M.L.Lustig, R.H.Toward a unifying hypothesis of metabolic syndromePediatricsPediatrics557-57012932012[ HYPERLINK \l "_ENREF_3" \o "Bremer, 2012 #58" 3]. The choice of lag length is to some degree arbitrary (2), but the findings above remained robust as we varied the lag length across the range of available observations.
Granger Causality TestsDependent VariableCovariateSugartDiabetes prevalencetObesity prevalencetDiabetes prevalencetDiabetes prevalencet-1-0.19(0.31)0.93***(0.0062)0.011(0.030)0.98***(0.0049)Sugart-10.98***(0.0043)0.00066***(0.000085)Obesity prevalencet-11.02***(0.00082)0.00042(0.0013)Significance at * p<0.05, ** p<0.01, *** p<0.001
Heckman model
Selection bias involves non-random assignment to a treatment and control group; that is, there is some unobserved factor relating to having high sugar availability that is also related to high diabetes prevalence in a way we have not observed and as a result for which we have not corrected.
We can address the issue of unobserved selection bias directly by constructing, and controlling for, a variable of the hazard of having high sugar availability (a Heckman-type selection model or control function approach) (4).
In medicine, randomized controlled trials include a random assignment pattern that can identify the unbiased average effect of an intervention. In observational studies, a Heckman selection model or control function approach can be used to model the selection into a high-sugar-availability group based on a set of unobservable and observable factors, and then correct for this selection in the outcome equation.
To do this, we define a system of equations:
(1) Diabetesit = 1 + 1SUGARit + 2Xit + 3lit + mi + it (the outcome equation )
and
(2) SUGARit = 2 + 4Xit + 5Wit + it (the selection equation )
Here, X is the set of controls used in the main equation 1, l is a selection coefficient (defined below), and W is a set of variables that affect sugar availability but are not included as predictors of diabetes prevalence (for example, the percentage of food that is imported is a significant determinant of sugar availability but is not used as a predictor of diabetes prevalence). The selection equation is a bivariate probit model in which high sugar availability was defined as at least 300 kcal/person/day (twice the recommended daily intake limit for men), though the results below remained robust when the definition of high availability was varied between 200 and 600 kcal/person/day.
A selection coefficient can be calculated as = (Z)/(1 " (Z)), where predictions of high sugar availability from the selection equation give the standard normal cumulative distribution function (Z), from which we can calculate the normal distribution (Z) of these predictions using the Gaussian function e-(^2/2)/"(2). Since hazard has the general definition H(x) = (Z)/(1 " (Z)), we are simply controlling for the hazard of high sugar availability in our models; if the selection coefficient has a significant relationship to diabetes prevalence, this would suggest that selection bias may be a problem for our model. As shown in the table below, the selection coefficient did not have a significant relationship to diabetes prevalence.
Key assumptions for applying the Heckman approach are: a L OPQRl
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Of note, while the Heckman-selection strategy is a standard approach to evaluating models for selection bias, it features important limitations, including sensitivity to the model specification. For these reasons, we regard the selection model as an additional robustness check on our basic finding, not as definitive proof that selection bias is not still possible in our experiment, as we have additionally specified in the main manuscript.
Association of Sugar Availability (kcal/person/day) with Log Diabetes Prevalence, Adjusted for Selection BiasEffect of Sugar Availability0.008 (0.0009)***Effect of Sugar Hazard Rate l -1.08 (0.66)* p<0.05, ** p<0.01, *** p<0.001
Note: Robust standard errors in parentheses clustered by country. Models include controls used in the basic specification in equation 1.
l is the hazard rate of sugar availability, also referred to as the Inverse Mills Ratio.
References
ADDIN EN.REFLIST 1. Hardin JW, Hilbe JM (2003) Generalized estimating equations. Boca Raton: CRC Press.
2. Maddala G (1983) Limited-Dependent and Qualitative Variables in Econometrics. Cambridge: Cambridge University Press.
3. Bremer AA, Mietus-Snyder ML, Lustig RH (2012) Toward a unifying hypothesis of metabolic syndrome. Pediatrics 129: 557-570.
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